Kreiss Initial Boundary Value Problems and Navier Stokes

& F f&>CC C-L:-A+S?Se1-C?S InApplied Mathematics siam Gul Initial-Boundary Value Problems and the Navier-Stokes Equations SIAM's Classics in Applied Mathematics series consists of books that were previously allowed to go out of print. These books are republished by S1AM as a professional service because they continue to be important resources for mathematical scientists. Editor-in-Chief Robert E. O'Malley, Jr., University of Washington Editorial Board Richard A. Brualdi, University of Wisconsin-Madison Herbert B. Keller, California Institute of Technology Andrzej Z. Manitius, George Mason University Ingram Olkin, Stanford University Stanley Richardson, University of Edinburgh Ferdinand Verhulst, Mathematisch Instituut, University of Utrecht Classics in Applied Mathematics C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences Johan G. F. 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Allgower and Kurt Georg, Introduction to Numerical Continuation Methods Heinz-Otto Kreiss and Jens Lorenz, Initial-Boundary Value Problems and the Navier- Stokes Equations iii This page intentionally left blank Initial-Boundary Value Problems and the Navier-Stokes Equations Heinz-Otto Kreiss Jens Lorenz Society for Industrial and Applied Mathematics Philadelphia Copyright © 2004 by the Society for Industrial and Applied Mathematics This SIAM edition is an unabridged republication of the work first published by Academic Press, 1989. 10 987654321 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permis- sion of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688. Library of Congress Cataloging-in-Publication Data Kreiss, H. (Heinz-Otto) Initial -boundary value problems and the Navier-Stokes equations / Heinz-Otto Kreiss, Jens Lorenz. p. cm. — (Classics in applied mathematics ; 47) Originally published: Boston ; Academic Press, c!989. Includes bibliographical references and index. ISBN 0-89871-565-2 (pbk.) 1. Initial boundary problems. 2. Boundary value problems. 3. Navier-Stokes equations. I. Lorenz, Jens, 1949- II. Title. III. Series. QA378.K74 2004 515'.35-dc22 is a registered trademark.2004045359 Contents Preface to the Classics Edition xi Errata xiii Introduction xv Chapter 1. The Navier-Stokes Equations 1 1.1 Some Aspects of Our Approach 2 1.2 Derivation of the Navier-Stokes Equations 9 1.3 Linearization and Localization 18 Chapter 2. Constant-Coefficient Cauchy Problems 23 2.1 Pure Exponentials as Initial Data 24 2.2 Discussion of Concepts of Well-Posedness 34 2.3 Algebraic Characterization of Well-Posedness 44 2.4 Hyperbolic and Parabolic Systems 55 2.5 Mixed Systems and the Compressible N-S Equations 62 Linearized at Constant Flow 2.6 Properties of Constant-Coefficient Equations 66 2.7 The Spatially Periodic Cauchy Problem: A Summary for 73 Variable Coefficients Notes on Chapter 2 79 Chapter 3. Linear Variable-Coefficient Cauchy Problems in ID 81 3.1 A Priori Estimates for Strongly Parabolic Problems 82 vii Vlii Contents 3.2 Existence for Parabolic Problems via Difference 87 Approximations 3.3 Hyperbolic Systems: Existence and Properties of 100 Solutions 3.4 Mixed Hyperbolic-Parabolic Systems 111 3.5 The Linearized Navier-Stokes Equations in One Space 113 Dimension 3.6 The Linearized KdV and the Schrodinger Equations 115 Notes on Chapter 3 118 Chapter 4. A Nonlinear Example: Burgers' Equation 121 4.1 Burgers' Equation: A Priori Estimates and Local 122 Existence 4.2 Global Existence for the Viscous Burgers' Equation 131 4.3 Generalized Solutions for Burgers' Equation and 138 Smoothing 4.4 The Inviscid Burgers' Equation: A First Study of Shocks 141 Notes on Chapter 4 156 Chapter 5. Nonlinear Systems in One Space Dimension 159 5.1 The Case of Bounded Coefficients 160 5.2 Local Existence Theorems 165 5.3 Finite Time Existence and Asymptotic Expansions 167 5.4 On Global Existence for Parabolic and Mixed Systems 172 Notes on Chapter 5 175 Chapter 6. The Cauchy Problem for Systems in Several 177 Dimensions 6.1 Linear Parabolic Systems 177 6.2 Linear Hyperbolic Systems 181 6.3 Mixed Hyperbolic-Parabolic Systems and the Linearized 188 Navier-Stokes Equations 6.4 Short-Time Existence for Nonlinear Systems 190 6.5 A Global Existence Theorem in 2D 198 Notes on Chapter 6 202 Chapter 7. Initial-Boundary Value Problems in One Space 203 Dimension 7.1 A Strip Problem for the Heat Equation 204 7.2 Strip Problems for Strongly Parabolic Systems 211 7.3 Discussion of Concepts of Well-Posedness 222 7.4 Half-Space Problems and the Laplace Transform 228 Contents IX 7.5 Mildly Ill-Posed Half-Space Problems 248 7.6 Initial-Boundary Value Problems for Hyperbolic 253 Equations 7.7 Boundary Conditions for Hyperbolic-Parabolic Problems 262 7.8 Semibounded Operators 268 Notes on Chapter 7 272 Chapter 8. Initial-Boundary Value Problems in Several Space 275 Dimensions 8.1 Linear Strongly Parabolic Systems 275 8.2 Symmetric Hyperbolic Systems in Several Space 283 Dimensions 8.3 The Linearized Compressible Euler Equations 302 8.4 The Laplace Transform Method for Hyperbolic Systems 306 8.5 Remarks on Mixed Systems and Nonlinear Problems 322 Notes on Chapter 8 323 Chapter 9. The Incompressible Navier-Stokes Equations: The 325 Spatially Periodic Case 9.1 The Spatially Periodic Case in Two Dimensions 325 9.2 The Spatially Periodic Case in Three Dimensions 337 Chapter 10. The Incompressible Navier-Stokes Equations under 345 Initial and Boundary Conditions 10.1 The Linearized Equations in 2D 348 10.2 Auxiliary Results for Poisson's Equation 349 10.3 The Linearized Navier-Stokes Equations under 355 Boundary Conditions 10.4 Remarks on the Passage from the Compressible to the 359 Incompressible Equations Appendix 1: Notations and Results from Linear 361 Algebra Appendix 2: Interpolation 365 Appendix 3: Sobolev Inequalities 371 Appendix 4: Application of the Arzela-Ascoil Theorem 389 References 395 Author Index 399 Subject Index 401 This page intentionally left blank Preface to the Classics Edition Due primarily to the proliferation of computers, many questions in science and engineering have become amenable to quantitative study, and modeling by partial differential equations (PDEs) is playing an ever increasing role. This book introduces a part of this vast subject: initial and initial- boundary value problems for systems of PDEs that are first order in the time variable. We emphasize the well-posedness theory of parabolic, hyper- bolic, and mixed parabolic-hyperbolic systems and address the difficulties due to boundaries. The results are illustrated with applications to the Navier-Stokes equations for compressible and incompressible flows. In parts, our approach to the subject is low-tech. For example, to show existence of solutions to parabolic systems we use difference schemes instead of semi-group theory or a parametrix. Functional analytical prerequisites are kept to a minimum. What we need in terms of Sobolev inequalities is developed in an appendix. Our main intent in writing the book was to give a rigorous introduction to a field of mathematics, important in the applications, which is accessible to many readers. The success of the first edition of the book leads us to believe that we have succeeded, at least to some degree. We are grateful to have the book reprinted in SIAM's series Classics in Applied Mathematics. Heinz-Otto Kreiss (Potsdam, Germany) Jens Lorenz (Albuquerque, NM) October, 2003 XI This page intentionally left blank Errata p. 44: In the 2nd line of section 2.3 replace ut — P(d/dx] by ut = P(d/dx)u. p. 51: In the last display equation replace |«j| by |Re«j|. p. 100: In the display e uation within Lemma 3.3.1 replace ||/||#p by ll/lilf,- p. 107: The point denoted by (x,t) in Figure 3.3.1 should be denoted by (x,t). p. 153: In Figure 4.4.12 replace u = I by u = 0. p. 156: In line 8 replace e —> oo by e —> 0. p. 191: In the last line the last sum extends over all multi-indices a ^ 0 with a < v (componentwise). p. 213: In formula (7.2.6) replace wx(l) by wx(j). p. 232: In the 6th line from below, the third matrix is A, not A. p. 239: In the 5th formula line from lo replace uxx(x,t) by uxx(x, s). p. 241: To obtain the estimate Re(pAfc) > 6\\p\ stated in the 9th line from below, it should be noted that |argAfc| < \. Xlll XIV Initial-Boundary Value Problems and the Navier-Stokes Equations p. 247: Delete the leading symbol e in the first display equation, p. 332: In formula (9.1.18) replace p. 343: The reference to von Wahl's paper made in the remark is incorrect. The fact that the spatial 1/3 norm of u becomes unbounded for 0 < t < T if there is blow-up at time T has only recently been proved by L. Escauriaza, G. Seregin, and V. SVerak, On L^,^-solutions to the Navier-Stokes equations and backward uniqueness, Uspekhi Mat. Nauk 58 (2003), pp. 3-44. Introduction The aim of this book is to develop a theory of initial-boundary value problems for linear and nonlinear partial differential equations. There are already many books available, and we shall list some of them after the introduction. How- ever, the area is vast, and any one book can only treat certain aspects of the theory. Our choice of material is very much influenced by the availability of fast computers. They have made it possible to solve rather complex problems for which the classical theory of second-order equations is not adequate. Exis- tence and regularity questions play a fundamental role in computations because the resolution required depends on the smoothness of the solution, and there is always the danger that one tries to compute things which do not exist. Another fundamental question concerns admissible boundary conditions, which we shall discuss in great detail. In computations the boundary conditions cause most of the problems. We believe that this book can fill a gap between elementary and rather ab- stract books. To illustrate our theory, we have chosen the compressible and incompressible Navier-Stokes (N-S) equations, which describe fluid flows rang- ing from large scale atmospheric motions to the lubrication of ball bearings. The choice was dictated by the desire to find a system which is so rich in phenomena that the whole power of the mathematical theory is needed to discuss existence, smoothness and boundary conditions. We hasten to add, however, that we only scratch the surface of the diversity in which its solutions can behave. For exam- XV XVI Initial-Boundary Value Problems and the Navier-Stokes Equations pie, turbulent flow is described by the N-S equations, and at present no adequate mathematical theory is available. The are different ways to develop the theory. One way is to start with weak soluti and then to discuss their smoothness. This approach can lead to dif- ficulties when it comes to boundary conditions and to nonlinear equations. For numerical calculations one is very much interested in knowing the exact smooth- ness behavior "up to" the boundary. Also, for nonlinear problems it is often difficult to show that the weak solutions have sufficient regularity (smoothness) in order to make them unique. We proceed instead in the following way: First we show, by using difference approximations for linear problems and linearization for nonlinear problems, that there is a set of C°°-smooth data, dense in LI, for which the equations we discuss have Cx-smooth solutions. These solutions and their derivatives can be estimated in terms of the data. Then we use the usual closure argument to define weak solutions if the data are less smooth. This process is much closer to computing than the previous one: If one wants to compute solutions with discontinuous data then one obtains better results if one approximates the discontinuous data by smoother dat Also, in computations of solutions of nonlinear problems, one adds terms in the equation (numerical dissipation) so that the solutions do not develop discon- tinuities. We proceed analogously in the analytic theory of hyperbolic equations and treat these as a limit of parabolic ones. The latter have Cx -smooth solu- tions for C°°-smooth data. For the Euler equations (which are identical with the N-S equations without viscous term) this corresponds to replacing the inviscid equations by the viscid equations. In general, this process of adding a formally small higher-order derivative term is doubtful: There is no assurance that one obtains anything meaningful in the limit as the term goes to zero. However, for problems coming from applications, the extra terms are often present in the full equations and have only been neglected - without mathematical justification - to formally simplify the equation. We would like to thank our students who partly read the manuscript and suggested valuable improvements. Thanks is due also to Linda Soha, who expertly typed a large part of the manuscript. We wish to acknowledge the support of our research by the National Science Foundation (contract number DMS-8312264), by the Office of Naval Research (contract number N-00014- 83-K-0422), and by the Department of Energy (contract number DE-AS03- 76ER72012). Introduction XVli Text and Reference Books Courant, R., and Hilbert, D. (1962). "Methods of mathematical physics", Vol II, Interscience. Friedman, A. (1964). "Partial differential equations of parabolic type", Reprint, Krieger, 1983. Garabedian, P.R. (1964). "Partial differential equations", Wiley. Hellwig, G. (1964). "Partial differential equations, an introduction", Blais- dael. Henry, D. (1981). "Geometric theory of semilinear parabolic equations", Lecture Notes in Math. 840, Springer. John, F. (1971). "Partial differential equations", Springer. Mizohata, S. (1973). "The theory of partial differential equations", Cambridge University Press. Petrovskii, I.G. (1954). "Lectures on partial differential equations", Inter- science. Treves, F. (1975). "Basic linear differential equations", Princeton University Press. Weinberger, H.F. (1965). "A first course in partial differential uations with complex variables and transform methods", Wiley. This page intentionally left blank 1 The Navier-Stokes Equations In this preliminary chapter e first outline some questions which will be treated in this book. Then we derive the Navier-Stokes equations. Though the deriva- tion will not be used later, it is of interest to understand the underlying logical and physical assumptions, because the mathematical theory of the equations is not complete. There is no existence proof except for small time intervals. Thus it has been questioned whether the N-S equations really describe general flows. If one changes the stress tensor such that diffusion increases when the velocities become large, then existence can be shown. This change of the equa- tions does not seem to be justified physically, however. For example, certain similarity laws - valid for the Navier-Stokes equations - are well-established experimentally, but the modified equations do not allow the corresponding sim- ilarity transformations. Possibly a lack of mathematical ingenuity is the reason for the missing existence proof, and the N-S equations are physically correct. The N-S equations form a quasilinear differential system, and much of our un- derstanding of such systems is gained through the study of linearized equations. These will, in general, have variable coefficients. By freezing the coefficients in such a problem, one obtains systems with constant coefficients. It is much easier to analyse the latter, as will be later shown in Chapter 2. However, the relation between variable-coefficient and constant-coefficient equations is not trivial. The fundamental ideas of linearization and localization are discussed in Section 1.3. 1 2 Initial-Boundary Value Problems and the Navier-Stokes Equations 1.1. Some Aspects of Our Approach 1.1.1. The Equations, Initial and Boundary Conditions Neglecting effects due to thermodynamics, i.e., assuming constant entropy, the full N-S equations consist of the three momentum equations the continuity equation and an equation of state Here the velocity u = (u,v, w), the density p, and the pressure p are the un- knowns, F is a given forcing, and r(p) with dr/dp > 0 is assumed to be a known function. Furthermore, we have used the common notations The term P is defined by where A denotes the Laplacian, and n and n' are nonnegative material constants, assumed to be known. In most applications one deals with simplified equations: Case 1. Incompressible flow, p = po a known constant, the equation of state is dropped. Most of the mathematical theory is done for this case. After choosing suitable units, one can assume p = 1 , and the equations read The Navier-Stokes Equations 3 Note that the momentum equation and the equation div u = 0 are of different type. This makes the existence theory more difficult. Indeed, the small-time existence theory for the compressible equations is somewhat easier than the theory for the incompressible case. However, as we will show, the vorticity formulation allows a systematic treatment of the incompressible equations. Us- ing the idea of initialization for problems with different time scales, one can also treat the incompressible case as a limit of compressible ones: Case 2. Compressible but inviscid flow, i.e., // = \i = 0. One obtains the full equations with P = 0. The resulting system - possibly augmented by an energy equation - is fundamental in gas dynamics; Case 3. Inviscid, incompressible flow. The equations are known as Euler equations: The differential equations have to be supplemented by initial and boundary conditions. The mathematically easiest case is the Cauchy problem with pe- riodic initial data, where one seeks for solutions periodic in space. Loosely formulated, the following results are known for the viscous equations: 1. If the initial data are sufficiently smooth, then there is a time T > 0 such that the N-S equations have a unique smooth solution for 0 < t < T. The time T depends on the initial data. 2. In the incompressible case one can say more: The solution is analytic for 0 < t < T. Furthermore, if the velocity u satisfies a bound in maximum norm* like then the solution can be continued for a time At > 0, which only depends on C. This latter result guarantees - for the viscous incompressible case - that the solution can be continued as long as the velocities stay uniformly bounded. In other words, it is not possible that the solution ceases to exist because just derivatives of u become large; u has to become large itself. In experiments extremely large velocities are not observed. The maximal velocity hardly ever exceeds twice the mean velocity. This nourishes the hope that it might be possible to derive uniform bounds for u and thus *By |u| = (u2 4- v2 4- w2)*/2 we denote the Euclidean length of a vector u. The notion maximum norm refers to maximization w.r.t. to the arguments x, y, z. 4 Initial-Boundary Value Problems and the Navier-Stokes Equations to prove existence at all times. Thus far, however, this has not been achieved. In practical applications one studies flows in finite domains and most of the in- teresting phenomena happen near the boundary. Therefore, it is of great interest to study initial-boundary value problems. Indeed, most of the results formulated for the Cauchy problem can be carried over to suitably posed initial-boundar value problems using the general theory of partial differential equations. It is one of the main aims of the second part of these notes (starting in Chapter 7) to describe techniques for choosing correct boundary conditions, i.e., boundary conditions which lead to a mathematically well-posed problem. 1.1.2. Shocks and Weak Solutions For the inviscid compressible case of the N-S equations the solution can cease to exist though u stays uniformly bounded. This can be illustrated by the inviscid Burgers' equation: Discontinuities (shocks) for u(x, 0 can develop in finite time though u stays uniformly bounded. We will show this by the method of characteristics. Thus a classical solution can cease to exist in a finite time. To obtain existence of a solution for all time, one can broaden the solution concept and allow for eak solutions. Let us illustrate this for equation (1.1.1) together with an initial ondition where f(x) is a smooth* function with compact support. Later we shall show that the (regularized) problems together with the initial condition (1.1.2) have a unique smooth solution uf(x, t) existing for all time. Furthermore, *A function is called smooth if it has sufficiently many continuous derivatives. We use this con- venient term whenever more precise smoothness specifications are unimportant for our arguments. The Navier-Stokes Equations 5 This allows us to show that there is a unique function UQ(X. t), satisfying the above inequalities, such that for every smooth test function 0(ar, t) with compact support The function UQ is called a weak solution of the problem (1.1.1), (1.1.2). It is desirable to introduce and characterize weak solutions in a more direct way, without studying a regularization. To this end, let </> = 0(x. 0 denote a smooth function with compact support. If we multiply (1.1.3) by 0 and integrate over we find after integration by parts Using the initial condition and sending e —> 0, one finds from (1.1.4) that UQ satisfies This relation does not involve the e-dependent equation (1.1.3), and thus it motivates the Definition. A function u(x,t) is called a weak solution of (1.1.1), (1.1.2) if (1.1.5) holds for all test functions <£. Our derivation shows that UQ(X, t) is a weak solution. It turns out, however, that the given definition does not characterize a unique function. There is, in general, more than one weak solution if we accept the definition given above. From our point of view (and for physical reasons) it is natural to define weak solutions of the inviscid equations as the limit of solutions of the viscous equa- tions. But unfortunately, for more general equations or systems of equations, one generally does not know whether this limit exists. Therefore, it seems to be advantageous to define weak solutions directly, without going through such a limit process. This is usually done by requesting an integral relation for certain 6 Initial-Boundary Value Problems and the Navier-Stokes Equations test functions, as in (1.1.5), for example. In this way one can make the con- cept of a solution so weak that solutions do exist. However, often uniqueness becomes a problem. For the N-S equations there is no solution concept known which leads to both existence and uniqueness. 1.1 J. Scaling Arguments and Limiting Processes One of the most powerful tools for discussing properties of the N-S equations are scaling arguments. As an example we shall discuss some aspects of these arguments for the 2D inviscid compressible equations, which we write in the form (1.1.6) Assume that for a given flow we can scale the dependent and independent variables by so that u', t/, p', p' and their first derivatives are of order 0(1). Here t/, Q, P, L, T are positive constants and Q, P are assumed to be com- patible with the equation of state, i.e., P = r(Q). If, for example, p = p"1 then P = Q~>, and thus p' = //"*; i.e., the equation of state remains unchanged by the transformation. Introducing the new variables into the equations above, we find Assume now that UT/L = 1 and that the equation of state remains unchanged by the transformation. If we drop the ' sign in the notation, the equations become The Navier-Stokes Equations 7 It follows from these equations - as we shall discuss later - that the type of the flow depends crucially on the so-called Mach number M introduced above. If M < 1, M = 1, M > 1, then the flow defined by the state (U,P = r(Q)) is called subsonic, transsonic, or supersonic, respectively. Consider, for example, the case M —>• oo. Formally we obtain in the limit the equations Here the pressure has been eliminated, and the velocity equations are uncoupled from the density equation. We can also obtain the incompressible equations by a limit process from the equations governing compressible flows. To this end, assume that Thus the density is almost constant. For the pressure we obtain that and therefore the pressure is correctly scaled if we write Introducing p = po 4- Qp', p = Po 4- Pp' and using the same scalings of u,v,x,t/,t as above, we obtain from (1.1.6) the following system, where again ' is dropped in the notation: Initial-Boundary Value Problems and the Navier-Stokes Equations Considering the limit process we obtain formally the Euler equations, where the density equation above goes over in ux + vy = 0. We shall discuss this limit process in Chapter 10. Another limit is obtained for The resulting equations are Combining these, we obtain the wave equation for p: (In the original coordinate system where L = T = 1 the quantity (dr(po)/df>)l'z denotes the sound speed.) The above scaling arguments show that the N-S equations describe vastly different types of problems; these appear when certain parameters approach extreme values. One could believe that these extreme cases are practically uninteresting. However, the opposite is true: they represent the usual behavior. The reason is that the parameters do not need to take extreme values for the flow to have qualitatively the same behavior as suggested by the limiting equations. For example, a flow with M = 2 is in certain aspects not much different from the limiting flow corresponding to M = oo. Therefore, the analysis of the limiting equations is important. 1.1.4. Remarks on Solution Estimates and Computations The Navier-Stokes Equations 9 From a purely mathematical point of view the theory of the viscous N-S equa- tions is in rather good shape, except for the question of long-time existence and the related question of blow-up. Once we assume a bound for the maximum norm of the solution, we can estimate all its derivatives and we know what boundary conditions to prescribe. Also, one understands various limit processes as long as v is fixed. In the incompressible case one can show that the j-th derivatives of the flow- variables are bounded in terms of where (-Dii)^ denotes the maximum norm of the velocity gradient. Let us dis- cuss the implications in two space dimensions first. In 2D, the values |Du(-, 0| are essentially bounded by the initial data |Du(-,0)|. If we scale the variables such that |Du(-.0)| = 1, then the j-th derivatives are bounded in terms of i/~J/2. In applications a value v = 10~6 is not uncommon. If the derivatives are of the order O(103) everywhere, then we need O(106) computational points in a two-dimensional region. This stresses the capacities of today's machines, but it is computationally feasible. In three space dimensions the velocity gradients are not bounded independently of v, but one can make a plausibility argument that \Du(-,t)\ = O(i/-'/2). For v = 10~6 one would need O(i/~9/4) = O(1027/2) computational points to resolve the flow; this is clearly beyond today's (and tomorrow's) machines. Fortunately, the situation is not always this bad. There are many flows for which the derivatives are bounded independently of v. Other flows, particularly those in (compressible) gas dynamics, have derivatives bounded independently of v except near isolated sharp fronts (shocks). In this case one can replace a very small v by a larger value, without changing the structure of the flow away from the fronts. Burgers' equation is a good example, and we shall discuss it in detail. To treat situations where derivatives are large in extended regions, we need a mathematical theory to identify functional which depend smoothly on v as v — > 0. These could be evaluated using larger ^-values for computations, and could be extrapolated. These questions are closely related to turbulence models and so-called subgrid modeling. 1.2. Derivation of the Navier-Stokes Equations In this section we give a brief derivation of the N-S equations. Our presentation relies heavily on Meyer (1971) and Chorin and Marsden (1979). For additional 10 Initial-Boundary Value Problems and the Navier-Stokes Equations information, see also Serrin (1959). The general logical assumptions underlying continuum mechanics - in particular smoothness assumptions - are not easily formulated and we do not discuss them here. Instead we give a rather uncriti- cal intuitive description of the ideas of particle paths and velocity fields. Both concepts can serve to describe flows. The relation between both descriptions is discussed first. Then the laws of conservation of mass and change of mo- mentum are formulated; these laws - in their differential form - constitute the fundamental differential equations discussed in this book. Specific assumptions about the stress tensor appearing in the momentum equations lead to the N-S system. The equations obey an important similarity law, which is strongly con- firmed experimentally. This justifies indirectly the assumptions made about the stress tensor. 1.2.1. Eulerian and Lagrangian Descriptions Consider a body of fluid which - at time t = 0 - occupies some open bounded domain HO with boundary df&o. We think of every point a € HO as a fluid particle whose motion we can follow for t > 0. Its position at a later time t is denoted by $(a, t). At this time the fluid elements occupy the domain The curve t — > $(a, t) describes the trajectory of the fluid particle which at time t = 0 has the initial coordinate a = $(a, 0). It is assumed that $(a, t) ^ $(b, t) for t > 0 if a ^ b; i.e., the fluid particles keep their identity. We formalize the basic assumption underlying the description of the flow in Assumption 1.2.1. There exists a smooth function $(a, t) with the following properties; 1. $(a,0) = a. 2. If a ^ b then $(a, t) ^ $(b, t) for t > 0. 3. The mapping a -»• $(a, 0 has a smooth inverse. These conditions arc assumed to hold in HO u dflq, i.e., the region including the boundary. Then, by general topological arguments, at any later time t the boundary d£l t of ft* is the image of the original boundary: This will be used below. Let u(x, t) denote the fluid-velocity at x € rz t and time t. Then The Navier-Stokes Equations 11 FIGURE 1.2.1. Trajectory with velocity vector. and thus we have In principle, if the mapping $ = $(a, t) describing the trajectories is known, then the velocity field u = u(x,t) can be obtained from (1.2.1). Conversely, if we know the velocity field u, then we can reconstruct the trajectories t —* $(a. 0 by solving - for each fixed a e ffo - the (ordinary) initial value problems (1.2.1) with initial condition 3>(a,0) = a. Thus one can describe the fluid motion in terms of t and the initial position a € fio (the so-called material coordinate) or one can use t and the spatial coordinate x € flt. The first way is known as Lagrangian description whereas the second is the Eulerian point of view. One can ask which description one should use in actual calculations. If discontinuity surfaces (e.g., density discontinuities) are present which move with the flow, then the Lagrangian way is advantageous: The location of the discontinuities in terms of the material coordinates remains fixed in time, and one can work with a description which is smooth with respect to t. On the other hand, watching the distortion which a blob of ink undergoes in a turbulent flow makes us wonder if the Lagrangian description is always useful. Here the initial coordinates lose their meaning. One prefers the Eulerian description if large distortions are present. Also, in numerical calculations it is quite common to use a combination of both descriptions. Let / = /(x, t) denote a scalar quantity depending smoothly on the Eulerian coordinates x e ftt and t. For example, / can denote the local fluid-density or 12 Initial-Boundary Value Problems and the Navier-Stokes Equations the temperature. One can ask how the quantity / changes in time if one follows a trajectory \(t) = $(a, t). The chain rule and (1.2.1) imply that (Here u = (u, v, w) and x = (x, y, 2).) This motivates us to introduce the operator D/Dt acting on functions / = /(x, t) in Eulerian coordinates: The function (D/Dt)f(\, t) is known as the material derivative of /. One easily shows the product rule: We consider now the time-change of integrals of the quantity /. Using the mapping $ to transform all integrals to the fixed domain !7o, one can show with the rules of calculus the Transport Theorem. Under appropriate smoothness assumptions the follow- ing holds: where A proof of the Transport Theorem is given, for example, in Hughes and Marsden (1976). With Gauss' Theorem one obtains further that where n(x, t) denotes the unit outward normal of dftt and dS is the surface element. The amount of the quantity f in 17 changes due to direct changes of f and due to the flow of f across the boundary. The Navier-Stokes Equations 13 If we apply the Transport Theorem with / = 1, we obtain that and thus each volume is preserved in time if and only if div u = 0. 1.2.2. Conservation of Mass and Balance of Momentum We start with the conservation of mass, which can be formalized as follows: Assumption 1.2.2. There exists a smooth function p = p(x, i) > 0 defined on each fluid domain ftf such that for all t > 0, The function p is the fluid-density, and the integral (1.2.4) is the mass contained initially in fio and conserved in time. Using the Transport Theorem, we find that holds for any domain J7t; therefore, the integrand must vanish, and we obtain for p the so-called continuity equation: This equation often takes a simpler form: A fluid is called incompressible if For an incompressible fluid the continuity equation reads divu = 0, thus volumes are preserved. Conversely, if the flow is volume preserving, then div u = 0 and the continuity equation becomes Dp/Dt = 0. Under this as- sumption the Transport Theorem implies that p is constant w.r.t. x and t if the density is homogeneous at t = 0, i.e., p(x,0) = const. Let us now discuss the balance of momentum. The momentum of the fluid in the domain Q t is 14 Initial-Boundary Value Problems and the Navier-Stokcs Equations By Newton's law, its time derivative equals the force which acts on the mass in Qt. We assume that this force is the sum of an external body force (e.g., gravity, the Coriolis force, or an electromagnetic force) and an internal force, which acts on the boundary of the fluid domain. The internal force is due to friction between fluid elements. We assume that it can be written in terms of a stress tensor field: Assumption 123. There exists a force field F = F(x, t) and a stress tensor field such that, for any fluid domain fjt, Note that this is a vector equation. Using Gauss' Theorem, we can write If we apply the Transport Theorem to each component of (1.2.6) we obtain that Under appropriate smoothness assumptions the integrand must vanish, and, using Ww continuity equation, we find that We refer to (1.2.7) as the momentum equation. 1.2.3. Forms of the Stress Tensor Any further analysis of the fluid motion depends on knowledge about the stress tensor S. We must relate S to the other flow variables. In many situations - in particular in gas dynamics - one can use a very simple form for S: Definition. A flow is called inviscid if the stress tensor S has the following fo The Navier-Stokes Equations 15 The scalar quantity p = p(\, t) is the pressure. For an inviscid flow the momen- tum equation (1.2.7) becomes Together with the continuity equation (1.2.5), the vector equation (1.2.9) repre- sents a first-order system of four equations in the five scalar unknowns p.p.u.v. w. There are two important different ways to complete the system. First, if the fluid is incompressible, i.e., if p(x, t) = po is a known constant, we have the Euler equations: A flow governed by these equations, i.e., an inviscid incompressible flow, is said to be ideal. Another way to complete the set of equations (1.2.5), (1.2.9) is to add an equation of state. In the absence of thermodynamic effects it takes the form where r is a known function. The resulting system describes inviscid compress- ible flow. Let us consider again the viscous case, which requires a more general form of the stress tensor S than (1.2.8). The appropriate dependence of S on the other flow variables can only be inferred from observations. If the motion is uniform, i.e., if all velocity gradients are zero, then experiments show that the simple form (1.2.8) for S is still appropriate. Thus it is reasonable to assume - at least as a first approximation - that S -f pi depends linearly on the matrix of velocity gradients: Heuristically one can think of S(T) as the stresses due to viscous phenomena. The assumption (1.2.11) is made in most of classical fluid mechnics. We want to argue now that S only depends on the symmetric part of T, i.e., on the deformation tensor 16 Initial-Boundary Value Problems and the Navier-Stokes Equations and not on the local rotation of the fluid, described by the antisymmetric part |(T - T*) of T. Consider the Couette experiment. Assume that the space between two long concentric cylin- ders is filled with fluid. Let the inner cylinder be forced to rotate slowly and let the outer cylinder be free. Then the fluid and the outer cylinder also begin to rotate. This can only be due to tangential stresses. After some time the outer cylinder and the fluid move with the same velocity as the inner cylinder. Thus, the motion has become like the rotation of a solid body; there is no momentum transfer anymore, and we have S = -pi. The matrix T corresponding to a solid rotation is antisymmetric. Therefore, it is reasonable to assume that S is zero for an antisym etric argument and depends on the symmetric part D = ^(T + T*) of T only. One can prove (see Gurtin and Martins (1976)) Theorem 1.2.1. Assume that S = S(£>) is a linear function of D which is invariant under all rotations of the coordinate system, i.e., for all orthogonal 3 by 3 matrices U and all symmetric 3 by 3 matrices D. Then S can be written as where n and y! are constants independent of D. An application of this result gives us for the stress tensor: If we substitute this e/tpreasion into the momentum equation ( 1 .2.7). we obtain - together with the continuity equation - the so-called Navier-Stokes system. The material constants n and \i' depend on the temperature and the chemical properties of the fluid. The coefficient // is known as the shear viscosity and C = //' 4- 1// is the bulk viscosity. Again, one distinguishes incompressible and compressible flow. In the first case one has p(\. t) = po, div u = 0; for compressible flow one uses an equation of state p = r(p) to complete the system. The resulting equations are listed in Section 1.1. The Navier-Stokes Equations 17 1.2.4. The Similarity of Flows and the Reynolds Number One of the most important tools in hydrodynamics is to scale the variables. This enables one both to simplify the equations by neglecting terms and to use one calculation or experiment to obtain results for another similar problem. We shall explain the procedure for incompressible flow without external forces. To this end, assume we have a fluid with density p and viscosity //. Its motion satisfies We consider (1.2.12) in a time-independent domain f2 with initial conditions in Q and boundary conditions on dfl. Suppose now that we want information about another fluid with density p and viscosity /} in a geometrically similar domain fi = Lfl. The functions u = u(x. t), p = p(x, f) satisfy and are subject to initial conditions in ft and boundary conditions on dft. Sup- pose further that we can find scaling transformations such that the transformed functions satisfy the same initial and boundary conditions as u and p. From and (1.2.12) we find that Thus the functions u'.p' satisfy the same equations as u.p if If these relations for the scaling parameters are satisfied and if the initial- boundary value problem r u,p has a unique solution, then we can conclude that 18 Initial-Boundary Value Problems and the Navier-Stokes Equations The requirements simplify if the coefficients in the boundary conditions do not depend on time, because we can choose T arbitrarily. If we take T = L/U, then (1.2.13) requires that Roughly speaking, flows are similar if their respective values for coincide. The number R is called the Reynolds number of the flow. As mentioned above, the similarity law has been confirmed by experiments over and over again. Indirectly, this experimental fact supports and justifies the basic assumption underlying the Navier-Stokes equations, namely that the stress tensor depends linearly on the velocity gradients and not on other fluid variables. 1.3. Linearization and Localization In this section we outline some basic mathematical questions which will be treated more thoroughly in the subsequent chapters. We want to solve the so- called Cauchy problem for systems of quasilinear partial differential equations with initial conditions Here x = (x\, • • • ,xs) e R3 is the vector of independent space variables and u = u(x,t) denotes the solution to be determined, which takes values in C". The operator P = P(x, t,u, d/dx) is a quasilinear differential operator of order m of the general form Here 1 1/ 1 = i/i H ----- h va for any multi-index v = (y\ , • • • , va). The coefficients The Navier-Stokes Equations 19 are given smooth matrix functions of the arguments Also, the forcing function F in (1.3.1) and the initial function / in (1.3.2) are assumed to be given. More specific assumptions and boundary conditions will be discussed later. From a purely mathematical point of view there are two fundamental questions: 1. When can one guarantee that the problem (1.3.1), (1.3.2) has a unique solution, at least for a small time interval? 2. Assuming that (1.3.1), (1.3.2) has a unique solution u for some fixed F, /, what is the influence on the solution of small perturbations added to F and/? The second question can be rephrased more precisely as follows: Replace F by F + 6F in (1.3.1) and / by / + 6f in (1.3.2), and thus consider a perturbed problem If (1.3.1), (1.3.2) is "physically reasonable", one can hope that the perturbed problems (1.3.3), (1.3.4) are also uniquely solvable, at least for sufficiently small perturbations, and that their solutions can be written as where 5u satisfies an estimate Here ||£w||(i), ||6F||(2), P/||(3> are certain norms which still have to be specified. The constant K in (1.3.5) should be independent of <5F and df as long as the perturbations are kept sufficiently small. This leads to the preliminary Definition. Assuming that (1.3.1), (1.3.2) has a unique solution u for some fixed F, /, we say that the nonlinear problem (1.3.1), (1.3.2) is well-posed at u if there is an € > 0 s.t. for all smooth functions 6F and 6f with the perturbed problem (1 .3.3), (1.3.4) is also uniquely solvable, and 6u := v — u satisfies an estimate (1.3.5) with K independent of 6F and df. Note that we still have to specify the norms. Indeed, different norms lead to different concepts of well-posedness, as we will discuss below. 20 Initial-Boundary Value Problems and the Navier-Stokes Equations We will try to show the existence of a solution of the nonlinear problem (1.3.1), (1.3.2) in a sufficiently small time interval by considering the sequence uk = uk(x, t) of functions defined iteratively through the linear equations For this reason, we first have to study the Cauchy problem for linear equations Here and the coefficients Av = Av(x, t) depend smoothly on (x, t). Also, the ques- tion of well-posedness at u, which we formulated above for the nonlinear prob- lem (1.3.1), (1.3.2), is closely related to well-posedness of linearized equations. Roughly speaking, the following linearization principle holds: A nonlinear prob- lem w well-posed at u if the linear problems which are obtained by linearizing at all functions near u are well-posed. Thus we will study linear equations first. In order to understand the ba- sic questions of existence, uniqueness, and well-posedness for them, it suffices to treat just the homogeneous case, i.e., one can take F = 0 in (1.3.6). In- homogeneous equations can then be treated by Duhamel's principle. More importantly, it is desirable to relate the well-posedness of the Cauchy problem for a v#r/a/?/i-coefficient equation to the well-posedness for the constant-coefficient equations which are obtained by freezing the coefficients The Navier-Stokes Equations 21 at arbitrary but fixed points (io, *o)- This idea is called localization. By Fourier transformation, the discussion of constant-coefficient problems can be reduced to purely algebraic questions; this will be carried out in Chapter 2. The localization principle one would like to have, can be formulated as fol- lows: // all frozen-coefficient problems are well-posed then the corresponding variable-coefficient problem is also well-posed. It should be noted in advance that this principle is not valid for general linear variable-coefficient operators. However, the differential operators which appear in the context with the N-S equations are of parabolic, of hyperbolic, or of mixed hyperbolic-parabolic type, and for these classes of operators the above localization principle turns out to be applicable. Formal linearization. To explain the process of linearization, we consider Burgers' equation as an example: Suppose U = U(x, t) is a smooth function. We may think of U as a known approximate solution of (1.3.10), but this is not essential. If we substitute into (1.3.10), then we obtain Thus far, no terms have been neglected. Since we consider u' as a small correction to U, we neglect the quadratic term u'u'x and write v ins of u'. Thus we obtain the linear equation This equation is called the linearization of (1.3.10) at U, or the linearized equa- tion. Note that F = 0 if and only if U solves (1.3.10) exactly. The linearized equation governs the influence of small perturbations on U. This is another important reason to study linear equations. This page intentionally left blank 2 Constant-Coefficient Cauchy Problems The main tool to discuss constant-coefficient problems (with initial data given on the whole space) is the Fourier transformation. It allows us to "decompose" general initial data into pure exponentials. One easily observes that - in case of constant coefficients - the time evolution of each pure (spatial) exponential can be treated separately. We introduce the symbol P(iu?) of a differential operator P(d/dx) and obtain as the solution for initial data u(x,0) = e'("-T) f(u>). Well-pose ess of the Cauchy problem can be characterized in terms of estimates for the ymbol. We will start out with some special cases, namely hyperbolic and parabolic systems in one space dimension. For these, the conditions for the symbol are easily checked, and one can solve the Cauchy problem. In Section 2.3 we characterize families of matrices A for which the exponential eAt. t > 0, is uniformly bounded. This result, which is central in a general theory of well- posedness, will be applied in Section 2.4 to characterize those first-order systems (in any number of space dimensions) which lead to well-posed Cauchy problems; the corresponding systems are called strongly hyperbolic. An important example is given, the compressible Euler equations linearized about a constant flow. Similarly, linearization about a constant flow of the viscous compressible Navier- Stokes equations leads to a mixed hyperbolic -parabolic system; applying the 23 24 Initial-Boundary Value Problems and the Navier-Stokes Equations same general principles, we obtain well-posedness of the Cauchy problem and can solve these linearized problems by Fourier transformation. Though constant-coefficient equations are of some interest by themselves, they are too restrictive for most applications. If one freezes variable coefficients at arbitrary points, then obviously constant-coefficient equations are obtained. Unfortunately however, it is possible that all frozen-coefficient problems are well-posed and the given variable-coefficient equation is ill-posed nevertheless. Roughly speaking, switching between variable and constant coefficients intro- duces lower-order terms. Naturally, this leads to the question: Which constant- coefficient problems can be perturbed by (arbitrary) lower-order terms without losing well-posedness? As we will show in Section 2.6, the answer is that only strongly hyperbolic and parabolic problems have this property. To obtain this result, we use a restrictive but simple concept of well-posedness: It must be possible to estimate the Z/2-norm of the solution at later times in terms of the Z/2-norm of the initial function; derivative terms of the initial function are not permitted in the estimate. This idea guides our discussion of well-posedness. At the end of this chapter, in Section 2.7, we will outline generalizations to variable-coefficient problems. The details will be presented in Chapters 3 and 6. 2.1. Pure Exponentials as Initial Data The solution of constant-coefficient Cauchy problems is particularly simple if the initial function is a pure exponential; it can be written down in terms of the symbol of the differential operator. The behaviour of the symbol for large wave-numbers will determine whether or not the Cauchy problem is well-posed. 2.1.1. Introductory Examples We start with some simple examples. Example 1. (The simplest equation with wave-solutions.) The differential equation (with a given constant a e R) is the simplest hyperbolic equation. Suppose that an initial condition Constant-Coefficient Cauchy Problems 25 is prescribed at t = 0. Here u; 6 R and /(u;) e C are constants; the factor /(u;) is added for later purposes. To solve the equation, we use an ansatz in separated variables, and obtain The solution of this ordinary initial value problem is Thus, if the problem (2.1.1), (2.1.2) has a solution of the form (2.1.3), it is necessarily given by One easily checks that the above function indeed solves the problem; the solution represents a wave of constant amplitude traveling with speed a; for a > 0 or a < 0 the wave travels to the right or left, respectively. Example 2. (The heat equation.) Another simple equation is the so-called heat equation Assuming the same initial condition (2.1.2), we use the same ansatz (2.1.3); the ordinary initial value problem for ii(u. •) reads Hence the problem is solved by Example 3. (The heat equation in backward time.) If we apply the transfor- mation t — » —t to the equation ut = ua:j., t < 0, we obtain Proceedin the same way as above, we find as a solution for the initial data (2. 1 .2). 26 Initial-Boundary Value Problems and the Navier-Stokes Equations If |u;| is large, the solutions (2.1.4) and (2.1.5) behave completely differently for increasing t: Whereas (2.1.4) decays rapidly, the solution (2.1.5) grows exponentially. For increasing |w|, there is no bound on the exponential growth rate in time. As stated more precisely below, the Cauchy problem for the heat equation in backward time is ill-posed. 2.1.2. The Symbol of a Constant-Coefficient Operator The previous examples can be generalized to systems of constant-coefficient equations in any number of space dimensions. Notations. With x = (x\, ...,xs) € R* we denote the space variable. A multi-index v is a vector with nonnegative integers as components: Its order is Each multi-index v determines a differential expression Furthermore, we use the following basic notations: fora;,* € Rs, If A € Cn'n is a matrix, then its norm is defined by If H\,H2 G Cn<n are Hermitian matrices, then Consider a differential equation Constant-Coefficient Cauchy Problems 27 with constant matrices Av € Cn>n and assume an initial condition where u> e Rs and f(u>) £ Cn are fixed. The unknown solution takes values in Cn. A basic observation is and therefore Thus, application of the differential operator P(d/dx) to e^w'x^/(o;) results in multiplication by the matrix This matrix P(iu>), which is formally obtained by substitution of iu)j for is called the symbol of the differential operator P(d/dx). We show Lemma 2.1.1. The initial value problem (2.1 .6), (2. 1 .7) /ias the solution Proof. The ansatz leads to This ordinary initial value problem has the solution and the formula (2.1.8) is obtained. 28 Initial-Boundary Value Problems and the Navier-Stokes Equations 2.1.3. Well-Posedness in Terms of the Symbol Thus far we have considered only initial data of the form i.e., the spatial behaviour of the initial function is essentially determined by a single wave-vector u; 6 R8; the constant vector f(uj) 6 Cn only allows us to multiply e1^^ by different constants in different components. If more general initial data f(x) are given, we can try to write these as where is the Fourier transform of f(x). (See Section 2.2 for more details.) According to Lemma 2. 1 . 1 , the evolution of each individual term of the integrand of (2. 1 .9), is known, and it is tempting to believe that describes the evolution to general data f(x). This formula expresses the prin- ciple of superposition. Indeed, as we will discuss in Section 2.2, this process is often justified. However, it is also clear that there might be serious convergence problems if is unbounded for |oj| — » oo. We want to give here an operational definition of the well-posedness of the Cauchy problem in terms of the symbols P(iuj). The definition is operational in the sense of providing conditions which can be checked in applications. In Section 2.2 we will show that this definition of well-posedness is equivalent to another one which might be more familiar, namely that Constant-Coefficient Cauchy Problems 29 (i) for all initial data / in a certain class there is a unique solution in a certain class; (ii) the solution depends continuously on / with respect to certain norms. The operational definition is Definition 1. The Cauchy problem (2.1.11) is called well-posed if there are constants a, K such that for all t > 0 and all u e R*. 2.1.4. Examples of Well-Posed and of Ill-Posed Problems Let us illustrate the previous defin on by some examples. Example 4. (The wave equation as a hyperbolic system*.) The wave equa- tion ytt = yxx leads to the system if one introduces the variables u\ = yt, U2 = yx. There is a unitary transfor- mation U such that Therefore, and thus |eF(Iu;)| = 1. According to Definition 1, the Cauchy problem for (2.1.12) is well-posed. Also note that the change of variables v = Uu reduces (2.1.12)to we obtain two uncoupled scalar equations of the form discussed in Example 1 . Example 5. A so-called weakly hyperbolic system is given by *The notion of a hyperbolic system will be introduced below. 30 Initial-Boundary Value Problems and the Navier-Stokes Equations Here Thus the problem is ill-posed since the term \u\t is not bounded independently of u>. The growth of |ep(lw)t| with increasing |w| is much less severe here than in the backward heat equation, where \ePiiu>)t\ = e*"*. In fact, the problem of Example 5 is weakly well-posed, which is defined as follows: Definition 2. The Cauchy problem (2.1.11) is called weakly well-posed if there are constants a, K, q such that for all i > 0 and all u; € Rs. In Section 2.2 we will discuss the difficulties involved with weak well- posedness: If a problem is only weakly well-posed but not well-posed, then a perturbation by a lower-order term can make the problem ill-posed in any sense; this makes it hard to go over to variable-coefficient problems. Hyperbolic systems in ID. Consider a first-order system where A € O". We show 2.1.2. Tht Cauchy problem far (2.1 . 13> M wcll-poxd if and only I/ «?// ft£tfiVd/Kro of A arc real and A has a complete set of eigenvectors. Proof. First assume that the eigenvalues of A are real and that there is a complete set of eigenvectors. There is a matrix S with From A = S\S~ l one obtains that since |e*wA'| = 1. The estimate of Definition 1 hold with a = 0, /f = Conversely, assume the problem is well-posed. Let Constant-Coefficient Cauchy Problems 31 denote an eigenvalue of A We must show 6 = 0. Since etulAt is an eigenvalue of ep(iw)t, we obtain Clearly, if b G R, 6 ^ 0, we cannot have a bound by /vTeat independent of w. This shows that 6 = 0, i.e., that the eigenvalues of A are real. Now suppose that A does not have a complete set of eigenvectors, and let denote a (nontrivial) Jordan block of the Jordan matrix J of A. Then in contradiction to the existence of a bound by Keat independent of u. Definition 3. A first-order system ut = Aux for which the Cauchy problem is well-posed is called strongly hyperbolic. For first-order systems in any number of space dimensions, we will define strong hyperbolicity in the same way, namely by requesting well-posedness of the Cauchy problem. In Section 2.4 we will give an algebraic characterization generalizing the previous theorem. Solution formula. Suppose ut = Aux is strongly hyperbolic, and S~}AS = A = diag(Aj). If one introduces new variables v by Sv = u, then one obtains the diagonal system vt = Avx, which is decomposed into n scalar equations. An initial condition transforms to 32 Initial-Boundary Value Problems and the Navier-Stokes Equations Thus and therefore where Sj denotes the j-th column of S. This shows that w(x, t ) is a sum of n waves which travel with the so-called characteristic speeds — Aj, . . . , — \n. Parabolic systems in ID. We restrict ourselves here to second-order systems Definition 4. The above system is called parabolic if the eigenvalues AI ..... An of A satisfy Clearly, the heat equation ut = uxx is a simple example. We want to apply Definition 1 and show Theorem 2.13. The Cauchy problem for a second-order parabolic syst is well-posed. Proof. \. By Schur's Theorem (Appendix 1) we can transform A by a unitary matrix U to upper-triangular form, If D = diag(l, d, • • • , dn~'), d > 0, denotes a diagonal matrix, then Constant-Coefficient Cauchy Problems 33 Therefore, if we choose d > 0 sufficiently large and set 5 = DU, then The reason is that \j + Aj > 26 > 0, and if d > 0 is sufficiently large, then the influence of the outer-diagonal entries is as small as we please. We define the positive definite Hermitian matrix H = 5*5 and rewrite the above matrix inequality as 2. Now consider the symbol We obtain with Q independent of w . To finish the proof of the theorem, we prove a lemma on matrix exponentials ept. It can be applied to each symbol P = P(zu;) separately. Lemma 2.1.4. Let P € C71-", and let Proof. Suppose that y(t) solves the initial value problem Then one obtains 34 Initial-Boundary Value Problems and the Navier-Stokes Equations Integration yields9" Therefore, Hence we have shown that for all yn. and the lemma is proved. 2.2. Discussion of Concepts of Well-Posedness In this section we introduce the space MO of all functions / = f(x) whose Fourier transforms are C00 -smooth and have compact support. If an initial condition u(x,0) = /(x), / € MO, is given, then the Cauchy problem for any constant-coefficient operator P = P(d/dx) is solvable, and the solution is unique within a certain class. For example, we can solve the backward heat equation for initial data in M0. Well-posedness of the Cauchy problem means more, namely an estimate of the L2-norm of the solution at all later times by the LI -norm of the initial data. We will prove that such an estimate is possible if and only if the problem is well-posed according to the definition (using the symbol) given in the previous section. One can weaken the concept of well-posedness and allow derivative-terms of the initial function / for estimates of the Z/2-norm of the solution at later times. Again, this concept of weak wfU-posedness can be characterized in terms of estimates for the symbol P(iu). However, if a problem is only weakly well- posed, then perturbations of the differential operator P = P(d/dx) by suitable lower-order terms will lead to arbitrarily fast exponential explosion, and the perturbed problem is ill-posed in any sense. We will illustrate this result by an example. 2.2.1. Solution for Smooth Initial Data via Fourier Transform The space MQ. Let CQ° denote the space of all functions </> : R* — > Cn which have derivatives of all orders and have compact support; i.e., each </> e CQ° vanishes outside a bounded set in R*; the bounded set will depend on 0. By *See Lemma 3.1.1 for a simple result on differential inequalities, which we ore using here. Constant-Coefficient Cauchy Problems 35 MO we denote the space of all functions / : R8 — > Cn which can be written in the form* One easily shows that / € C00, and one can differentiate under the integral sign since </> has compact support: Also, for / € MO the Fourier transform f(u>) is defined, and Therefore, the representation (2.2.1) is nothing but the Fourier representation of/. For a function / : Rs — > Cn we say that / € Z/2 if a: — » \f(x)\2 is (Lebesgue-) integrable over Ra. On La one defines an inner product and a norm by We remind the reader of Parseval's relation Here we will need this relation only for functions /, g e MQ. (One can show that the space MO is dense in LI. Then / —>• / is a densely defined bounded linear operator from MO C 1/2 —»• 1/2, and one can obtain the Fourier transform / e LI for any / € L2 by continuous extension; see Theorem 2.2.4 below. With this extension, Parseval's relation becomes valid for all /, g e L2.) Solution formula. Consider the Cauchy problem where / € MO; thus ''"The integral is defined componentwise. Initial-Boundary Value Problems and the Navier-Stokes Equations Since a solution of the differential equation for initial data is given by it follows that solves the given Cauchy problem. The reason is that the boundedness of the support of / ensures that we can differentiate under the integral sign. Before we can state a uniqueness result, we must make the concept of a solution more precise. A convenient concept* is the following: Definition 1. A function u = u(x, t) is called Mo-solution of (2.2.3), (2.2.4) if (ii) the function u(u>, 0 is continuous, and u(u>, t) = 0 for \u)\ > K with some K independent of t\ (iii) u is a classical solution; i.e., ut exists and u satisfies (2.2.1), (2.2.2) at each point x e Rs, t > 0. We prove: Lemma 2.2.1. For any / 6 M0 the Cauchy problem (2.2.3), (2.2.4) has a Mo solution. It is given by (2.2.5). Proof. It is easy to show that (2.2.5) is an Mo-solution; thus it remains to prove uniqueness. To this end, assume that u is an arbitrary Mo-solution and note that Integration in t yields *We do not aim for generality here, but merely want to illustrate that one can obtain existence and uniqueness results for ill-posed problems, too. The estimates are essential for well-posed s. Constant-Coefficient Cauchy Problems 37 Since the Fourier representation of u(x. t) - f(x) is unique, it follows that Therefore, and we tain This shows that u has the representation (2.2.5), and uniqueness is proved. Solution operator. The solution formula (2.2.5) shows that u(-.f) lies in M0 at each later time, and u(-,£) depends linearly on /. Thus we obtain linear operators So(f) : MO — * M0 which assign to the initial data w(-,0) = / the solution at time t > 0. Instead of determining u(-,£) directly from u(-,0) we can first determine and use u(-.t\) as new initial data to calculate w(-,0- We have Thus the one-parameter family (5o(t), t > 0} of linear operators on M0 has the properties: One says that the family {5o(0t t > 0} forms a semigroup on MQ. 2.2.2. Estimates of the Solution by the Initial Data If (2.2.3), (2.2.4) shall describe the evolution of a physical process, it is rea- sonable to require that the solutions are stable with respect to perturbations of the initial data, i.e., if v(x, t) is the solution of vt = P(d/dx)v with perturbed initial data 38 Initial-Boundary Value Problems and the Navier-Stokes Equations then we must be able to estimate v(x, t) - u(x, 0 in terms of g. Linearity of the equation implies that such an estimate is possible if and only if one can estimate any solution w(-, 0 in terms of its initial data «(•, 0) = /. In this section we want to show how such estimates are related to bounds of the symbol P(iu). Basically, the results follow from Parseval's relation. Throughout we denote by u = u(x, t) the Mo-solution of (2.2.3), (2.2.4) for initial data / € MO- If the problem is well-posed; i.e., if then (2.2.5) and Parseval's relation yield In this way, we have an estimate of u at all later times by the initial data. The converse is also valid. Theorem 2.2.2. Given a constant-coefficient operator P(d/dx). For any real K, a the following conditions are equivalent: 1. The symbols satisfy (2.2.6). 2. For all u(-, 0) = / € MO it holds that Proof. It remains to show that (2) implies (1). To this end, let t > 0 and wo € R* be fixed. There is a vector v e Cn, |v| = 1, with For 6 > 0, define the function f(u>) by (To be precise, we must approximate / by functions in CQ°.) For any t > 0 there is 6 > 0 with Using (2.2.5), we obtain Since c > 0 was arbitrary, the estimate of the symbol follows. Constant-Coefficient Cauchy Problems 39 Now we allow derivative-terms of / to bound the solution at later times. For q = 0. 1,2,... define i.e., || • || HI measures all derivatives of order < q. By Parseval's relation, thus There is a constant c — cq independent of u; with Using this inequality, one obtains a characterization of weak well-posedness. Theorem 2.2.3. Given a constant-coefficient operator P(d/dx); let q £ {1.2,...} and a € R. The following conditions are equivalent: 1 . There is K\ with 2. 7/iere w /^2 w/f/j To summarize, we can distinguish equations ut = P(d/dx)u of three different types. Accordingly, the Cauchy problem is 1. well-posed: the solutions satisfy ||w(-,t)|| < tfeat||u(-,0)||; 2. weakly well-posed but not of type 1: the solutions satisfy for some positive integer q but not for q = 0; 3. not weakly well-posed: the symbols |ep(lw)< | grow faster than any polyno- mial in |u>|. We say, there is exponential explosion and call the problem ill-posed in any sense. 40 Initial-Boundary Value Problems and the Navier-Stokes Equations Problems of type 1 or 2 can usually be treated with standard numerical meth- ods. (Higher-order methods might be necessary for type 2 problems.) Since numerical calculations introduce round-off errors - which correspond to high wave-number oscillations - there is no hope to calculate solutions of type 3 problems with standard methods. (Admitting a data error and requesting addi- tional restrictions for the solution, one can regularize ill-posed problems, how- ever.) From this perspective, type 1 and type 2 problems are both adequate for applications. However, if one wants to go over to variable coefficients, then the concept of weak well-posedness leads to serious difficulties. These will be illustrated by the next two examples. 2.2.3. A Perturbation Leading to Exponential Explosion Recall that Example 5 of Section 2.1.4 is weakly well-posed but not well-posed. If we add the zero-order term then the equation becomes Here the symbol has the eigenvalues Hence, for large |u>|, there is an eigenvalue with real part The Fourier transforms of the solutions can grow like e'w' *, and there is exponential explosion. This demonstrates that the estimate (2.2.7) with q = 1 is not invariant under pertur- bations by lower-order terms of the equation. One can prove that this example is typical: If the solutions of a given differ- entia] equation satisfy an estimate (2.2.7) with some q > 1 , but not with 9 = 0, Constant-Coefficient Cauchy Problems 41 then one can perturb the equation by a lower-order term such that the perturbed system shows exponential explosion. 2.2.4. Variability of Coefficients Corresponds to Perturbations The system (2.2.9) can also serve as an example that variability of coefficients is related to adding lower-order terms. Consider wh If one freezes A(t) at an arbitrary t — to, then one obtains a constant-coefficient equation, which behaves like Example 5, Section 2.1.4. Thus, for all frozen- coefficient problems one cannot estimate the L2-norm of the solution by the Z/2-norm of w(-,0), but one can estimate the L2-norm of the solution as in (2.2.8) with q = 1. Now let us discuss the given var/aWe-coefficient problem. The transformed function satisfies the constant-coefficient equation discussed above. Thus the v-equation can show arbitrarily fast exponential growth, and - transforming back - the same holds for the given w-equation. Summarizing, to allow estimates of ||u(-.t)|| as in (2.2.8) with q > 1 is not useful if variable-coefficient problems are to treated via localization. 2.2.5. ^Extension of the Solution Operator So(t) Up to this point we have only allowed initial data in MO. We shall now extend the admissible initial data to all functions / € 1/2 provided the initial value problem is well-posed. To this end, let / € 1/2 be given. There is a sequence fj € MO with *This section might be omitted on first reading. We will use - without proof - completeness of the space LI and density of MO in Z/7. Since C^ is dense in L-> (mollification), the latter result can be shown by Fourier transformation. 42 Initial-Boundary Value Problems and the Navier-Stokes Equations From it follows that the sequence Uj(-,t) = 5o(0/j converges in L2 for every fixed t. Also, the limit does not depend on the specific choice of the approximating sequence f}: If fj 6 MO is another sequence with /, —» /, then Hence, the construction defines a unique function w(-,0 € 1/2 for given initial data / € Z/2: The function u is called the generalized solution of the initial value problem (2.2.3), (2.2.4). We write and call the one parameter family (S(t), t > 0} the semigroup of generalized solution operators. For fixed t, the above construction is nothing but the usual extension of a densely defined bounded linear operator to the whole space. With arguments as given above one can prove the following theorem of functional analysis: Theorem 2.2.4. Let B\, BT. denote normcd spaces, let M denote a dense subspace of B\, and let Bi be complete. If SQ : M —» BI is a bounded linear operator, then there is a unique bounded linear operator S : B\ —> B^ with Sf = SQ/ for all / e M. The operator S is called the extension of So. By our construction, the generalized solution u(x. t) is just an Z/2-function with respect to x for each fixed t. It is often possible, however, to obtain smoothness properties of u(x, t) with respect to x and t by further investigations. Let us consider two simple examples. Example /. Consider the differential equation Constant-Coefficient Cauchy Problems 43 FIGURE 2.2.1. Discontinuous initial function. with initial data We write f(x) in the form and approximate / by (The functions /j have Fourier transforms fj with compact support. The piece- wise smooth function fj can be approximated with arbitrary accuracy by func- tions in CQ°.) For the solution with initial data fj it holds that and therefore Hence, the generalized solution consists of the box traveling with speed 1 to the right. Here u is not smooth, but at least t —> w(-, 0 G LI is continuous. Example 2. Consider the heat equation subject to the initial condition given in Example 1. Here we obtain 44 Initial-Boundary Value Problems and the Navier-Stokes Equations FIGURE 2.2.2. Smooth approximation. Hence, in any interval 0 < 6 < t < oo the sequence Uj(x, t) converges uniformly to In this case, the generalized solution is a C°° -function for x € R, t > 0, which satisfies the differential equation in the classical (i.e., pointwise) sense for t > 0. The discontinuities of the initial data disappear for t > 0. This behavior is typical for parabolic equations. 2.3. Algebraic Characterization of Well-Posedness According to Definition 1, Section 2.1, the Cauchy problem for a constant- coefficient system ut = P(d/dx) is well-posed if and only if there is a 6 R such that with K independent of uj e Rs and t > 0. Thus, for fixed a, one has to consider the set of matrices and establish a uniform bound of the matrix exponentials: In the main result of this section, Theorem 2.3.2 below, we will characterize the validity of such an estimate by other algebraic conditions. The case of a single matrix. A simple result on matrix exponentials is shown first. Constant-Coefficient Cauchy Problems 45 Lemma 2.3.1. For any A € Cn>n the following conditions are equivalent: 1. There is a constant K such that \eAi\ < K for all t > 0. 2. All eigenvalues K of the matrix A have a real part Re K < 0. Furthermore, if Jr is a Jordan block of the Jordan matrix J = SAS~l which corresponds to an eigenvalue K with Re« = 0, then Jr has dimension 1x1. (In other words, if K is an eigenvalue with Re K = 0, then the dimension of its eigenspace equals the multiplicity of K as a root of the characteristic polynomial of A.) Proof. First note that Also, if is an arbitrary Jordan block, then the exponential eJrt stays bounded for t > 0 if and only if either Re K < 0 or (Re K = 0 and Jr has dimension 1x1). Thus the result follows. The Matrix Theorem and its proof. Let F denote an infinite set of matrices A G Cn •". The uniform boundedness is not as easily discussed. It is not sufficient to request condition 2 of the previous lemma for each A € F separately because |5~'||5| can depend on A and can become arbitrarily large. The characterization given next is useful if one wants to derive necessary and sufficient conditions for well-posedness. Theorem 2.3.2. Let F denote a set of matrices A £ Cn-n. The following four conditions are equivalent: 1 . There is a constant K\ with \eAt \ < K\ for all A e F and all t > 0. 2. For all A e F and all s e C with Re s > 0 the matrix A — si is nonsingular, and there is a constant KI such that (This condition is known as the Resolvent Condition.) 46 Initial-Boundary Value Problems and the Navier-Stokes Equations 3. There are constants K$\ , Kyi with the following property: For each A 6 F there is a transformati n S = S(A) with such that SAS~l is upper triangular, the diagonal is ordered, and the upper diagonal elements satisfy the estimate 4. There is a positive constant K* with the following property: For each A G F there exists a Hermitian matrix H = H(A) with For our applications, the most interesting implication is (4) =>(!). It allows us to show well-posedness by constructing a suitable Hermitian matrix H = H(u) for each symbol P(iv). This part of the Matrix Theorem follows immediately from Lemma 2.1.4. The only difficult part in the proof of the theorem is to show (2) => (3). This part will only be used to show necessary conditions for well-posedness, and its proof might be omitted on a first Proof of Theorem 233. e show the implications "(1) =* (2)": If A € F then \eAi\ < K\ for all * > 0, thus by Lemma 2.3.1 all eigenvalues K of A have a real part Re « < 0, Therefore, if Re a > 0, the matrix A - si is nonsingular and Furthermore, This shows that (2) holds with KI — K\ . Constant-Coefficient Cauchy Problems 47 "(2) =>• (3)": For any matrix A € C" " there is a unitary transformation U such that UAU~* is upper triangular; the ordering of the diagonal entries can also be prescribed. This is Schur's Theorem, see Appendix 1. Since unitary transformations do not change the Resolvent Condition (2.3.1), it is no restriction to assume that the matrices A G F already have the form It suffices to show the following lemma by induction on n. Lemma 2.3.3. Given a set F of matrices A € C" " of the form (2.3.6) which satisfy the Resolvent Condition (2.3.1), there are constants A'.u. A'32 depending only on n and A'2 which have the following property: For each A £ F there is a transformation S = S(A) of the form with \S\ + \S~l\ < A'3! such that the transformed matrix SAS~{ (see (2.3.3)) fullfills the estimates (2.3.5). Proof. The statement of the lemma is obvious for n — \. To simplify the induction step below, we first consider the case n = 2. Thus let We want to transform A to Note that |m/j| < |M| for any matrix A/, and that Hence assumption (2.3.1) yields the bound 48 Initial-Boundary Value Problems and the Navier-Stokes Equations which we can rewrite as For s — + — KI , one obtains that If K2+K] — 0, we do not have to transform A. Thus we define the transformation matrix where and obtain Here by (2.3.7). A bound |5| + |S-'| < A'3| also follows from |->j < 2A"2, and the result of the lemma is shown for n = 2. Now let n > 3, and assume the statement of the lemma is true for all sets of matrices of order n - 1. Let F denote the given set of n by n matices A of the form described above. Any A e F can be partitioned as Since (A — s/)~' is of the form the Resolvent Condition (2.3.1) also holds for the set of n - I by n - 1 matrices {A\, A G F}. By the induction hypothesis, there are constants A'^.A'^ - depending only on n and K-i - and transformations Constant-Coefficient Cauchy Problems 49 with |S| + \S~]\ < such that the transformed matrix satisfies the estimates With we transform the matrix A itself and find Except for the last column, the elements of A1 are known to satisfy the de- sired estimates. The set of matrices {A'. A e F} again fulfills the Resolvent Condition: We partition ,4' in the form As before, we can apply the induction hypothesis to the matrices A'2 and find transformations 5' = S'(A'2) with 50 Initial-Boundary Value Problems and the Navier-Stokes Equations Here Furthermore, Using the upper-triangular form of 5 ~ ' and the fact that the elements of except a'ln have already been estimated (see (2.3.8)), we also have that Except for a'/n, all outer-diagonal entries of A" arc known to satisfy the de- sired estimates. We now apply an additional transformation of the form with This is motivated by the 2 by e discussed above. Note that and that Constant-Coefficient Cauchy Problems 51 It remains to prove an inequality of the form in order to bound 7. To this end, note that A" satisfies again the resolvent condition, Let (A" — s/)"1 = (cjj), and compute c)n by Cramer's rule: with and To treat £>2, we can use the proven estimates and find 52 Therefore,Initial-Boundary Value Problems and the Navier-Stokes Equations As in the 2 by 2 case we let s — > -«i and obtain the bound This gives a uniform estimate for the quantity 7 defined in (2.3.9) and ends the proof of Lemma 2.3.3. Hence the implication "2. =j> 3." of Theorem 2.3.2 is proved. We proceed with the proof of the theorem. "3. => 4.": Let A e F be arbitrary and let 5 = S(A) be determined as in 3.; i.e., where < and Define a diagonal matrix and set thus Constant-Coefficient Cauchy Problems We note that53 As a consequence, let us show that the Hermitian matrix C + C* is negative definite, if d = d(n, KM) is sufficiently large. The i-th row of C -I- C* reads Using the ordering 0 > Re«i > ••• > RCK,(, we find that the sum of the absolute values of the outer-diagonal elements in the z-th row of C + C* is bounded by Thus, by Gerschgorin's Circle Theorem, all eigenvalues of C + C* are nonpos- itive if Now set Si = DS, where D is defined in (2.3.10) and d fulfills the estimate above. The Hermitian matrix satisfies Furthermore, and condition 4 is proved. "(4) =$> (1)": This implication follows immediately from Lemma 2.1.4 with a = 0. Thus we have proved Theorem 2.3.2. Applications to the question of well-posedness. Let us note two simple im- plications of the theorem. As before, we consider the Cauchy problem for a constant-coefficient system ut = P(d/dx)u. The problem is well-posed if and only if there are constants a, K e R with 54 Initial-Boundary Value Problems and the Navier-Stokes Equations This holds if and only if for each u € Ra there is a Hermitian matrix H(UJ) € Cn'n with where K$ does not depend on u. For later reference we summarize this result: Corollary 23.4. The Cauchy problem for ut = P(d/dx)u is well-posed if and only if for each u; € Ra there is a Hermitian matrix H (a?) € Cn-n with where K* and a are independent ofui. Let us note again that the "difficult" part of the Matrix Theorem is only needed to ensure the existence of H(u>) for a well-posed problem. The converse result, namely that the existence of H(UJ) implies well-posedness, is elementary and follows immediately from Lemma 2. 1 .4. In another application of the Matrix Theorem we show that well-posedness does not depend on the zero-order term. (This result should be compared with the example in Section 2.2.3, which demonstrated that weak well-posedness does depend on the zero-order term.) Lemma 23.5. Let P(d/dx) denote a constant-coefficient operator, let B 6 C"'n, and let P0(d/dx) = P(d/dx) + B. The Cauchy problem is well-posed for ut = P u if and only if it is well-posed for m — PQU. Proof. Assume that the Cauchy problem is well-posed for ut = Pu, thus Here we use the notations of Corollary 2.3.4. Consequently, Another application of Corollary 2.3.4 shows the well-posedness of the Cauchy problem for P0 = P + B. Constant-Coefficient Cauchy Problems 55 Remark. Suppose that the matrices H(u/') are constructed for a given operator P(d/dx). Then the above corollary shows the well-posedness of ut = Pu + Bu for any matrix B. To obtain this result, only the elementary part 4. =*» 1. of the Matrix Theorem is needed. 2.4. Hyperbolic and Parabolic Systems In this section we define strong hyperbolicity and parabolicity for constant- coefficient equations in any number of space dimensions. The Cauchy problem for these equations is well-posed. As an application, we consider the compress- ible Euler equations linearized at a constant flow. One obtains a system which is strongly hyperbolic if dr/dp > 0, where p = r(p) is the equation of state. If one adds viscosity, i.e., goes over to the Navier-Stokes equations, then the linearized system is neither parabolic nor hyperbolic, but "almost" parabolic. This motivates us to treat certain mixed systems in Section 2.5; these can be considered as coupled hyperbolic-parabolic equations. 2.4.1. Hyperbolic Systems Consider a first-order equation in s space dimensions, We want to characterize all equations of the above form for which the Cauchy problem is well-posed. Note that the symbol depends in a linear way on the length |u;| of uj: for u; ^ 0, we set a/ = u;/|u/-| and obtain P(iu;) = |u;|P(iu;'). This simple observation and Theorem 2.3.2 lead to Theorem 2.4.1. The Cauchy problem for the first-order equation (2.4.1) is well-posed if and only if the following two conditions hold: \. For all u/ € Rs, |u/| = 1, all eigenvalues of P(iu/) are purely imaginary. 2. There is a constant KT,\, and for each u/ G Rs. |u/| = 1, there is a trans- formation S(u/) with 56 Initial-Boundary Value Problems and the Navier-Stokes Equations such that the transformed matrix has diagonal form. Proof. First assume conditions (1) and (2) to hold. For u; ^ 0, the diagonal matrix has purely imaginary entries. Therefore, Thus the problem is well-posed. (The wave-vector a? = 0 has the symbol P(0) = 0 which causes no problem for well-posedness.) Now assume conversely that the problem is well-posed, and let denote an eigenvalue of P(iu/). We first show that a = 0. The matrix P(iu), (jj = \uj\uj', has the eigenvalue |u;|(a + ib) with real part a|u;|. If a > 0 then cannot be bounded by Keat with a. K independent of a;. If a < 0 then we onsider P(—i\u\u)'), and also arrive at a contradiction. This shows that all eigenvalues of P(iw'} are purely imaginary. To prove the second condition, note that well-posedness yields witn some A, A independent of *> And t > 0, Therefore, We fix t' = |u>|t and obtain for |u;| -* oo, Now we apply Theorem 2.3.2 to the set of matrices Constant-Coefficient Cauchy Problems 57 and use condition 3. of that theorem. Since we have already shown that Re K = 0 for the eigenvalues K of P(iu/), the estimate (2.3.5) implies that SP(iu')S~{ is diagonal. This finishes the proof of the theorem. Concepts of hyperboticity. Theorem 2.4. 1 characterizes those first-order (con- stant-coefficient) equations for which the Cauchy problem is well-posed. These equations ut = Pu are called strongly hyperbolic. The definition of other concepts of hyperbolicity, which play a role in applications, is summarized in Definition 1. A first-order equation ut = P(d/dx)u with is called: weakly hyperbolic if for all u; G Rs all eigenvalues of P(iu>) are purely imagi- nary; strongly hyperbolic if the conditions of Theorem 2.4. 1 are met; symmetric hyperbolic if A j — A * . j = 1 . • • • . s ; strictly hyperbolic if for all u; £ Rs, w ^ 0. all eigenvalues of P(?'u;) are purely imaginary and distinct. As we have seen, the Cauchy problem for a weakly hyperbolic equation is generally not well-posed. In the case of a symmetric hyperbolic equation, the symbol has the form P(iu) = i ]T u/'j A j with a Hermitian matrix £3 ^j A) • Therefore, P(zuO can be diagonalized by a unitary transformation, and the Cauchy problem is well-posed. If the equation is strictly hyperbolic, then there is a complete set of eigenvectors for P(iu/). |u/| = 1, and the eigenvectors can be chosen as analytic functions of u/. If S~'(u/) contains the eigenvectors as columns, then S(u/)P(iu/)S'~l(u;') is diagonal, and one obtains a bound Consequently, strictly hyperbolic equations are strongly hyperbolic, and the Cauchy problem is well-posed. No exponential growth. The solutions of a strongly hyperbolic system (2.4. 1 ) (without zero-order term) do not show any exponential growth: For each u; there is a transformation S = S(u>) such that SP(iuj)S~l = A is diagonal with purely imaginary entries. Thus, for the Hermitian matrix H = S*S it holds that 58 Initial-Boundary Value Problems and the Navier-Stokes Equations Lemma 2.1.4 applies with Q = 0, and therefore |ep(iu;)t| < K. Here K is independent of u; and of t > 0. For later reference we formulate an algebraic result, which summarizes a part of the above considerations. Lemma 2.4.2. For each A € Cn'n the following two conditions are equiva- lent: \. The eigenvalues of A are real, and A has a complete set of eigenvectors. 2. There exists a positive definite Hermitian matrix H with HA = A*H. Proof. First assume that (1) holds and set P — iA. There is 5 such that SPS~l is diagonal with purely imaginary entries. We define H = S*S and obtain, as above, Conversely, let HA = A*H, H = #* > 0. We can write H = H1'2 = (#'/2)*, and obtain For the Hermitian matrix Hl/2AH~1/2 there is a unitary matrix U such that is real diagonal. Consequently, if we set S — UWf2t then SAS~l = D, and (1) is shown. Smoothness of H(u>). For general strongly hyperbolic systems, the matrices H = H(u) with HP(iu>) + P*(iui)H = 0 cannot be chosen as globally smooth functions of a; 7^ 0. This leads to technical difficulties if one wants to go over to variable coefficients. We will comment on this further in Section 3.3.1 below. Of course, if the system is symmetric hyperbolic, then P(iu>) + P*(iu;) = 0, and one can choose H (u;) = /. Also in the strictly hyperbolic case the matrix H (u/0 can be chosen as a smooth function of uj ^ 0, because the eigenvectors of P(iu) depend smoothly on the matrix elements. However, strictly hyperbolic systems hardly ever appear in applications. One can prove, for example, that a Constant-Coefficient Cauchy Problems 59 strictly hyperbolic system in three space variables must have at least dimension n = 7. 2.4.2. The Compressible Euler Equations Linearized at Constant Flow The full equations without forcing read where Let U = ([/, V, W). R, P denote a constant state of u, p, p with P = i.e., U, R. P are independent of x and t. With - supposedly small - corrections u'. p', p' we substitute the ansatz into the above equations; after neglecting all terms which are quadratic in the corrections we obtain the linear equations where D/Dt denotes now the constant-coefficient operator To simplify the notation, we drop ' in the expressions for the perturbation terms u', etc. If we set and use the equation p = Kp to eliminate the pressure from the momentum equations, we find that In matrix form, this first-order system reads 60 Initial-Boundary Value Problems and the Navier-Stokes Equations Let us assume « = c2 > 0 . Then the system can be symmetrized by a simple scaling. Using the new variable p = cp/R, we obtain the symmetric hyperbolic system Since this system leads to a well-posed Cauchy problem, the same is true for the equations (2.4.2). The assumption is crucial. If K < 0 then, for example, the matrix has the nonreal eigenvalues Hence the system (2.4.2) is not hyperbolic, and the Cauchy problem is ill-posed. For K = 0 the Cauchy problem is ill-posed, too. Constant-Coefficient Cauchy Problems 61 2.4.3. Parabolic Systems Consider an even-order operator of the form where and p2m(d/dx) is called the principle part of P(d/dx). Parabolicity of ut = P(d/dx)u is defined in terms of the principle part of F as follows: Definition 2. The equation ut = P(d/dx)u is called parabolic if for all (jj E Ra the eigenvalues KJ(U;), j = 1,..., n, of P2m(zu;) satisfy with some d > 0 independent of uj. A simple but important example is the equation ut = Aw with the Laplacian Here F2(iu;) = -(w2 H h u;2), and one can take 6 = 1. We show Theorem 2.4.3. The Cauchy problem for a parabolic system ut = P(d/dx)u is well-posed. Proof. The proof proceeds along the same lines as the proof of Theorem 2. 1 .3; the Matrix Theorem is not needed. First consider the operator P2m(d/dx) with the symbol Transform P2m(iu>') to upper-triangular form by a unitary matrix U = U(u>'): 62 Initial-Boundary Value Problems and the Navier-Stokes Equations The elements of Pim(iu'), |u/| = 1, are uniformly bounded; therefore, for some constant K it holds that The parabolicity assumption yields We set D = diag(l, d, . . . , dn~'), d > 0, 5 = Dt/, and obtain, for sufficiently large d, Here d can be chosen independently of u/, |u/| = 1. As in the proof of Theorem 2.1.3, we set H = #(u/) = 5*5. Then and the result follows from Lemma 2.1.4. Roughly speaking, in the present context of well-posedness, the main feature of parabolic equations is the following: The dissipativity of the principle part - which is expressed by the estimate (2.4.3) - will force all high wave-number components el^'x^ \uj\ large, to decay in time, no matter what the lower-order terms of P(d/dx) look like. The formal analog in the above proof is that we can dominate the term const (jo;!2"*"1 + 1) by £|u>|2rn if |u>| is sufficiently large. 2.5. Mixed Systems and the Compressible N-S Equations Linearized at Constant Flow The compressible N-S equations without forcing read where Constant-Coefficient Cauchy Problems 63 As in Section 2.4.2, we let U = (J7, V, W), R, P denote a constant state with P = r(R) and use an ansatz as described in 2.4.2. Neglecting all terms which are quadratic in the corrections, we find linear equations for u', /£>', p'. Again, we drop ' in the notation and eliminate the pressure using As a result, we obtain the linear constant-coefficient equations These read in matrix form Here £>i = d/dx, etc. The matrices A\,A2,A3 can be read off from (2.4.2). Let us denote the second-order operator on the right side of the above equation by P2 — P2(d/dx); thus ^2 acts on (w, v, w, p)T . Obviously, the symbol has 0 as an eigenvalue. The reason is that the continuity equation does not contain a second-order term. Consequently, the above system is not parabolic. We now rewrite the above system in block form and separate the momentum equations from the continuity equation: 64 Initial-Boundary Value Problems and the Navier-Stokes Equations The second-order operator P2 acts only on u, not on (u, p). Let us show that ut = P2u is parabolic. The symbol is real and symmetric; it reads For any y € R3 it holds that It follows that all eigenvalues of the matrix P2(iuj) are < -j/|u;|2. Hence, according to Definition 2, Section 2.4.3, the equation u( = P2u is parabolic. To study the full linear system for (u, p) given above, we consider the system in its block form, Here P\ is the first-order scalar operator as can be read off from (2.5.1). Clearly, the scalar equation pt — P\p (with real coefficients) is strongly hyperbolic. The operators Rij are of order one. Their precise form will not be important for the discussion. Since both F2 and P\ lead to well-posed Cauchy problems, it is clear that the Cauchy problem for the completely uncoupled system is also well-posed. We want to show that the coupling term does not destroy well-posedness. This follows from the result proved in the next theorem, where we consider a slightly more general situation. It might be Constant-Coefficient Cauchy Problems 65 worth while to note again that - on the Fourier side - the discussion is purely algebraic; the relevant estimate for the coupling term becomes quite transparent. Theorem 2.5.1. Consider vector functions where u(x,t) has m components and v(x,t) has n components. Let P = P(d/dx) denote a constant-coefficient differential operator whose corresponding block form is Assume that ut = P2u is second-order parabolic, that vt = P\v is first-order strongly hyperbolic, that the Rjk are of order one, and that RQ is of order zero. Then the Cauchy problem for wt = P(d/dx)w is well-posed. Proof. We can assume that PI coincides with its principle part. With suitable constants 6 > 0, K* > 0 and suitable Hermitian matrices //i(u;). HI(U}} we have, for all u; € Rs, We define the Hermitian matrix and consider the quadratic form belonging to For all u € Cm, v e Cn, all u> e R", and some constant K it holds that 66 Initial-Boundary Value Problems and the Navier-Siokes Equations For any two real numbers a, 6 we know that 2ab < a2 + 62 , and therefore, Using this inequality, we arrive at the estimate If we choose e2 = 6/K then Hence, with some constant K\ independent of u>, u, v it holds that and therefore, Now well-posedness follows from Corollary 2.3.4. 2.6. Properties of Constant-Coefficient Equations As mentioned previously, the switching between a variable-coefficient problem and the constant-coefficient equations obtained by freezing coefficients intro- duces lower-order terms. Thus, it is natural to ask which constant-coefficient equations can be perturbed by arbitrary lower-order terms without destroying wcll-poscdncsi. We shall show thai only strongly hyperbolic and parabolic ays terns have this property. (Of course, for other equations, a restricted class of perturbing lower-order terms might not destroy well-posedness.) Another technically important result of this section can be described as fol- lows: If the Cauchy problem for ut = P(d/dx)u is well-posed, then one can construct an inner product and a norm such that In other words, the estimate of Theorem 2.2.2, which expresses well-posedness, holds with K = 1 if we replace the L2-norm by || ||//. If one uses the £/2-norm only, then - in general - a constant K > 1 is required; however, if this is the Constant-Coefficient Cauchy Problems 67 case, the solution-estimate becomes locally (in time) useless: it does not even express continuity in time. Therefore, the construction of norms || ||# with property (2.6.1) is important if one wants to treat variable-coefficient problems by localization. Furthermore, we will establish Duhamel's Principle for inhomogeneous equa- tions 2.6.1. Perturbation by Lower-Order Terms and Well-Posedness Consider the Cauchy problem for a system where If the principle part Pm is of first order and strongly hyperbolic or of even order and parabolic, then the Cauchy problem for P = Pm + Q is well-posed, no matter what the lower-order terms Q look like. For the strongly hyperbolic case the operator Q is of order zero, and the result follows from Lemma 2.3.5; the parabolic case has been treated in Theorem 2.4.3. We show the following converse: Theorem 2.6.1. Consider the Cauchy problem for (2.6.2) with some fixed principle part Pm, and assume it is well-posed for any choice of the lower-order terms Q. Then either m = 1 and (2.6.2) is strongly hyperbolic or m is even and (2.6.2) is parabolic. Proof. For m = 1, strong hyperbolicity and well-posedness are the same; see Theorem 2.4.1 and the definition following the theorem. Thus we can assume m > 2 and must show that m is even and (2.6.2) is parabolic. First assume m > 3 to be odd. We show that the real parts of the eigenvalues of Pm(iuj) are zero. Since the Cauchy problem is well-posed for Q = 0, we have As before, let u> = |u;| u/, |u/| = 1, and let K denote an eigenvalue of Pm(iu>'). Then Pm(zu>) has |u;|mK as an eigenvalue and (since m is odd) Pm(-iuf) has — \(jj\mK as an eigenvalue. The estimate (2.6.3) implies that Re « = 0. 68 Initial-Boundary Value Problems and the Navier-Stokes Equations Now let thus The real part of each eigenvalue of P(iu) equals u\. This function of u; is unbounded, and therefore the Cauchy problem for P cannot be well-posed. Second, assume that m is even, but ut ~ Pmu is not parabolic. Let KJ(U>') denote the eigenvalues of Pm(iw'), \u>'\ = I. The function depends continuously on u/, and if ut = Pmu is not parabolic, then there exists a vector 77, (77! = 1, and an eigenvalue K(rj) of Pm(iTj) with Let the symbol of P = Pm + Q reads For the wave-vectors where 77 is chosen with (2.6.4), the real parts of the eigenvalues of P(iui} cannot be bounded. This contradicts well-posedness. 2.6.2. Symmetrization and Energy Norms Suppose that the Cauchy problem for ut = P(d/dx)u is well-posed where P is a general constant-coefficient operator, see (2.2.3). There are constants a and K with Constant-Coefficient Cauchy Problems 69 According to the Matrix Theorem 2.3.2, one can construct Hermitian matrices H(u) with Here K4 > 0 is independent of u> 6 R". Using these matrices, we define the linear (pseudo-differential) operator H mapping LI into 1/2 by (The symbol P(iu;) depends analytically on u;, and an examination of the con- structive proof of the Matrix Theorem shows that H(UJ) can be chosen as piece- wise smooth. Therefore, the above integral exists for u € MQ. For general u € 1/2, the image Hw can be obtained by the usual extension process; see Theorem 2.2.4.) The operator H is used to define an inner product and a norm by Theorem 2.6.2. The constructed norm is equivalent to the Li-norm; i.e., Furthermore, if u(x, t) denotes a (Mo or generalized) solution of the Cauchy problem for ut = Pu, then Proof. With Parseval's relation one finds that and similarly (u,Hu) > 70 Initial-Boundary Value Problems and the Navier-Stokes Equations The following equations hold for any Mo-soiution u = u(x, 0: Thus the solution-estimate follows. For general initial data u(-,0) e L2, the result follows by approximation. The inner product (u, v)// is called an energy inner product for P = P(d/dx) and || || # is called an energy norm. The transformation by //(u;) in Fourier space is called symmetrization. The above proof shows that the operator P satisfies the following estimate for all sufficiently smooth functions w = w(x), w € £2: For this reason the operator P is called semibounded w.r.t. the inner product (-, •)//• With this definition we can summarize our result in the following way. eorem 2.6.3. The Cauchy problem for ut = P(d/dx)u is well-posed if d only if the operator P is semibounded w.r.t. an inner product (•, •)// which corresponds to a n equivalent to the Li-norm. The above construction of the operator H is important in the general theory of partial differential equations. Many proofs concerning the well-posedness of linear problems with variable coefficients proceed technically by constructing suitable inner products such that the differential operators with frozen coeffi- cients become semibounded. For the Navier-Stokes equations one can generally work with the usual La -inner product. 2.63. Inhomogeneous Systems: Duhamel's Principle We stan with an ordinary initial value problem Constant-Coefficient Cauchy Problems 71 Here A(i) € Cn>n and F(t) 6 C" are assumed to be continuous in t. For any T > 0 consider the homogeneous system The solution depends linearly on VQ; thus we can write This defines the solution operator 5(t,r) e C"'n. Duhamel's principle for (2.6.5) states Lemma 2.6.4. The solution of the inhomogeneous system (2.6.5) is given by Proof. By definition of the solution operator S(t, r) we have that and therefore Since S(t, t) = I, we find, by differentiation of (2.6.6), This proves the lemma. Consider a Cauchy problem and assume the problem is well-posed for F = 0. The homogeneous system is solved by 72 Initial-Boundary Value Problems and the Navier-Stokes Equations where 5(£) is the generalized solution operator defined in Section 2.2.5. Then Duhamel's principle states that (2.6.7) is solved by On the right-hand side, one has to integrate the LI -valued function r —> S(t - r)F(-, r) from r = 0 to r = t. With a proper concept of a generalized solution and a proper definition of the integral, one can indeed prove formula (2.6.8) under mild assumptions on F. We will restrict ourselves here to Mo- solutions (see Section 2.2.1) and assume that i) / € M0; ii) F(-,0€M0forall t > 0; iii) the function F(u),t) is continuous, and F(u>, t) = 0 for |u;| > K with some K independent of t. Then Fourier transformation yields By Duhamel's principle for ordinary differential equations, Lemma 2.6.4, Transforming back, we obtain Here we have changed the order of integration in x and t', this is justified under our assumptions on F. If we now define as denoting the integral on the right-hand side of (2.6.9), we can drop the variable x in (2.6.9) and write the equality of functions Thus we have shown Duhamel's principle (2.6.8) under our restrictive assump- tions. Constant-Coefficient Cauchy Problems 73 We can also derive solution-estimates for inhomogeneous equations. Let (-.-)// denote an inner product which makes P semibounded, and let || ||// denote the corresponding norm. Under the same assumptions on / and F as above, More general data / and F can be treated by approximation. It is not difficult to deduce that for 0 < t < T. An inequality of the above type will be used below as the definition of well-posedness for problems with variable coefficients. 2.7. The Spatially Periodic Cauchy Problem: A Summary for Variable Coefficients Thus far we have assumed a constant-coefficient operator and initial data on the whole space. The purpose of this section is to state briefly the corresponding results for the spatially periodic case. For constant coefficients, the proofs could be given as above simply by replacing Fourier transforms with Fourier expansions. For later reference we state definitions and results for variable-coefficient equations. The proofs are carried out in Chapter 3 for one space dimension and in Sections 6.1 and 6.2 for more than one space dimension. In the last section of this chapter we present two counterexamples to a naive localization principle. 2.7.1. Solution Concept and Well-Posedness A function v = v(x), x e R% is 1 -periodic in the j-th coordinate if We call v = v(x), x € R'9, 1 -periodic in x if v is I -periodic in each coordinate Xj, j = 1 . . . . , s. Also, a function v(x, t) is called 1 -periodic in x if x — > r(x, t) is 1 -periodic in x for each fixed t. These functions are also called 1 -periodic, for short; we will not deal with periodicity in time. 74 Initial-Boundary Value Problems and the Navier-Stokes Equations If w, v : Ra —> Cn are C°° -functions which are 1-periodic in x, then their LI -inner product and norm are defined by We refer to the following problem as the spatially periodic Cauchy problem. Suppose that are C°°-functions of their arguments, which are 1-periodic in x. We try to find a C00 -solution u = u(x, t) of which is 1-periodic in x. Definition 1. The above spatially periodic Cauchy problem is called well- posed if: i) for each / = /(x) and each F = F(x, t) (satisfying the above conditions) there exists a unique solution u = u(x, t) (satisfying the above conditions); ii) for each T > 0 there is a constant K(T) independent of /, F with If the coefficients Av = A^x, t) are constant, then one can use the symbol P(iu) to decide the question of well-posedness. Similarly to Theorem 2.2.2, one obtains: The spatially periodic Cauchy problem with a constant-coefficient operator P(d/dx) is well-posed if and only if there are constants K and a such that (One could restrict the components of u> to be integer-multiples of 2?r. This would not change the possibility of an estimate of the above form, however.) Constant-Coefficient Cauchy Problems 75 2.7.2. Strongly Hyperbolic and Second-Order Parabolic Equations Hyperbolic systems. Consider a first-order operator with variable coefficients We define its symbol by Suppose all frozen-coefficient problems are strongly hyperbolic as defined in Section 2.4.1. It follows from Lemma 2.4.2 that there is a positive definite Hermitian matrix with We define strong hyperbolicity of the variable-coefficient equation as follows: Definition 2. The equation (2.7. 1) is called strongly hyperbolic if there exists a Hermitian matrix function which is C°° -smooth in all arguments, is 1 -periodic in x, and satisfies (If a zero-order term and a forcing function are added in (2.7.1), strong hyper- bolicity is defined in the same way.) T matrix function H(x,t,uj) is called a symmetrizer. Thus, except for the smoothness and periodicity of the symmetrizer, we have defined strong hyperbolicity by adopting the constant-coefficient concept. 76 Initial-Boundary Value Problems and the Navier-Stokes Equations The most important cases are symmetric hyperbolic systems, where For such systems the symbol is always antisymmetric, and one can take H = I as the trivial symmetrizer. The hyperbolic systems which occur in the context of the Euler equations are symmetric hyperbolic after a simple transformation of the density; a first example was given already in Section 2.4.2. One can show Theorem 2.7.1. The spatially periodic Cauchy problem for a strongly hyper- bolic equation is well-posed. For a proof in one space dimension, see Section 3.3.1; for the general case, see Section 6.2. Parabolic systems. Now consider a second-order equation where The principle part of P is and we define its symbol Let denote the eigenvalues of P^(x, t, icu). Suppose all frozen-coefficient equations of (2.7.2) (with F = 0) are parabolic, as defined in Section 2.4.3. Then, for each fixed (x, t), there is 6(x, t) > 0 with Constant-Coefficient Cauchy Problems 77 Smoothness and periodicity in x imply that we can choose a uniform 6 = 6(T) > 0 for 0 < t < T. Therefore, the following definition of parabolicity requires nothing but parabolicity of all frozen-coefficient problems. Definition 3. The equation (2.7.2) is called parabolic if for each T > 0 there is 6(T) > 0 with One can show Theorem 2.7.2. The spatially periodic Cauchy problem for a second-order parabolic system (2.7.2) is well-posed. This result will be proved in Sections 3.1 and 3.2 for one space dimen- sion and in Section 6.1 for more than one space dimension. If we ignore the requirement of the smoothness of the symmetrizer in the strongly hyperbolic case, we can summarize Theorems 2.7.1, 2.7.2 by saying: for strongly hy- perbolic and second-order parabolic systems the localization principle is valid. These variable-coefficient systems inherit their well-posedness from the frozen- coefficient equations. It is not known whether the requirement of smoothness of the symmetrizer in the strongly hyperbolic case is really necessary. In the next section we present two counterexamples to localization for equa- tions which are neither hyperbolic nor parabolic. 2.7.3. Counterexamples to Localization Our first example is a 27r-periodic system for which all frozen-coefficient equa- tions are well-posed; the variable-coefficient system is ill-posed nevertheless. Example 1. Consider the second-order system where For any frozen-coefficient problem 78 Initial-Boundary Value Problems and the Navier-Stokes Equations we can introduce new variables and obtain The Cauchy problems for w and for v are well-posed. Now we introduce as a new variable into the given variable-coefficient problem; then we obtain The system for v has constant coefficients, and therefore we can decide the question of well-posedness. The eigenvalues «i, «2 of the symbol P(iu) are the solutions of Thus,0 = det(P(iu;) - K) = det and consequently the problem is ill-posed for /? ^ 0. In the following Schrodinger equation the converse happens: The variable- coefficient problem is well-posed although the problems with frozen coefficients are ill-posed. Example 2. Consider the scalar equation where p(x) is a real, smooth 1 -periodic function with p(x) > po > 0. The Cauchy problem for a frozen-coefficient equation Constant-Coefficient Cauchy Problems 79 is ill-posed if Pa;(xo) / 0, because is not bounded from above. We only sketch the proof that the given variable- coefficient equation leads to a well-posed spatially periodic Cauchy problem. The equation can be written as If we assume that u = u(x, t) is a smooth 1 -periodic solution, then Thus we obtain a priori To obtain the existence of a solution, we can consider the parabolic equations and send e — > 0. This technique to prove existence is illustrated in Chapter 3. Notes on Chapter 2 The first paper dealing with the Cauchy problem for general systems of partial differential equations with constant coefficients is due to Petrovskii (1937). He uses Hadamard's (1921) definition of well-posedness, which is equivalent to our definition of weak well-posedness. He proved that the Cauchy problem for ut — P(d/dx)u is well-posed if and only if the eigenvalues K of the symbol P(iu>) satisfy an inequality Later Carding (1951) proved that one can choose Ci = 0. The Matrix Theorem is proved in Kreiss (1959); its application to the Cauchy problem is discussed in Kreiss (1963). The latter paper also shows that the Cauchy problem for hyperbolic first-order systems is well-posed if the eigenval- ues of P(iui) are purely imaginary and their algebraic multiplicity is constant. Our perturbation example in Section 2.2.3 is typical. Yamaguti and Kasahara (1959) have proved the following theorem: If the Cauchy problem for a first- order system is weakly well-posed for all lower-order perturbations, then it is well-posed. In connection with our counterexamples to localization, W.G. 80 Initial-Boundary Value Problems and the Navier-Stokes Equations Strang (1966) gave a necessary condition for the well-posedness of Cauchy prob- lems: If the Cauchy problem is well-posed for a system ut = P(x, d/dx)u with x-variable coefficients then the problem is also well-posed for all problems ut = Pm(xo,d/dx)u with frozen coefficients. Here Pm denotes the principle part of P. 3 inear Variable- oefficien Cauchy Problems ID In this chapter we treat second-order parabolic and first-order strongly hyperbolic systems in one space dimension. Instead of considering the Cauchy problem with initial data in 1/2, we deal with problems which are 1-periodic in x. The periodic problem has the technical advantage that the behavior at x = ±oc need not be specified, but the arguments for initial data in Z/2 would be essentially the same. Assuming the existence of a smooth solution for the parabolic equation, we first prove estimates of the solution and its derivatives. In Section 3.2 we write down a simple difference scheme and prove analogous estimates for it. By sending the step-size to zero, we obtain a solution for parabolic systems in ather elementary and constructive way. Strongly hyperbolic equations are treated by adding a small second-order term, whose coefficient is sent to zero. In a similar fashion as in the constant-coefficient case, we also treat certain mixed hyperbolic-parabolic systems and give an application to the linearized N-S equations. In the parabolic case we use as a guiding principle: first, assume the existence of a solution and show estimates for it and its derivatives; second, write down a difference scheme and show analogous estimates which imply the existence of a solution. This principle is very useful for equations of different type, also. We demonstrate this in Section 3.6 with an application to the linearized Korteweg- 81 82 Initial-Boundary Value Problems and the Navier-Stokes Equations de Vries equation. The linear Schrodinger equation will be treated as a limit of parabolic equations. The estimates derived for the parabolic systems in Section 3.1 are very ele- mentary. They do not express the smoothing property of the parabolic operator. In some applications this smoothing is important, however, and it will be shown in Section 3.2.6. Also, we demonstrate some important properties of strongly hyperbolic systems in Section 3.3: there is a finite speed of propagation, and discontinuities travel along the characteristics. 3.1. A Priori Estimates for Strongly Parabolic Problems We consider a second-order system together with an initial condition The matrices and the vector functions are assumed to be of class C°° for simplicity. It is an essential assumption, however, that all functions are taken as 1-periodic in x for each fixed t, Fur- thermore, to begin with, we assume that This is slightly more restrictive than the assumption of parabolicity for all frozen- coefficient problems. If AI + A\ satisfies a lower bound of the above form, the equation (3.1.1) is called strongly parabolic. In this section we will assume that u(x, t) is a C^-sol ion of the above problem which is 1-periodic in x for each fixed t. The existence of such a solution will be proven in Section 3.2. Here we will derive estimates for u and its derivatives. Notations. For vectors u, v e Cn and matrices A e C11-" we remind the reader of the notations Linear Variable-Coefficient Cauchy Problems in ID 83 For (smooth) 1 -periodic vector functions u = w(x), v = v(x) our basic inner- product and norm are For nonnegative integers p we also use the Sobolev inner-product and norm given by For (smooth) 1 -periodic matrix functions A = A(x) let It will be frequently used that and similarly that If u — u(x, t) and v = v(x, t) are functions of (ar, t) we often abbreviate and thus leave off the arguments to simplify the notation. The main tool we will use is "integration by parts": If u, v e C1 are 1 -periodic, then 3.1.1. The Basic Energy Estimate Let T > 0 denote an arbitrary but fixed time. For 0 < t < T we have 84 Since A2UXX = (A2UX)X — A2Xux, integration by par yields and similarly Therefore, For any two real numbers a, 6 we know that ab < |a + ^6 ; thus for any a ^ 0, and therefore, Choosing a = 0(6, €2) > 0 appropriately, we find from (3.1.3) that The derivation shows that c\, 02,03 only depend on d and on bounds for in the interval 0 < t < T. Using the abbreviations we have shown the differential quality Now (a simple version of) GronwalPs Lemma, formulated next, allows us to estimate y(t).Initial-Boundary Value Problems and the Navier-Stokes Equations Linear Variable-Coefficient Cauchy Problems in 1D 85 Lemma 3.1.1. Suppose y e C'[0,T], ^ e C[0,T] satisfy for some c > 0. Then Proof. For the function z(t) = e~ciy(t) it holds that Thus integration yields and the desired estimate for y(t) = eciz(t) follows. An application of this result gives us Lemma 3.1.2. Let C\ denote a constant which bounds the norms (3.1.5) in 0 < t < T. There is another constant €2 which depends only on C\, 6, and T such that Proof. If c = 03 denotes the constant in (3.1.4), the result follows from the previous lemma with Ci — ecTmax {l,c}. 3.1.2. Estimates for Space-Derivatives of u If we differentiate the given differential equation (3.1.1) with respect to x we find that, for v = ux, with 86 Initial-Boundary Value Problems and the Navier-Stokes Equations The function v satisfies the initial condition v(x, 0) = ux(x, 0) = fx(x). Since we have an estimate for \\u\\ and thus for G, we can apply the previous lemma and obtain an estimate for v = ux: Here €3 depends only on 6, on T, and on a bound for in 0 < t < T. It is obvious that we can proceed in the same way and further obtain estimates for uxx,uxxx, etc. Just differentiate the above equation for v = ux with respect to x, etc. Lemma 3.13. Given a positive integer p and a time T > 0, then the solution of (3.1.1), (3.1.2) satisfies Here C depends only on p, on T, on 6, and on a bound for the coefficients Aj(x, t) and their derivatives of order < p in 0 < t < T. 3.1 J. Estimates for Time-Derivatives and Mixed Derivatives Using the given differential equation (3.1.1), we can always express time deriva- tives of W by Space derivatives, For example, if the differential equation readE then differentiation gives us To express uttt, we need six space derivatives of u, four space derivatives of F, two space derivatives of Ft, and the function Ftt. In general, we can express q time derivatives of u by 2q space derivatives of u and 2(q — 1 — k) space derivatives of Linear Variable-Coefficient Cauchy Problems in ID 87 Further differentiation with respect to x allows us to express mixed derivatives of u also. Since the space derivatives of u are already estimated in Lemma 3.1.3, we have Theorem 3.1.4. Given any nonnegative integers p and q and a time T > 0, there is a constant C with in 0 < t < T. The constant C is independent of f and F, and only depends on 6, on T, and on a bound for the derivatives of the coefficients AQ, A\, A2 of order <2q + pinQ<t<T. For p — q = 0 one also needs a bound for ^x- 3.2. Existence for Parabolic Problems via Difference Approximations In this section we prove existence of a C00-solution u(x. t) of the space-periodic initial value problem (3.1.1), (3.1.2). As in Section 3.1, the main assumption is We relax this condition in Section 3.2.5, however, where we treat general parabolic problems. All functions are assumed to be 1-periodic in x, and the coefficients of the differential equation are assumed to be C00-smooth. First we also assume that the initial function is in Cx; general /^-initial data and the smoothing property of parabolic equations are treated in Section 3.2.6. It follows immediately from the basic energy estimate stated in Lemma 3.1.1 that the Cauchy problem (3.1.1), (3.1.2) has at most one classical solution, i.e., at most one solution in C'(0 n C2(x). If there were two solutions w, v then w = u - v would be a solution of (3.1.1), (3.1.2) with F = / = 0. Then our basic energy estimate implies w = 0. 3.2.1. The Main Result and Outline of the Proof We want to prove the existence of solutions using a difference approximation. For that reason we introduce a gridlength h = 1 /N, N a natural number, grid- points xv = *//i, v 6 Z, and gridfunctions with function values vv := v(x^ e C". 88 Initial-Boundary Value Problems and the Navier-Stokes Equations We shall always assume that the gridfunctions are 1 -periodic; i.e., As usual, the translation operator E is defined by Clearly, w := Ev is also 1 -periodic. The j-th power of E (j an integer) is and we shall use the notation With the help of the translation operator E, we define the forward and backward divided difference operators D+, D_: The usual second-order accurate approximations to d/dx and d*/dx2 are respectively. If v = v(x) is a smooth function of x 6 R, then its restriction to the grid is (vv)v^i = (^(z,,)),^. This gridfunction is also denoted by v, for simplicity. A similar notation applies to matrix functions A — A(x). Often A — A(x) will be a 1 -periodic matrix function and Av a gridfunciion with function values We need some simple error formulae for the above difference operators. E.g., if we apply the forward difference operator D+ to the restriction of a smooth function v, then Taylor expansion gives us where Linear Variable-Coefficient Cauchy Problems in ID Correspondingly, where89 These error formulae will be applied below. We now describe a simple difference analog of (3.1.1), (3.1.2), which we use for our existence argument. Only space is discretized, time is left continuous. The difference analog reads The equations (3.2.5) represent an infinite system of ordinary differential equa- tions for v(t) = (tv(0)i/ez subject to initial conditions. However, we are only interested in 1-periodic soiutions and therefore e need to consider (3.2.5) only for v = 0.1.2..... N — I. Using the periodicit conditions FIGURE 3.2.1. Gridlines for difference scheme. 90 Initial-Boundary Value Problems and the Navier-Stokes Equations we can eliminate vx(t) and V-\(t) and obtain a system of N differential equa- tions (with initial conditions) for the AT unknowns v0(t),..., v.v-i(f). Any solu- tion of this initial value problem generates the corresponding solution of (3.2.5) by periodic extension. Therefore, the existence of a unique periodic solution of (3.2.5) follows from standard theorems for ordinary differential equations. It is clear that the solution v(t) = vh(t) depends on the step-size h, but we often suppress this in our notation. We are going to show in Section 3.2.3 that we can estimate all difference- differential quotients D+dqvh(t)/dtq independently of h. Then we shall use a theorem from approximation theory which states that we can interpolate the gridfunction v = vh with respect to x in such a way that the interpolant wh(x, t) is as smooth as vh; i.e., we can estimate all derivatives dp+qwh(x,t)/dxpdtq in terms of Dp+dqvh(t)/dtq. The next step is to show that wh satisfies (3.1.1), (3.1.2) with F, / replaced by F + hPf, f + hff. For h -» 0, the functions wh converge to the solution of (3.1.1), (3.1.2). 3.2.2. The Basic Energy Estimate for the Solutions of the Difference Approximation Before we derive the basic energy estimate for the system (3.2.5), let us first introduce a discrete scalar product and norm and let us establish some simple rules. If v, w are 1-periodic gridfunctions, then we define their discrete scalar product and norm by The scalar product for gridfunctions has the same general properties as the L^- scalar product of functions of a continuous variable; i.e., it is a sesquilinear form, and we have the rules Also, where Linear Variable-Coefficient Cauchy Problems in ID 91 Note that is bounded independently of h if A = A(x) is a smooth 1 -periodic matrix function. Another simple rule is: If Av + A*v > 281 for all V, then Corresponding to integration by parts, there are rules of summation by parts in the discrete case. For periodic gridfunctions v. w, This is easily checked: The second relation follows from the first. The identity gives us a discrete analog of Leibniz' product rule: Now we can derive the basic energy estimate. The solution v — vh of the system (3.2.5) satisfies 92 Initial-Boundary Value Problems and the Navier-Stokes Equations Here thus (Observe that (D+A2)v — h~l $*"** AI* dx.) Correspondingly, where RQ satisfies a similar estimate as RQ. Introducing these expressions into (3.2.7) and observing (3.2.6), we obtain an inequality of the same type as for the continuous case, and therefore the following basic energy estimate holds: Lemma 3.2.1. Let C\ denote a constant which bounds the norms (3.1.5) in 0 < t < T. There is another constant C2 which depends only on C\ , 6, and T such that 3.23. Estimates for Higher Order Divided Differences As an auxiliary result we first show Lemma 3.2.2. For any smooth I -periodic function g it holds that Proof. For p = 0 the estimate is true. Assume it is valid up to the order p — 1. Let thus /(£„) = (D+g),,. Therefore, This proves the lemma. We begin with an estimate for w := D+v where v is the solution of (3.2.5), as before. Applying D+ to (3.2.5) and using the discrete Leibniz' rule, we find the following system for w: Linear Variable-Coefficient Cauchy Problems in ID 93 where The system (3.2.8) is of the same form as (3.2.5) and - using Lemma 3.2.2 to estimate coefficients - we obtain an estimate for w = D+v, which is of the same type as for the continuous case. Repeating this process, we obtain estimates for all forward difference quotients D^v. As in the continuous case, we can also differentiate with respect to time and obtain bounds for D^_dqv(t)/dtq. If one introduces the discrete analog of the continuous Hp-norm by one can summarize the result as follows: Theorem 3.2.3. The estimate of Theorem 3.1.4 holds for v = vh if one replaces the Hp-norm by its discrete version. The constant is independent of the step- size h. 3.2.4. Convergence as h —> 0 The following result about Fourier interpolation is proved in Appendix 2. Theorem 3.2.4. Let h = l/(2m +1) where m is a positive integer. Ifvh(x) is a 1 -periodic gridf unction, then there is a unique Fourier polynomial w/i/c%/i interpolates vh at all gridpoints: For a/ry p = 0,1,2,... it holds that It is remarkable that the constant in the estimate does not depend on h. In our application, the gridfunction vh(x, t) depends smoothly on t. If wh(x, t) denotes the Fourier interpolant w.r.t. x, then wh is a C°°-function of (x,t) since the 94 Initial-Boundary Value Problems and the Navier-Stokes Equations coefficients o£ are C°° -functions of t. (This follows from the construction of the interpolant in Appendix 2.) Also, differentiation of wh w.r.t. t gives us the Fourier interpolant of dvh/dt. Therefore, Theorems 3.2.3 and 3.2.4 imply bounds with constants independent of h. To obtain bounds in maximum norm, the following result - an example of a Sobolev inequality - will be applied. (More general Sobolev inequalities, which we need below, are proved in Appendix 3.) Lemma 3.2.5. Suppose u € C'[0, 1]. Then Proof. There are points XQ, x\ with Let XQ < x\ for definiteness. Then Since |U(XQ)| < ||w||, the result follows. Using the estimates (3.2.9) and the previous lemma, we obtain bounds for all derivatives of wh(x, t) in maximum norm: with constants independent of h. For short, we say that the family of functions wh is uniformly smooth in each finite time interval. Let us show next that wh(x,t) solves the Cauchy problem (3.1.1), (3.1.2) up to terms of order h. Lemma 3.2.6. The functions Ff(x, 0, ff(x) defined by Linear Variable-Coefficient Cauchy Problems in ID 95 satisfy estimates w/iere C does no/ depend on h. Proof. Let x,, < x < xv+\. Then where |^(x, t)| < const, j = 1, 2, 3. If we use these expressions to replace wh in the above definition of F/1 and observe the equation (3.2.5) for v\ then the estimate for F1 follows. Also, and the lemma is proved. Since the family of functions wh = wh(x, t) is uniformly smooth in 0 < t < T, we can apply the result of Appendix 4 (which is based on the Arzela- Ascoli Theorem) and obtain the following: There is a 1 -periodic C°° -function M = w(x, t), x € R, 0 < t < T, and a sequence /i = hj — > 0 such that for all derivatives dp+q/dxpdtq. By the previo lemma it follows that w solves the given initial value problem in 0 < t < T. Here T > 0 is arbitrary, and thus we have shown Theorem 3.2.7. The strongly parabolic initial value problem (3. 1 . 1 ). (3. 1 .2) with l-periodic (in x) C°°-data has a unique Cx -solution u = u(x, t) which is \-periodic in x. 96 Initial-Boundary Value Problems and the Navier-Stokes Equations Remark. To obtain existence of a solution u, it was sufficient to establish convergence for some subsequence wh, h = hj -* 0, via the Arzela-Ascoli The- orem. In view of numerical applications one might ask, however, whether in general wh —* u as h —> 0. This convergence does indeed hold as follows from the basic energy estimate of Section 3.1.1 applied to the difference wh — u. Ac- tually, the (nonconstructive) Arzela-Ascoli argument can be avoided altogether in an existence proof: From the basic energy estimate it follows that wh is a Cauchy sequence in LI as h = hj —» 0, and smoothness of the limit function can be deduced from the uniform (in h) decay rate of the Fourier coefficients of wh. Such a uniform rate follows from the uniform smoothness. 3.2.5. General Parabolic Systems Thus far we have considered parabolic systems (3.1.1) under the restriction We want to replace this by the weaker assumption that all frozen-coefficient problems are parabolic in the sense of the definition given in Section 2.4.3. To this end, we fix T > 0 and assume there exists 6 > 0 independent of x. t with (The assumption AI + A\ > 261 implies Re K > 6 for all eigenvalues K of AI.) A variable-coefficient problem (3.1.1) satisfying the eigenvalue condition (3.2.10) is called parabolic. The discussion given below of a var/oW^-coefficient problem - via prop- erties of equations with constant coefficients - is very important in a general theory of partial differential equations. Basically the technique is the follow- ing; wall pooedness of each constant coefficient prebliffl - Obtained by freezing coefficients at an arbitrary point P = (o?0, *o) - is expressed via the Matrix The- orem in Fourier space. One obtains existence of a certain Hermitian matrix HP for each point P — (#o, to)- These matrices HP are used to define time dependent norms || limn, which are all equivalent to the Z/2-norm. Applying II II//(t) instead of the L2-norm, one can again prove the basic energy estimate, where the energy is now measured in the new norm. In the present situation we do not have to rely on the Matrix Theorem to construct Hp. Instead, using the first part of the proof of Theorem 2.1.3, we have Lemma 3.2.8. Let A denote a constant matrix and let Re K > 6 > 0 for all eigenvalues K of A. There exists a Hermitian matrix H = H * > / such that Linear Variable-Coefficient Cauchy Problems in ID 97 Note that the bound H > I can be enforced by multiplying H with a scalar, if necessary. We proceed with the construction of the norms || \\HW- The idea of the construction is due to Carding (1953). Lemma 3.2.9. Under assumption (3.2.10) there exists a smooth \-periodic matrix function H(x,t) — H*(x.t)> I such that The norms \\ \\H(t) defined by are equivalent to the Li-norm. Proof. For each point P = (XQ, to) we construct a matrix HP with using the previous lemma. By continuity, the point P has a neighborhood N(P) with for all (x,t) e N(P). By the Heine-Borel Theorem there are finitely many points Pi,---.Pj whose neighborhoods N(Pj) cover the set [0,1] x [O.T], T > 0 being fixed. Choose a partition of unity 0j = 0j(ar, t) of 1 -periodic functions subordinate to these neighborhoods: (To be precise, the Heine-Borel Theorem and the Partition of unity argument are being applied to the compact space 5l x [0, T] where Sl denotes the circle. Functions on this space are identified with 1 -periodic functions.) With the help of the locally supported functions 0,, define the matrix function and note that 98 Initial-Boundary Value Problems and the Navier-Stokes Equations The first property ensures that || \\H(t) is equivalent to the L2-norm; the second property implies the estimate (3.2.11) if we sum over j. To illustrate how the basic energy estimate can be obtained under the condition (3.2.10), we consider an equation We have Thus the proof of the basic energy estimate can be finished in the same way as above. The other estimates follow similarly. We summarize the result in Theorem 3.2.10. The spatially periodic Cauchy problem (3.1.1), (3.1.2) ij well-posed if the system is pointwise parabolic. 3.2.6. Smoothing Properties of Parabolic Systems For simplicity we neglect lower-order terms and consider a strongly parabolic system If the initial data u(-,0) = / are given in £,2, there is a unique generalized solution u since the problem is well-posed. We want to show in this section that u G C°° for t > 0 even if the initial data are just in £2 and have no smoothness properties. To show this, we first derive sharper a priori estimates for the solutions to initial data in C°°. Recall that This gives us the usual bound for ||u(-,t)||. which depends only on ||-u(-,0)||. The function v = ur satisfies Linear Variable-Coefficient Cauchy Problems in ID 99 and therefore Thus Integration from 0 to t yields Hence we have a bound for ||ua;(-,0|| which depends only on ||u(-,0)|| and not on ||wx(-,0)||. This process can be continued. We obtain bounds for all expressions tp\\dpu/dxp\\2 which do not depend on derivatives of the initial data, but only on the L2-norm of w(-,0) = /. Now consider the case where the initial data w(-,0) € LI are not necessarily smooth. We construct the corresponding generalized solution as the limit of regular solutions w(j)(x, f) with initial data For 0 < r < t < T we find that with a constant C independent of j. Similar estimates also hold for all time derivatives of u^. Using the Sobolev inequality formulated in Lemma 3.2.5, one finds that This implies smoothness of the limit function u, and we have proven Theorem 3.2.11. Consider (3.2.12) with initial data in LI. The correspond- ing generalized solution is a function belonging to C°° for t > 0. The same result holds for general parabolic systems. 6 Iterative Methods for Solving Linear Systems the set of (complex) eigenvalues using a fcth-degree polynomial with value 1 at the origin? When A is nonnormal, however, the bound (1.7) may not be sharp. If the condition number of V is huge, the right-hand side of (1.7) may also be large, but this does not necessarily imply that GMRES converges poorly. It may simply mean that the bound (1.7) is a large overestimate of the actual GMRES residual norm. An interesting open question is to determine when an ill-conditioned eigenvector matrix implies poor convergence for GMRES and when it simply means that the bound (1.7) is a large overestimate. If eigenvalues are not the key, then one would like to be able to describe the behavior of GMRES in terms of some other characteristic properties of the matrix A. Some ideas along these lines are discussed in section 3.2. Finally, since the full GMRES algorithm may be impractical if a fairly large number of iterations are required, one would like to have theorems relating the convergence of some nonoptimal methods (that do not require extra work and storage) to that of GMRES. Unfortunately, no fully satisfactory theorems of this kind are currently available, and this important open problem is discussed in Chapter 6. 1.1.3. Preconditioners. The tools used in the derivation of precondition- ers are much more diverse than those applied to the study of iteration methods. There are some general results concerning comparison of preconditioners and optimality of preconditioners of certain forms (e.g., block-diagonal), and these are described in Chapter 10. Many of the most successful preconditioners, however, have been derived for special problem classes, where the origin of the problem suggests a particular type of preconditioner. Multigrid and domain decomposition methods are examples of this type of preconditioner and are dis- cussed in Chapter 12. Still other preconditioners are designed for very specific physical problems, such as the transport equation. Since one cannot assume familiarity with every scientific application, a complete survey is impossible. Chapter 9 contains two example problems, but the problem of generalizing application-specific preconditioners to a broader setting remains an area of active research. 1.2. Notation. We assume complex matrices and vectors throughout this book. The results for real problems are almost always the same, and we point out any differences that might be encountered. The symbol i is used for \/—I, and a superscript H denotes the Hermitian transpose (A^ = Aji, where the overbar denotes the complex conjugate). The symbol || • || will always denote the 2-norm for vectors and the induced spectral norm for matrices. An arbitrary norm will be denoted HI-HI- The linear system (or sometimes the preconditioned linear system) under consideration is denoted Ax = 6, where A is an n-by-n nonsingular matrix and b is a given n-vector. If x^ is an approximate solution then the residual b — Ax^ Linear Variable-Coefficient Cauchy Problems in ID 101 Proof. We have Integration by parts gives us and therefore This shows the desired estimate for ||u||2. The function w = vx satisfies We have already a bound for v, and therefore we can consider Bxv + Fx = F as a new forcing function. Thus (3.3.3) is of the same form as (3.3.2), and we obtain the corresponding estimate. Repeating this process, we obtain the estimates for the higher derivatives, and the lemma is proved. Bounds for time derivatives and mixed derivatives of v — v( can be shown as previously by using the differential equation. We now prove that (3.3.1) has a unique solution and start with uniqueness. If u and v are two smooth solutions then their difference w = u — v satisfies the homogeneous system (3.3.1). By the estimate of the above lemma (with e = 0) we have hence w = 0, and uniqueness is shown. To prove existence, we consider a sequence of solutions vf for c — > 0. The functions vf are uniformly smooth, and by the general argument given in Appendix 4 we can select a subsequence which converges along with all its derivatives. The limit is clearly a solution of (3.3.1), which also satisfies the estimates of Lemma 3.3.1. To summarize, Theorem 3.3.2. The symmetric hyperbolic system (3.3.1) has a unique solu- tion u = u(x, t). The solution is C°° -smooth and satisfies the estimates of Lemma 3.3.1. The spatially periodic Cauchy problem for a symmetric hyperbolic system is well-posed. We shall generalize the result to systems (3.3.1) for which the matrix A is not necessarily Hermitian. For equations with constant coefficients, we defined 102 Initial-Boundary Value Problems and the Navier-Stokes Equations strong hyperbolicity by the condition that the eigenvalues of A are real and that there is a complete set of eigenvectors. According to Lemma 2.4.2, a matrix A has this property if and only if there is a positive definite Hermitian matrix H with For a variable-coefficient equation we can "almost" use this condition pointwise, and obtain a well-posed problem. More precisely, we make the Definition. The variable-coefficient system (3.3.1) is called strongly hyper- bolic if there exists a smooth 1-periodic "symmetrizer", i.e., a smooth 1-periodic positive definite matrix function H(x, t) such that (3.3.4) is valid for all x, t. For a strongly hyperbolic system we can proceed in the same way as for a symmetric one, if we replace the I/2-inner product by The estimates of the solution and its derivatives follow; see also Section 3.2.5. One obtains Theorem 3.3.3. The results of Theorem 3.3.2 hold if the system is strongly hyperbolic, i.e., if we can find a smooth I-periodic symmetrizer. Note that if the problem is only "pointwise" strongly hyperbolic then a matrix H(x, t) with (3.3.4) exists at each point (x, t) according to Lemma 2.4.2. The only extra assumption we have made is that //(x, t) depends smoothly on (x, t). The existen of such a smooth symmetrizer is guaranteed if, for example, for all (x, t) the n x n matrix A(x, t) has n real distinct eigenvalues; in this case the system (3.3.1) is called strictly hyperbolic. Therefore, as in the case of constant coefficients, strict hyperbolicity implies strong hyperbolicity. 33.2. Properties of Scalar Equations The simplest case. Let a be a real constant. The simplest hyperbolic differen- tial equation is given by the scalar equation Its solution for 1-periodic smooth initial data Linear Variable-Coefficient Cauchy Problems in ID 103 can be written down explicitly: Thus the solution is constant along the so-called characteristic lines a: = (The important concept of a characteristic is defined below for more general equations.) This shows that there is no smoothing effect; the solution is as smooth as the initial data. For generalized solutions, whose initial data have discontinuities, this fact is particularly important. Consider, for example, the periodic function which jumps at all integers and half-integers. We approximate / by functions ft € C°° by "rounding the corners." For every fixed e > 0 our problem is solved by uc(x, t) = fc(x - at). Therefore the generalized solution is u(x, t) = f(x-at); in particular, the discontinuities move along the characteristics through i and i + 5, i € Z. The case a = a(x, t). Suppose a = a(x, t) is a smooth function, 1 -periodic in x. A parametrized line is called a characteristic for the equation Since a(x, t) is bounded in every finite time interval, it is clear that a unique characteristic exists through any given point (x,t). If u solves (3.3.8) and (x(t), t} is a characteristic, then Thus the solution u of 104 Initial-Boundary Value Problems and the Navier-Stokes Equations carries the value /(XQ) along the characteristic (x(t), t) starting at (x(0), 0) = Cro,0). If the initial function /(x) has a jump discontinuity at x = XQ, then the generalized solution w(x, t) will jump when crossing the characteristic In other words, discontinuities move along the characteristics. The inhomogeneous equation. We can also solve inhomogeneous problems Characteristics are defined as for the case F = 0, i.e., by condition (3.3.9). If (x(t), t] is a characteristic and u(x, t) solves (3.3.10), then and therefore Suppose F is only piecewise smooth with jump discontinuities along lines which are nowhere tangent to a characteristic. In this case the integral in (3.3.1 1) mol- lifies the jumps of F. The generalized solution u is continuous if /(x) is continuous. If / has jumps, these travel along the characteristics; the disconti- nuities of F do not introduce new jumps in the function u, but - in general - in the derivatives of u. The general linear case. For equations the characteristics are again defined by condition (3.3.9). If u solves the above equation, then along each characteristic. Thus we can obtain u(x(t),t) by solving a linear ordinary differential equation. 333. Properties of Hyperbolic Systems The case A = const, B — 0. Consider a strongly hyperbolic system Linear Variable-Coefficient Cauchy Problems in ID 105 with a constant coefficient matrix A. There is a nonsingular transformation 5 such that If we introduce the new variables v = 5~lu, then the equation (3.3.12) trans- forms to Thus we obtain n uncoupled scalar equations and the considerations of the previous section apply. The wave equation. An important example is obtained from the wave equation with 1 -periodic initial data We introduce a new variable v by The wave equation then gives us hence integration in t yields For v we have the initial condition and if we choose h(x) = /IQ = J0 5(0 <^» tnen is a 1 -periodic initial condition. To summarize, if y solves the wave equation with initial conditions (3.3.13), then 106 Initial-Boundary Value Problems and the Navier-Stokes Equations solves the hyperbolic system Conversely, if u solves the above system, then the first component y = u\ solves the original problem. The eigenvalues of A are AI = +1, A2 = -1. If then the transformation w = S~lu leads to and we can obtain a solution in closed form. As a result, one finds that Strongly hyperbolic systems. We will explain here how to obtain the solution of a general strongly hyperbolic system by solving a sequence of scalar equations and going to the limit. The scalar equations themselves can be solved by the method of characteristics, as discussed above. In this way one can study in detail how discontinuities propagate and one can prove that information travels at a finite speed. Consider a general linear system and assume that there is a smooth transformation S = 5(x, t) such that Introducing new variables v = S~lu, we obtain and therefore Linear Variable-Coefficient Cauchy Problems in 1D 107 Solution by iteration. System (3.3.14) suggests the iteration starting with v°(x,t) = 0. For every fc = 0,1,2,... the initial value problem (3.3.15) represents n scalar equations which one can solve by the method of characteristics. Accordingly, one makes the Definition. The lines with are called characteristics for the system (3.3.14) (and also for the given system written in u-variables). Through any point (x,0 there are n characteristics (xj(t),t) with These are distinct if the system is strictly hyperbolic, i.e., but otherwise two characteristics can be identical. To study the iteration (3.3.15), we fix a point (x,t). The characteristics through this point intersect the x-axis in #i(0). . . . , xn(0). FIGURE 3.3.1. Characteristics through a given point (x,7). 108 Initial-Boundary Value Problems and the Navier-Stokes Equations Note that the characteristics do not depend on the iteration index k. We obtain from (3.3.15) Integration yields Let us show that the sequence vk converges to the solution v. We will use the following auxiliary result, which we refer to as Picard's Lemma. Lemma 3.3.4. Let 77*(<), k = 0, 1,..., denote a sequence of nonnegative continuous functions which satisfy the inequalities with nonnegative constants a, 6. Then for 0 < t < T and fc = 0, I,... In particular, the sequence rjk(t), 0 < t < T, is uniformly bounded. Ifa = 0, then the sequence converges uniformly to zero. Proof. For k = 0 the estimate is true. Assume it is valid up to the index k. Then and the lemma is proved. We use this to show the following convergence result. Lemma 3.3.5. Suppose v solves (3.3.14), and the sequence vk is defined by (3.3.15). Then Linear Variable-Coefficient Cauchy Problems in ID 109 for any fixed T. Here it is assumed that all data are smooth and 1 -periodic in x. Proof. The solution v satisfies Therefore we obtain, for the difference wk = vk - v, thus If we set then, for 0 < t < T, An application of Picard's Lemma (with a = 0) yields This proves the convergence. Propagation of discontinuities. We can use (3.3.17) to deduce properties for the sequence vk = vk(x,t) and then go to the limit as fc —* oo. Consider, for example, the discontinuous initial data where / is given in (3.3.7). For simplicity assume F = H = 0 and assume the system is strictly hyperbolic, i.e., (3.3.16) holds. The function v[(x, t) represents the pulses whose edges are moving along the characteristics (x\(t),t) starting at (xi(0),0) = (i,0), (rri(O).O) = (i + |,0), i € Z. All other components 110 Initial-Boundary Value Problems and the Navier-Stokes Equations By our considerations for scalar equations, it follows that v\(x, t) has jumps when crossing the characteristics (x](t), t) mentioned above, whereas vj(x, 0, j = 2,..., n, is continuous. The reason for the continuity of the latter functions is that the characteristics (xj(t), t}, j = 2, ...,w, are nowhere tangent to the discontinuity lines (x\(t), t) of the forcing. This is the qualitative behavior for all iterates vk, and therefore of the solution itself. There are no difficulties in discussing the behavior for more general piecewise smooth initial data because we can split these up into a smooth function plus simple jumps. A discontinuity of the initial data at x = XQ propagates along the characteristics which start at The finite speed of propagation. Another important property of hyperbolic differential equations is the so-called "finite speed of propagation" which we explain in the next theorem. Consider again the system (3.3.14) under assump- tion (3.3.16). Theorem 3.3.6. For a given point (x, t) let (x\(t), t)...., (xn(t), t) denote the characteristics through (x,t); see Figure 3.3.1. Let ft denote the region bounded by the two characteristics and the piece of the initial line Then the restriction of the solution v(x,t) to the region ft only depends on the restriction of H to ft and of g to the interval x\($) < x < arn(0). In other words, we can change H and g outside these regions without affecting v inside ft. Proof. Consider any point (x, t) € ft. Then the characteristics through (x, t) also belong to ft. Now consider the iteration (3.3.15). It is clear that the restriction of t> ' to ft depends only on the corresponding restrictions of H and g. By induction we obtain that all functions vk have this property, and therefore it is valid for the limit also. This proves the theorem. In Section 6.2 we shall prove a similar theorem for hyperbolic systems in several space dimensions using a difference approximation. Linear Variable-Coefficient Cauchy Problems in ID 111 3.4. Mixed Hyperbolic-Parabolic Systems Consider a system of the following block form It is assumed that all coefficients and data are C°°-smooth and 1-periodic in x. Also, let If B\2 = C\2 = 0, BI\ = C2\ = 0, then we have a strongly parabolic u-equation and an uncoupled symmetric hyperbolic v-equation. Assuming (3.4.2), we want to show that the spatially periodic Cauchy problem for the mixed system (3.4.1) with coupling terms is well-posed. First consider the parabolic systems According to Theorem 3.2.7, there is a unique solution w = u>e of the Cauchy problem for e > 0. We shall show that the solutions wc and their derivatives can be estimated independently of c > 0; we start with the basic energy estimate. Lemma 3.4.1. For any T > 0 there is a constant K(T), independent ofe > 0, with Proof. We write w = we. The norms | \x below refer to the interval 0 < t < T. We have where 112 Initial-Boundary Value Problems and the Navier-Stokes Equations The term I can be estimated as explained in Section 3.1.1 where we treated parabolic systems: The term II can be treated as explained in Section 3.3.1 for the symmetric hyperbolic case: Integration by parts gives us for III: Finally, Summarizing, we have shown that where c4 does not depend on e. This proves the lemma. It should be remarked that the estimate of Lemma 3.4.1 is also valid for e = 0 if we assume the existence of a smooth solution for this case. Thus uniqueness for (3.4.1) is shown. To obtain existence, we estimate all derivatives of w( independently of £. Differentiation of the system (3.4.3) with respect to x yields an equation for with the same structure. Observing that we already have estimates for |jw< ||, we can bound ||u^|| in the same way as above. Estimates for higher space derivatives are obtained by repeated differentiation. This proves Lemma 3.4.2. For any T > 0 and any p = 1,2...., there is a constant K(p, T), independent of e > 0, with Linear Variable-Coefficient Cauchy Problems in ID 113 Here K(p, T) depends only on the maximum norm of the coefficients and their derivatives of order < p. Time derivatives and mixed derivatives of w* can always be expressed by space derivatives using the differential equation. Thus we have estimates for all derivatives. Now consider a sequence e —» 0. According to Appendix 4, we can select a uniformly smooth subsequence of wf which converges along with all its derivatives. The limit solves (3.4.1) and obeys the estimate (3.4.4). This proves Theorem 3.4.3. The spatially periodic Cauchy problem for the mixed system (3.4.1) is well-posed if (3.4.2) is true. Thus far, the underlying uncoupled equations were assumed to be strongly parabolic or symmetric hyperbolic. We can generalize our result to the case where instead of (3.4.2) it is assumed that there are smooth symmetrizers: To obtain the estimates for the more general case, we use the scalar product (w. W)H = (u. H\u)+(v, H2v) instead of the I/2-scalar product (w. u>) = (u, u)+ 3.5. The Linearized Navier-Stokes Equations in One Space Dimension Suppose the variables u, v, w. p, p do not depend on y, z. Then the Navier- Stokes equations read 114 Initial-Boundary Value Problems and the Navier-Stokes Equations This system ncouples into a system for (u, p), and two scalar equations The equations for v and w are linear if u, p are considered as known functions. Clearly, these equations are parabolic in the viscous case (n > 0), and strongly hyperbolic in the inviscid case (p- — 0). We now linearize the system (3.5.1) at a smooth flow U - U(x, t), R = R(x, t), P = P(x, t) with P = r(R). Substitute u = U + w', p = R + p', p = P + p' into the equations (3.5.1) and neglect terms quadratic in the corrections. The equation of state gives us thus p' = Kp' to first order. Therefore we can eliminate p' and obtain for w', p1 a linear system Here C and G are determined by £7, 72, P, and F. For /i > 0, p.' > 0, the system is of mixed hyperbolic-parabolic type. In the inviscid case /i = //' = 0 we obtain, upon neglecting the zero-order term and the forcing function, Now suppose that Then the eigenvalues A 1,2 of A are real and distinct: Linear Variable-Coefficient Cauchy Problems in ID 115 The system (3.5.2) is strictly hyperbolic and has the symmetrizer This symmetrization simply corresponds to the introduction of a scaled density in (3.5.2); if then the system for q becomes One easily confirms directly that the new system matrix is symmetric, The positivity assumption (3.5.3), which is crucial for hyperbolicity, is physi- cally reasonable because one expects increasing pressure with increasing density. Furthermore, the interpretation of the eigenvalues A 1,2 = U ±a as characteristic speeds (see Section 3.3) makes it plausible that is the sound speed corresponding to the base flow with density R. It is the speed of propagation of small disturbances in the base flow. 3.6. The Linearized KdV and the Schrodinger Equations We have shown the well-posedness of the spatially periodic Cauchy problem for linear variable-coefficient equations in case of a parabolic, a strongly hyperbolic or a mixed system. Basically, integration by parts applied to the differential equation and the differentiated differential equation gave us estimates of any possible solution and its derivatives. For the parabolic case we then used a dif- ference scheme to obtain existence; the strongly hyperbolic and mixed systems were considered as limits of parabolic ones. These techniques can be applied to other equations also, as will be demonstrated here for the linearized KdV and the Schrodinger equations. 116 Initial-Boundary Value Problems and the Navier-Stokes Equations 3.6.1. The Linearized Korteweg-de Vries Equation The nonlinear equation is known as the KdV equation. We linearize about a smooth function U = U(x, t). Upon neglecting quadratic terms in u, the ansatz w = U + u leads to We assume that all functions are real, C°° -smooth, and 1-periodic in x. If u solves (3.6.1) then Here and thus Also, If T > 0 is fixed and | («, refers to the interval 0 < t < T, then This gives the basic energy estimate Lemma 3.6.1. For any T > 0 there exists K(T) > 0 such that any solution o/(3.6.1) satisfies the estimate The above lemma implies uniqueness of a solution for given initial data Linear Variable-Coefficient Cauchy Problems in ID 117 To estimate v = ux, we differentiate the equation (3.6.1) and obtain Thus we can estimate v = ux following the same arguments as above. With repeated differentiation one obtains Lemma 3.6.2. For any T > 0 and any p — 1,2,... there exists K(p, T) with The constant K(p, T) depends only on bounds for a and b and their derivatives up to order p. Estimates of time derivatives and mixed derivatives of any solution follow from the differential equation. To obtain existence of a solution of (3.6.1), (3.6.2), we can use the difference scheme and mimic the previous estimates. The Fourier interpolants wh(x, t) of vv = v*(t) w.r.t. x are uniformly smooth. Exactly as in Section 3.2.4, one obtains for a sequence h = hj•, —>• 0. The limit u solves the Cauchy problem. Theorem 3.6.3. The spatially periodic Cauchy problem for the linearized KdV equation is well-posed. The solution satisfies the estimates of Lemma 3.6.2. 3.6.2. A Schrodinger Equation Consider the spatially periodic Cauchy problem The equation is neither parabolic nor hyperbolic. For e > 0, the parabolic problem has a unique solution u = u*. For u = uc it holds that 118 Initial-Boundary Value Problems and the Navier-Stokes Equations and thus ||w(-,t)|| < ||w(-,0)||. This estimate is valid for € = 0 also; thus uniqueness of a solution of (3.6.3), (3.6.4) follows. We now show how to bound derivatives of u = we. We let v = ut. The differential equation (3.6.5) yields and therefore (see Lemma A.3.2 in Appendix 3) Thus we have the estimates where C = C(T). Differentiation of (3.6.5) w.r.t. t gives us By (3.6.6) we find Thus we can bound independently of e in any finite time interval 0 < t < T. Higher derivatives can be estimated in the same way by repeated differentiation. The functions uf are uniformly smooth, and by the same arguments as before, we obtain a solution of the Schrodinger equation for e — > 0. Notes on Chapter 3 Parallel to the theory of differential equations, the theory of difference approxi- mations was developed; see Osher (1972), Michelson (1983, 1987), Richtmeyer Linear Variable-Coefficient Cauchy Problems in ID 119 and Morton (1967), Kreiss and Oliger (1973), Kreiss (1968), Gustafsson, Kreiss, and Sundstrom (1972). It is not new to prove existence of solutions by difference approximations; see, for example, Friedrichs (1954). Instead of using /^-estimates, parabolic differential equations are often treated via the construction of fundamental solutions and maximum norm estimates. The classical books are Friedman (1964) and Eidelman (1964). We develop the L.2 -theory because we are interested in coupled hyperbolic-parabolic systems. For hyperbolic systems in more than one space dimension, estimates in the maximum norm can, in general, only be obtained under "loss of derivatives"; see Brenner (1966). A more detailed discussion of the method of characteristics can be found in many books, for example in Petrovskii (1954), Courant and Hilbert (1962), Carrier and Pearson (1976). This page intentionally left blank 4 A Nonlinear Example: Burgers' Equation The Cauchy problem for the viscous and the inviscid Burgers' equation will be discussed in detail in this chapter. The techniques we apply can mostly be generalized in a straightforward way to more complicated nonlinear parabolic or hyperbolic equations, to systems in one or more space dimensions. These generalizations will be carried out in Chapters 5 and 6. In other words, we will use Burgers' equation as a simple example to illustrate a number of general techniques for the treatment of nonlinear evolutionary systems. Intentionally, we do not apply the celebrated Cole-Hopf transformation, which reduces Burgers' equation to the heat equation. (Though the Cole-Hopf transformation does have interesting generalizations also, these seem to be too limited, at present, to discuss the Navier-Stokes system.) We emphasize in this chapter: 1. local (in time) existence of solutions via a linear iteration; 2. local existence together with global (in time) a priori estimates leads to global existence; 3. smoothing properties for the parabolic, i.e., viscous case; 4. breakdown of smooth solutions for the hyperbolic, i.e., inviscid case in finite time. Concerning these points, our discussion of Burgers' equation is representative for parabolic and hyperbolic systems. 727 122 Initial-Boundary Value Problems and the Navier-Stokes Equations With regard to other aspects, namely global a priori estimates, the maximum principle, and our discussion of shocks, the techniques for Burgers' equation do not readily generalize. The difficulties involved will become apparent in Chapters 5 and 6. 4.1. Burgers' Equation: A Priori Estimates and Local Existence In this section we start our discussion of Burgers' equation with periodic boundary conditions. We assume that all functions are real-valued. Burgers' equation is of interest to us, since it shares its mathematical structure ut = quadratic first order term + diffusion term with the Navier-Stokes system. After showing the uniqueness of a classical solution, we prove a priori estimates in some time interval 0 < t < T. To obtain these estimates, we intentionally do not make use of the maximum principle. Therefore, the proof generalizes to systems for which maximum principles are generally not available. Also, we show a priori estimates which are independent of e. Thus we can use the same arguments later to discuss the inviscid case (e = 0). To show existence of a solution in some time interval, one can proceed in different ways. A first possibility is to use a difference scheme as in Chapter 3, and to show estimates of its solutions vh independent of the step-size h. We will write down a suitable difference approximation, but leave the necessary estimates to the reader. These estimates can be obtained along the same lines as the a priori bounds for the solution u. In the same way as in the linear case, one can Fourier interpolate the functions vh w.r.t. x and obtain uniformly smooth functions wh(x, t); these converge to a solution u(x, t) as h —>• 0. In the text we shall use another approach to show existence, namely the iteration where u°(x, t) = f(x). The sequence un will be shown to converge, and its limit will solve (4.1.1). Again, we will not apply the maximum principle, and thus the local existence proof generalizes to systems. A Nonlinear Example: Burgers' Equation 123 4.1.1. Uniqueness A classical solution of (4.1.1) is a function u £ Cl(t) n C2(x) which satisfies (4.1.1) pointwise. We first show Lemma 4.1.1. The \-periodic Cauchy problem (4.1.1) has at most one clas- sical solution. Proof. Let u and v be solutions. Their difference w = u — v satisfies From one finds that and therefore The initial condition w(x,Q) = 0 implies w = 0. 4.1.2. A Priori Estimates Let u denote a C°°-solution of (4.1.1) defined for 0 < t < T. First note that The first term on the right side is zero since Thus we find -^(u, u) < 0 and obtain the bound 124 Initial-Boundary Value Problems and the Navier-Stokes Equations To show estimates for space derivatives, we use the notation Uj = The function HI satisfies and therefore From Lemma 3.2.5, Furthermore, the estimate follows from Fourier expansion. Hence we have (For c > 0, we could now derive a bound of ||ui(-,0||; the timeinterval where the bound holds would depend on c, however. Therefore we proceed differently and show first an estimate for 112-} The function u2 satisfies and thus Since we obtain A Nonlinear Example: Burgers' Equation 125 Using the abbreviations we have shown the differential inequality To estimate y(t), we need a nonlinear extension of Lemma 3.1.1. Lemma 4.1.2. Let </> € C'[0, oo), and let y(t) and yo(£) denote nonnegative C1 -functions defined for 0 < t < T. If Proof. For any two arguments y, yo of 0 we have We set and obtain If we introduce then z'(t) < 0 and z(0) = 7/(0) < 0. Therefore z(t) < 0, and thus 7?(0 < 0. This proves the lemma. To apply this result, we note that the solution y0(t) of 126 Initial-Boundary Value Problems and the Navier-Stokes Equations is increasing, and if yo(0) > 0, then there is a time T^ > 0 with Therefore, the previous lemma gives us a bound for ||w2(sO|| in any interval 0 < t < T < r^. By (4.1.4) and (4.1.5), ||uj(-,0|| and ||u(-,t)|| are also bounded for 0 < t < T < Too. The proof clearly shows that the time T^ and the bound do not depend on e > 0. The value e = 0 is allowed. We summarize: Lemma 4.U. Consider (4.1.1) with e > 0. There exists a time T > 0 and a constant KI, both depending only on ||/||//2, with the following property: If a solution u(x, t) is defined for 0 < t < T then We proceed by showing estimates for all space derivatives Uj = d^u/dx* ofu. Interestingly, the time interval for which the estimates are valid does not depend on j nor on c > 0. Lemma 4.1.4. Suppose u is a Cx-solution o/(4.1.1) defined for 0 < t < T, T being fixed as in the previous lemma. For j = 2, 3, ... there is a constant KJ, depending only on ||/||#j, with Proof. We use induction on j. The case j = 2 has been treated in the previous lemma; thus let j > 3. The identity implies that Here The terms (ttj, uvUj+\-v) can be estimated as follows: For v = 1,..., j - 2, A Nonlinear Example: Burgers1 Equation 127 thus If Cj is a suitable numerical constant, we find that Thus the induction step can be completed, and the lemma is proved. Since we have bounded all space derivatives of u in 0 < t < T, we can use the differential equation ut = uux + euxx to bound all time derivatives and mixed derivatives. Each term can be written as a sum of products of space derivatives, and hence it can be bounded. The bounds are uniform for 0 < e < e0, eQ > 0 being fixed. 4.1.3. Existence of Solutions via Difference Approximations As in Section 3.2, we can discreti in space and approximate (4.1.1) by a system of ordinary differential equations: Here DQ = 1/2 (D+ + D_), and we have approximated uux = \/3(uux -f (u2)x) by 1/3 (v^Dov,, + D0vl). This approximation has the technical advan- tage that 128 Initial-Boundary Value Problems and the Navier-Stokes Equations and therefore we obtain the basic energy estimate from Bounds for \\D+ dqv/dtq\\h independent of h can be obtained along the same lines as the a priori estimates for u in the previous section. Thus we can proceed as in Section 3.2: Fourier interpolate v = vh with respect to x and send h — > 0. We obtain existence in some interval 0 < t < T, T depending only on 4.1.4. Existence via Iteration To prove existence via the iteration (4.1.2), we estimate the functions un inde- pendently of n. Again, our estimates will not take advantage of the diffusion term ewlx, and so will not depend on t > 0. All of the arguments are without reference to the maximum principle, thus generalize from scalar equations to systems. To show uniform smoothness of the sequence un, we start with the following analogue of Lemma 4.1.3. Lemma 4.1.5. For a suitable time T\ = T\ (H/H/^) > 0 it holds that Proof. We use induction on n; the case n = 0 is trivial. As abbreviations, let w = un, v = un+l, thus Note that For the derivative vx = v\ it holds that A Nonlinear Example: Burgers' Equation 129 and therefore Finally, implies that Adding the three inequalities, we obtain from the induction assumption that and therefore Thus we can complete the induction step if T\ > 0 is so small that This proves the lemma. We show now that all space derivatives of un can be estimated in the same interval 0< t < T\. Lemma 4.1.6. For each j = 2, 3, . . . there exists Kj with The constant Kj depends on \\f\\Hj, but is independent ofn and e > 0. Proof. As previously, let w = un, v = un+1. We can assume j > 3 and 130 From it follows that HereInitial-Boundary Value Problems and the Navier-Stokes Equations For v = 1,..., j - 2 it holds that (Note that ||v/|| < ||vj|| for / < j, by Fourier expansion and Parseval's relation.) For v = j - 1, j it holds that (see (4.1.5)) For i/ = 0, integration by parts gives us Thus, there is a numerical constant c = c, with By Gronwall's Lemma 3.1.1, Now the desired estimate follows from Picard's Lemma 3.3.4. Summarizing, we have estimated the 1/2 -norms of all space derivatives of the sequence un in 0 < t < TI, T\ = T, (\\f\\H2)- By the Sobolev inequality stated in Lemma 3.2.5, the functions diun /dx3 are also bounded in maximum norm. Since we can always use the differential equation (4.1.2) to replace time derivatives by space derivatives, it follows that A Nonlinear Example: Burgers' Equation 131 Here C(p, q) is independent of n and 0 < e < CQ, where CQ is arbitrary but fixed. We finally show Theorem 4.1.7. Let T\ =T\ (\\f\\Hi) be determined as in Lemma 4.1.5. For any e > 0 Burgers' equation has a C°° -solution u(x, t) defined for 0 < t < T\. Proof. Consider the sequence un defined by the iteration (4.1.2) and abbreviate Then implies that Therefore (by Gronwall's Lemma 3.1.1), where K is independent of n. Thus, by Lemma 3.3.4, the sequence wn(-, t) converges to a function w(-, t) € L2- The smoothness estimates (4.1.6) imply that u € C°°, and the convergence un — > u holds pointwise and also holds for all derivatives. (See Appendix 4.) Then the equation (4.1.2) implies that u solves Burgers' equation. 4.2. Global Existence for the Viscous Burgers' Equation In this ction we use the Cauchy problem for Burgers' equation to illustrate an important principle. Loosely formulated it says: Local existence together with global a priori estimates leads to global existence. We will formulate the a priori estimate in the following theorem. Throughout this section, e > 0 is arbitrary but fixed. 132 Initial-Boundary Value Problems and the Navier-Stokes Equations Theorem 4.2.1. Let f = f(x) denote \-periodic C°° initial data, and let u € C°° denote a solution of (4.1.1) defined for 0 < t < T. There is a constant K, depending on \\f\\H-i (and e) but independent ofT, with (The smoothness assumption f € C°° is unnecessarily restrictive and will be relaxed later.) Assuming this result is proven, we can show all-time existence for the problem (4.1.1) as follows: According to the Local Existence Theorem 4.1.7, there is a time T, > 0 with a C°° -solution u defined for 0 < t < T\. By (4.2.1) we have We can use the function x — > u(x, T\ ) as new initial data and apply the Local Existence Theorem 4. 1 .7 again. The solution starting with the initial data u(-,T\) exists in a time interval 0 < t < T2, T2 depending only on K (and e). Clearly, putting the two solutions together, we have a C°° -solution u of (4.1.1) defined for 0 < t < T\ + T2. The important point is that the a priori estimate (4.2.1) implies that with the same constant K as before. Thus, using the Local Existence Theorem, we can again extend the solution for a time interval of length T2, etc. This shows Theorem 4.2.2. Ut f € (7°° and c > 0, The \-pcriodic Cauchy problem (4.1.1) Has a unique solution u € C'00 defined for 0 < t < oo. It remainc to prove the 3 priori «timatt, ThS Pffrftf is bitted on the maximum principle. Lemma 4.2 3. Let u 6 C°° solve (4.1.1) in 0 < t < T. Then \u(-,t)\x < Proof. Let a > 0 be a constant and set v(x, t) = e~atu(x, t). The function v satisfies A Nonlinear Example: Burgers' Equation 133 FIGURE 4.2.1. Restarting process. Fix T > 0 and assume that v(x, t) attains its maximum over in contradiction to (4.2.2). Thus to = 0, i.e., a positive maximum can only be attained at t = 0. Similarly, a negative minimum can only occur at t = 0. Therefore, and if we let a — > 0, then the corresponding estimate for u follows. Proof of Theorem 4.2.1. The basic energy inequality implies that For the function u\ — ux, the equation gives us 134 Initial-Boundary Value Problems and the Navier-Stokes Equations Therefore, Thus, by (4.2.3), we obtain an estimate for independent of T. For w2 the equation implies that Therefore, (4.2.4) gives us bounds for independent of T. This finishes the proof of Theorem 4.2.1. Remark. To obtain all-time existence, we only needed an a priori estimate for INI//*- The reason is that the time interval of the Local Existence Theorem 4.1.7 depends on \\f\\H2 only, and does not depend on bounds of higher derivatives of /. Nevertheless, one can ask for a priori estimates of higher derivatives ||u,||, j > 3, in terms of ||/||w>. These can indeed be derived. Just note that and thus A Nonlinear Example: Burgers1 Equation 135 We can make an induction argume sing bounds for 4.3. Generalized Solutions for Burgers' Equation and Smoothing* We have shown that the 1 -periodic Cauchy problem has a unique C00 -solution u(x, t) existing for all times t > 0. In this section we first define a generalized solution for any initial function For example, / could be piecewise continuous with finitely many jumps. After- wards, we show that the generalized solution is always C°° -smooth for t > 0 4 J.I. Construction of Generalized Solutions Let /, g denote 1-periodic C"*5-functions. We estimate the difference of the solutions u and v of in terms of / - g. It is convenient to work with the LI-norm The difference w = u — v satisfies the linear equation *This section might be omitted on the first reading.and 136 The key result isInitial-Boundary Value Problems and the Navier-Stokes Equations Lemma 4 J.I. Let a = a(x, t), w = w(x, t) denote I-periodic C°°-functions with Proof. 1) To obtain smooth approximations to the sign function, we choose a fixed function Sgn € Cl with FIGURE 4.3.1. Approximation for the sign function. Then we set For 6 — » 0 we have that and thus A Nonlinear Example: Burgers' Equation 137 Therefore, we have expressed \w\\ approximately as an integral of a smooth function. This is the main technical point of the proof. 2) For 8 > 0 we find that where Therefore Here 6 = wt — awx is independent of 6. Integration from 0 to t yields and we send 6 —> 0. The left-hand side converges to The integrand R(,(T) of the right-hand side is If w(x, r) ^ 0 then sgn'^ (w(x, r)) -^ 0 as 6 -> 0. Thus the integrand of (4.3.2) converges pointwise to zero as 6 —* 0. Also, sgn^(w)w = Sgn' (w/8) (w/6) is bounded independently of 6 and x, because the (fixed) function y —* Sgn'(y) y is bounded. Therefore, Lebesgue's Dominated-Convergence Theorem yields 0 as 6 —» 0 for each r. The convergence follows by another application of Lebesgue's theorem and finishes the proof of the lemma. 138 Initial-Boundary Value Problems and the Navier-Stokes Equations Remark. A different proof could be given, which uses a monotonic difference scheme. This proof would not require Lebesgue's theorem. The estimate shown in Lemma 4.3.1 allows one to obtain generalized solutions of for nonsmooth initial functions / by the usual approximation process: Let and approximate / by 1 -periodic C°° -functions f(t/): If u("\x, t) denotes the solution for initial data u(>/)(x, 0) = f(v)(x) we have Thus, u(l/)(-, t) — » u(-, t) with respect to | • |j; the limit function u is independent of the specific choice of the approximating sequence /(</). By definition, u is the generalized solution of (4.3.3). Remark. Let us note another application of Lemma 4.3.1. Suppose u = uf solves (4.3.1), and set w = ux. Differentiation of (4.3.1) yields Hence, we can apply Lemma 4.3. 1 and obtain that This c-independent bound of the LI -norm of ux = utx is the key estimate to obtain a weak solution of the inviscid (e = 0) equation by a compactness argument. We will not present the details, however. 4.3.2. Smoothing for Burgers' Equation We want to show that the generalized solution H(X, t) defined above is a C°°- function for t > 0 if / € LOO- This is remarkable and not self-evident from linear theory. To show smoothing for parabolic linear equations, we used the smoothness of coefficients. In the present case we could think of u as solution of the linear equation A Nonlinear Example: Burgers' Equation 139 with a(x, t) = u(x, t), but t coefficient a(x, t) is rough initially. To obtain smoothness of the generalized solutions, we first show estimates of derivatives of u in terms of the maximum norm of the initial data u(-,0) = / where / 6 Cx. We remind the reader of the maximum principle and the notation Uj = &>u/dxJ. Lemma 4.3.2. For any interval 0 < t < T and any j = 0, 1, 2,... there exists a constant C = C(j, c, \f\x, T) with The constant C does not depend on bounds for derivatives of f. Proof. Recall the basic energy inequality which implies that For u\ we have that Integration yields By (4.3.5) it follows that Similarly, (see the computations in the proof of Theorem 4.2.1) 140 Initial-Boundary Value Problems and the Navier-Stokcs Equations and therefore, Using (4.3.6), we obtain bounds for Clearly, this process can be continued and the lemma follows. We have established bounds for the L2-norms of all derivatives Uj = &ufdxi in terms of l/l^. Applying the Sobolev inequality of Lemma 3.2.5, we find bounds for Uj in maximum norm, and - using the differential equation to express time derivatives by space derivatives - we obtain that The constant does not depend on derivatives of /, but only on \f\x. Thus'far we have assumed that / e C°°. Now let and approximate / by a sequence fM € C°°, f("\x) = f(t/\x + 1), All derivatives \6P^u(lJ)/dxPdt<i\ are bounded independently of v in 0 < r < t < T. This shows that the limit function u is Cx -smooth for t > 0. Also, the derivatives of w(l/) converge pointwise to the corresponding derivatives of u for t > 0. We summarize: Theorem 4.3.3. Burgers' equation (4.3.3) has a \-periodic generalized solu- tion u = u(x, f)for each initial function For t > 0, the solution u is Cx -smooth and satisfies the differential equation pointwise. A Nonlinear Example: Burgers' Equation 141 4.4. The In viscid Burgers' Equation: A First Study of Shocks The viscous equations* have unique C00-solutions u = ue, existing for all time t > 0. The correspond- ing inviscid equation shall be treated as the limit of (4.4.1) as e —> 0. The viscous solutions u(, e > 0, are uniformly smooth in some interval 0 < £ < T, T>0 independent of e, as shown in Section 4.1. Thus we can send e —> 0 and obtain a smooth solution u of the inviscid equation in the same time interval. (Indeed, if u = wf,. v = u(2, then we can argue as in the proof of Lemma 4.1.1 and show that Then, by Appendix 4, convergence uf —» u follows.) According to Lemma 4.1.1, this limit u is the unique classical solution in 0 < t < T. It is natural to ask whether the functions ut(x, t) converge for all t > 0 and whether we can extend the smooth solution u(x, t) for all times. Both questions have indeed affirmative answers. However, the smoothness of the solution u breaks down, in general, at a certain time 7},, at which one or several shocks form. This will be proved below using the method of characteristics. In gas dynamics one treats systems of equations which exhibit a similar mathe- matical structure as the inviscid Burgers' equation. In view of these applications, it is desirable to extend the solution u beyond the time TJ,, but this is possible only as a nonsmooth function. Of course, the concept of a nonsmooth solution of a differential equation requires some explanation. Concerning this question, we will restrict ourselves to Burgers' equation, and will first treat the special case of limits of traveling waves. Afterwards, in Section 4.4.3, we discuss more general piecewise smooth weak solutions. 4.4.1. A Solution Formula via Characteristics Suppose u = u(x. t) is a smooth function solving *We alter here the sign of the uux-temv, this corresponds simply to the transformation x —> -x. The nonlinear term appears now with the same sign as the convection term in the Navier-Stokes equations. 142 Initial-Boundary Value roblems and the Navier-Stokes Equations i0 < t < T. We can consider u(x, t) = a(x, t) as a coefficient in (4.4.3) ad u as the solution of ut + aux = 0. This suggests applying the idea of characteristics discussed for linear equations in Section 3.3. Accordingly, we call a characteristic line of the equation ut + uux = 0 for the specific solution u, if his agrees completely with the definition for the linear equation ut + aux = 0 if a = u. (In the nonlinear case, the characteristics depend on the solution under consideration, however.) As in the linear case, it follows that u is constant along each characteristic: i.e., u(x(t),t) = /(XQ) if x(0) = XQ. Now (4.4.4) implies that the characteristic (x(£), t] is a straight line: We have shown Lemma 4.4.1. Suppose u solves Then in any t-interval 0 < t < T where u is smooth. Note that this result is completely independent of spatial periodicity. Hence- forth we drop the periodicity assumption and consider (4.4.5), where / 6 Cx is a given function with We can use the formula (4.4.6) to obtain a solution u = u(x, t) as follows: Suppose x € R, t > 0 are given and suppose that there is a unique XQ € R with A Nonlinear Example: Burgers' Equation 143 Then define w(z, t) = f(xo). Let us explain why u solves (4.4.5). First note that, since / is bounded, the equation (4.4.7) always has at least one solution XQ. If there are two different solutions XQ and x\, then Thus, if we let then (4.4.7) has a unique solution XQ = XQ(X, t) for given .F e R, 0 < t < Tb, and consequently the function is well-defined. By the Implicit Function Theorem (see below), it follows that XQ(X. t) is C^-smooth. Assuming this is known, we prove now that (4.4.8) solves the inviscid Burgers' equation. Lemma 4.4.2. The Cx -function u = u(x, t) defined by (4.4.8) is the unique classical solution of (4. 4.5). Proof. It is clear that XQ(X.O) = x, and thus u(x,0) = f(x). To show that ut + uux = 0, we differentiate with respect to x and t, and find that Thus the definition u(x, t) = /(XQ(X, t)) gives us For completeness we state the Implicit Function Theorem. Let U C Rn. V C Rm be open, and let F : U x V — > Rm be a Cp -function, p > 1 . Let OQ € U, 60 € V and assume that 144 Initial-Boundary Value Problems and the Navier-Stokes Equations (Here Fb is the Jacobian matrix of F w.r.t. the b-variables.) Then there exists an open neighborhood UQO/OQ, and a unique Cp-function g : UQ -* Rm with Roughly speaking, the equation F(a, 6) = 0 defines 6 implicitly as a function of o, 6 = <?(a), and g is (at least) as smooth as F. This result applies to the previous situation with n = 2, m = \, and in the neighborhood of any point and shows that XQ depends smoothly on (x, 0- Here is essential. Let us illustrate the solution formula (4.4.8): If /'(£) > 0 for all f, then XQ < X[ implies f(xo) < f(x\)\ thus the characteristics never cross. We have Tb = oo, and a smooth solution exists for all time t > 0. On the other hand, if /'(£) < 0 for some £, then there are values and the characteristics (4.4.9) cross. The solution steepens up with time. The time Tb is finite. We cannot extend the solution u beyond Tj, as a smooth solution. See figures 4.4.1 to 4.4.3. 4.4.2. Traveling Wave-Solutions for Burgers' Equation Before we discuss the extension of the inviscid solution u as a nonsmooth function, let us study explicitly some simple solutions of the viscous equation. We start with the steady state problem under boundary conditions A Nonlinear Example: Burgers' Equation 145 FIGURE 4.4.1. Characteristics for f(x) increasing. FIGURE 4.4.2. Crossing characterises. FIGURE 4.4.3. Solution steepens up. 146 Initial-Boundary Value Problems and the Navier-Stokes Equations If 0(x) is a solution, then so is the translated function 0(z + c) for any real constant c. To fix the constant, we require that and prove Theorem 4.43. Ifa>0the problem (4.4. 10)-(4.4. 12) has a unique solution given by If a < 0 there is no solution. Proof. First suppose 0 is a solution. Integration of (4.4.10) yields It follows that either 0 is constant or 0x(z) / 0 for all x. The monotonicity of 0 and (4.4.11) imply that 0X tends to zero as x —> ±00, Thus we can replace the constant in (4.4.14) by a2/2: Clearly, if a ^ 0, then 0X(0) < 0, and therefore 0 is decreasing. The boundary conditions (4.4.11) can only be fulfilled for a > 0. The o.d.e. (4.4.15) with initial condition (4.4.12) has a unique solution, and by separation of variables one arrives at formula (4.4.13). The stationary profiles (4.4.13) are shown in figure 4.4.4. In the limit as e —> 0, the functions 0(x) approach a jump discontinuity. Now consider the time dependent problem under boundary conditions We use an ansatz with a constant wave-speed s. It leads to A Nonlinear Example: Burgers' Equation 147 FIGURE 4.4.4. Stationary profile. and the differential equation (4.4.16) requires If we set the wave-speed then we obtain the previous stationary differential equation for 0. The boundary conditions for 0 are those given in (4.4.11) with a = (6 — c)/2. We have shown Theorem 4.4.4. The time dependent problem (4.4.16), (4.4.17) has traveling wave-solutions For e —» 0 these solutions converge to the piecewise constant function 4.4.3. Weak Solutions of the Inviscid Burgers* Equation The inviscid equation 148 Initial-Boundary Value Problems and the Navier-Stokes Equations FIGURE 4.4.S. Limit of traveling waves. has a unique smooth solution u = u(x, t) only in 0 < t < T^. Here T& is finite unless /'(£) > 0 for all £. This result has been shown in Section 4.4.1. We now broaden the solution concept and first define weak solutions. To this end, let CQ° denote the set of all test functions 0 = <£(#, t), (x, t) € R2: Suppose u is a classical solution of (4.4.18) and (f> € CQ°. If we integrate over — oo < a: < oo, 0<t<oo, then integration by parts gives us The double integral makes sense even if u is not smooth. We make the Definition. A bounded measurable function u — u(x, t) defined for x e R, t > 0, is called a weak solution of (4.4.18) if equation (4.4.19) holds for all 0 € C0~. Clearly, we have broadened e solution concept. Using the solutions u( of the viscous equation and sendi e —> 0, one can show that there always exists a weak solution. It turns out, however, that weak solutions are not uniquely determined by their initial data. Before discussing these issues further, we ask for conditions characterizing a piecewise smooth function as a weak solution. First note: If u is a weak solution, A Nonlinear Example: Burgers' Equation 149 which is smooth in a neighborhood U of a point (XQ. to), then u satisfies the differential equation ut + uux — 0 in the usual sense at (XQ, to). To show this, we take an arbitrary $> e CQ° with support in U and find that thus ut + MWj = 0 in (#o, *o)» since </> is arbitrary. Next assume there is a smooth line (parametrized over time) and that the weak solution u has a jump discontinuity along F. The left and right limits are assumed to exist: and we suppose that these limits depend smoothly on t. Except for this jump, we assume that u is smooth in a neighborhood of F. Taking a test function <j> with support in a neighborhood U along F we find that The integrals over t// and Ur on the right-hand side vanish, and therefore Since 4> can take arbitrary values along F, it follows that 150 Initial-Boundary Value Problems and the Navier-Stokes Equations FIGURE 4.4.6. Discontinuit rve. This is a special case of the so-called Rankine-Hugoniot jump condition, which relates the shock speed s'(t) to the states ui and ur of u along the shock line: A particular case of this condition occurred already in Theorem 4.4.4 for the limit of traveling waves. In that example the shock speed s was given by Thus far, we assumed that u is a piecewise smooth weak solution and that the initial function f(x) is smooth. However, if / is only piecewise smooth, we obtain the same result. Conversely, reversing the arguments given above, one can show that a piecewise smooth function it is a weak solution of (4.4. 1 ) if (a) the initial condition u(x, 0) = f(x) is satisfied at the points of continuity of/; (b) the differential equation is satisfied in the pointwise sense in all regions of smoothness of w; (c) the jump condition (4.4.22) is satisfied along the lines of discontinuity of u. We summarize: Theorem 4.4.5. Suppose u is a piecewise smooth function with jump discon- tinuities along finitely many smooth lines. Then u is a weak solution o/(4.4.18) if and only if the conditions (a), (6), (c) above are met. A Nonlinear Example: Burgers' Equation Example 1. (A propagating shock.)151 FIGURE 4.4.7. Discontinuous initial function. FIGURE 4.4.8. Propagating shock. Here is a weak solution. The shock speed s = 1/2 = (l/2)(w/ + ur) fulfills the jump condition (4.4.22). This solution is a limit of traveling waves; compare Theorem 4.4.4. 152 Initial-Boundary Value Problems and the Navier-Stokes Equations Example 2. (A rarefaction.) FIGURE 4.4.9. Discontinuous initial function. FIGURE 4.4.10. Rarefaction wave. Here the characteristics starting at i = 0 do not enter the region If one uses an ansatz then the differential equation requires that A Nonlinear Example: Burgers' Equation and one obtains153 as a smooth solution in R, which connects the constant states This solution is called a rarefaction wave. The discontinuity at t = 0 is resolved instantaneously. Example 3. (A pulse.) FIGURE 4.4.11. Initial pulse. FIGURE 4.4.12. Rarefaction and shock. 154 Initial-Boundary Value Problems and the Navier-Slokes Equations At time t = 2 the rarefaction wave x/t starting at x = 0 catches up with the shock wave starting at x = 1. We can use the jump condition (4.4.22) to determine the location s(t) of the shock for t > 2: The equation is solved by In these examples we have used the characteristics, the jump condition, and the form of a rarefaction to construct weak solutions. Unfortunately weak so- lutions are not unique. This follows from Example 4. (A non-physical shock.) Thus we take the same initial condition which led to the rarefaction wave of Example 2. The function is a piecewise smooth weak solution since the conditions of Theorem 4.4.5 are met. Examples 2 and 4 describe two weak solutions to the same initial function. FIGURE 4.4.13. Non-physical shock. A Nonlinear Example: Burgers1 Equation 155 Which solution should be preferred? There are two reasons why the shock solution of Example 4 is unphysical: 1 ) It is not true that two characteristics meet at the shock. 2) The solutions uf of the corresponding viscous problem converge to the rarefaction wave and do not converge to the shock solution of Example 4. We do not prove the second statement here. The result is plausible, however, since we have shown in Section 4.4.2 that there are no traveling waves for € > 0 which are increasing in x; the shock solution of Example 4, however, corresponds to such a wave for e = 0. Let us now outline the results for the general case. The viscous problems have unique smooth solutions uf for all e > 0. These exist for all time. (We have shown this for periodic /. The present case can be treated as the limit where the period tends to infinity.) If 0 is any test function, it follows that Using a compactness argument for uf, one can show that there is a sequence e — Ck and a bounded measurable function u with (Compare the remark at the end of Section 4.3.1.) The convergence holds pointwise almost everywhere. By the maximum principle, the functions u( are uniformly bounded. Thus the integrals in (4.4.23) converge as e = ejt —» 0, and one obtains (4.4.19); therefore the limit u is a weak solution. This shows the existence of weak solutions. As we have seen, weak solutions are not unique. However, if one considers only piecewise smooth weak solutions and requires that at each discontinuity, then - in this restricted class - weak solutions turn out to be unique. Condition (4.4.24) is a plausible requirement at each shock: The characteristic speed should be larger to the left of the shock than to the right. For practical purposes, one can restrict attention to piecewise smooth weak 156 Initial-Boundary Value Problems and the Navier-Stokes Equations solutions. The physically relevant one is characterized by (4.4.24). This weak solution is the limit (almost everywhere) of the viscous solutions u€ as e —> 0. Roughly speaking, one has the following picture: The physically relevant weak solution u of the inviscid equation is a piecewise smooth function. The shock speeds are governed by the Rankine-Hugoniot condition (4.4.22). The function u jumps downwards if one crosses a shock from left to right. For small € > 0, the viscous solutions ut are uniformly close to u away from shocks. The convergence is of order O(e) as e —> oo. In a small neighborhood of the shocks, the viscous solutions behave like traveling waves connecting the state u/(x, t) with ur(x, t). Notes on Chapter 4 Burgers' equation has been discussed in many books; for example in Whitham (1974). We have discussed the estimates in detail because the same proce- dure can be used - as we will see - in very general situations (systems, many space dimensions). One can make the global estimates more precise and prove l&Pu/dxvioo = O(e~p). E. Hopf (1950) discussed the properties of the solu- tions of Burgers' equation by transforming the equation into the heat equation (Cole-Hopf transformation). In particular, he proved that the solutions converge for 6 —> 0 to a weak solution of the limit equation. The solutions of parabolic equations have the property that the number of sign changes does not increase with time; see Matano (1982), Protter and Weinberger (1967). This can be used to obtain very precise information about the solutions of Burgers' equation The derivative v = ux satisfies Assume first fx > 0; then v > 0 for all times, and Thus the solution u "becomes flat", i.e., we have a rarefaction. If fx < 0 then This page intentionally left blank 5 Nonlinear Systems in One Space Dimension For Burgers' equation we used an iteration solving linear equations to obtain a local existence result. This technique carries over to first-order hyperbolic, to second-order parabolic and to mixed systems. The important point is again to obtain a priori estimates of the solution and its derivatives; these lead in turn to uniform estimates for the iteration sequence mentioned above. Not to obscure the underlying principles, we shall not consider the most general situation, however, but restrict ourselves mainly to equations where A = A(u) is a given smooth matrix function. As before, all (vector- and matrix-) functions are assumed to be real, for simplicity, and 1 -periodic in x. In Section 5.1 we treat the case where A(u) and all its derivatives are globally bounded. For e > 0, one obtains existence for all time. If the system is hyperbolic, one obtains short time existence for e = 0. The assumption of global boundedness of A(u) is, in general, not fulfilled in applications. In Section 5.2 we use a simple cut-off technique to treat more general cases and obtain short time existence results if A(u) is smooth in a neighborhood of the initial data. 759 160 Initial-Boundary Value Problems and the Navier-Stokes Equations If a finite time To > 0 is given (To is not necessarily small), how can one decide whether a solution of (5.1.1) exists in 0 < t < TO? As for ordinary differential equations, this problem is often of a quantitative nature. We will sketch in Section 5.3, how asymptotic expansions and numerical calculations can be used to answer the finite time existence question, at least in principle. Finally, in Section 5.4, we discuss a global existence result under more spe- cific assumptions. 5.1. The Case of Bounded Coefficients We first settle the uniqueness problem. Lemma 5.1.1. If e > 0 or if(e — 0 and A(u) = A*(u)) then the problem (5.1.1) has at most one classical solution. Proof. Suppose w, v are solutions and w = u — v. We write and note that Therefore, and again it follows that In both cases the initial condition w(x, 0) = 0 implies w = 0. A priori estimates. We assume in this section that A = A(u) and all its deriva- tives are globally bounded: Nonlinear Systems in One Space Dimension (This assumption, which is very restrictive for applications, will be relaxed below.) As in the uniqueness lemma, the case e = 0 requires an additional assumption, namely hyperbolicity. Definition. The first-order system is called hyperbolic if there is a smooth symmetrizer H = H(u), i.e., a smooth matrix function H(u) with To simplify the proof below, we will restrict ourselves to symmetric hyperbolic systems where Let u = u(x,t) denote a smooth solution of (5.1.1), and let u3 = d^u/d x^. We first consider the parabolic case e > 0 and estimate u-j in terms of the initial data and its derivatives. Lemma 5.1.2. Assume the solution u = u(x,t) o/(5.1.1) is defined for 0 < t < T, and the global bounds (5. 1 .3) hold. For each e > 0 there exists a constant Cj =Cj(T,e)with The constant Cj depends on \\fj\\. Proof. Equation (5.1.1) implies that As usual, this yields 162 Initial-Boundary Value Problems and the Navier-Stokes Equations Thus we have proved (5.1.6) for j = 0. Now assume (5.1.6) is proved up to j - 1 . Clearly, Here* etc. In general, where a bound follows from (5.1.3), the Sobolev inequality and the induction hypothesis. Therefore, Now we can complete the induction step, and the lemma is proved. Under the global assumption (5.1.3), we have shown a priori estimates of all derivatives of a solution for any given interval 0 < t < T. The estimates depend on e > 0, however. Existence results. Existence of a solution can be shown by either using the difference approximation ^Formally, A'(u) is a bilinear map acting an two vectors, A"(u) is a trilinear map acting on three vectors, etc. This calculus is not essential here, however. It is only important that each matrix element of A(u) has bounded partial derivatives of all orders. Nonlinear Systems in One Space Dimension 163 or by considering the iteration Since the time interval 0 < t < T for the a priori estimate was arbitrary, one obtains global existence. We summarize: Theorem 5.1.3. Given the global bounds (5 .1.3), the parabolic system (5.1.1) has a unique smooth solution ut = w£(x, t) defined for 0 < t < oo. Here e > 0 15 arbitrary. To treat the first-order system we assume symmetric hyperbolicity (5.1.5) and the boundedness assumption (5. 1 .3). We will show e-independent bounds for the solutions uc of the parabolic equation in some time interval 0 < t < T. Sending e -» 0, we obtain a solution of (5.1.7) in this interval. Theorem 5.1.4. Suppose that (5.1.3) and (5.1.5) hold. There exists a time T > 0 and a constant C > 0, both depending on ||/||//2 but independent of e > 0, with The symmetric hyperbolic system (5.1.7) has a smooth solution u = u(x,t) in 0<t<T. Proof. 1) We fix e > 0 and write u — uf for the solution of (5.1.1). Equation (5.1.1) gives us For u\ — du/dx we have and therefore 164 Initial-Boundary Value Problems and the Navier-Stokes Equations We now estimate u2 to close the system. For w2 it holds that and therefore, Here The assumption A = A* implies cancellation of the terms One finds that Using Sobolev's inequality to bound |ui|oo by ||u2|| + \\u\\\, we arrive at a closed system of differential inequalities for || u ||2, ||ui||2, ||u2||2- Therefore the estimate (5.1.8) follows with T and C independent of €. 2) Estimates of higher derivatives follow in the same way from The time interval 0 < t < T determined in part 1 of the proof does not have to be diminished. This shows that the solutions u( of the parabolic problem are uniformly smooth in 0 < t < T. The existence result for the hyperbolic case follows if we let e —* 0. Nonlinear Systems in One Space Dimension 165 5.2. Local Existence Theorems The assumption of global boundedness (5.1.3) is too restrictive for applications; for example, it is even violated for Burgers' equation. In this section we show how to localize the assumptions to a neighborhood of the initial data and how to obtain existence in a small time interval. The basic idea is simple, namely the solution stays close to the initial data, at least for a short time. For a given initial function let denote the ^-neighborhood of the initial data. We choose a scalar cut-off function 0 € C°°(R") with If A = A(u), A G COC(C/T?), is the given coefficient matrix in (5.1.1), then set The altered systems satisfy the boundedness assumption (5.1.3). First consider the case e > 0 without structural assumptions on A. By continuity, the solution fif of (5.2.1) stays in [7,;/4 for some interval 0 < t < Tf. Here uf solves the original system. This shows Theorem 5.2.1. If A = A(u) is a C00 -function in a neighborhood of the initial data, then the parabolic systems (5.1.1) have unique smooth solutions u = u( in some time interval 0 < t < Tt . The time T( > 0 depends on € > 0. We add now the (local) assumption of symmetric hyperbolicity: Clearly A(u) = A*(u) holds; thus the e-independent estimates of Theorem 5.1.4 apply to ut. There is a time T > 0 and a constant C > 0, both independent of e, with 166 Initial-Boundary Value Problems and the Navier-Stokes Equations and thus Therefore, in some time interval 0 < t < T, T > 0, independent of e > 0, the functions uc solve the original system (5.1.1) and are uniformly smooth. For 6 — > 0, one obtains a solution of the hyperbolic problem. We summarize: Theorem 5.2.2. If A(u) satisfies the symmetry condition (5.2.2) in a neigh- borhood of the initial data, then the problems (5.1.1) have unique smooth solu- tions u£ = u£(x, t)for e > 0 in some time interval 0 < t < T. The time T > 0 can be chosen independently ofe. Generalizations. It is not difficult to generalize the local Existence Theorem 5.2.1 to parabolic equations where in a neighborhood of the initial data. Also, more general first-order hyperbolic systems can be treated as in Theorem 5.2.2. Finally, we can proceed as in Section 3.4 and obtain results for mixed hyperbolic-parabolic systems Here AI, B,-j, C are smooth functions of w, v, a;, f and One obtains Theorem 5.23. Consider the 1 -periodic Cauchy problem for a parabolic sys- tem (5.2.3), a hyperbolic system (5.2.4) or a mixed hyperbolic-parabolic system (5.2.5). A unique solution exists in a sufficiently small time interval Q <t <T. The time T depends on the initial data. 168 Initial-Boundary Value Problems and the Navier-Stokes Equations the previous techniques to the rewritten equation. In this way, the guaranteed existence interval can often be improved considerably. Example. Let us demonstrate the foregoing by the following example: where a € R is a given constant. The general techniques yield short-time existence in an interval 0 < t < T, T — T(\\f\\H2). Now suppose For e = 0, the problem has the solution existing for all time; this suggests the ansatz For w one obtains that It should be emphasized that no terms have been neglected, and therefore, if we obtain existence for w in 0 < t < T, we also can solve the original problem in 0 < t < T. Clearly, the nonlinear term in (5.3.2) is multiplied by the small parameter e. (In generalizations of this example, the value of e is a measure of the accuracy to which UQ solves the given problem.) Let us discuss (5.3.2). Proceeding as in Section 4.1.2, we obtain Consequently, if y(t) is the solution of then w exists in any interval 0 < t < T where y(t) is bounded. It is, of course, important to note that the nonlinear term y3/2 - which may cause a singularity in y - appears multiplied by e. Explicit discussion shows the following: Nonlinear Systems in One Space Dimension 169 Case 1. Q < 0. If then y(t) decays exponentially, and thus w exists for all time. Case 2. a = 0. Now y(t) has a singularity at a time Thus we have long-time existence for small e. Case 3. a > 0. In this case, y(t) grows exponentially and has a pole at Again, if \\gxx\\ = O(l), the blow-up time TB —» oo as e —> 0, but the increase of TB with decreasing e is very slow. (In order to obtain long-time existence in this case, the perturbation term e||<7x;E|| has to be "exponentially" small: The behaviour of the blow-up times TB as functions of € can be predicted on basis of the linearized equation. Linearization of (5.3.1) at UQ gives us If Q < 0, then 0o decays exponentially; the solution u(x, t) and the approxima- tion UQ(X, t) stay close for all time if the pertubation eg is sufficiently restricted. If Q = 0, then the solution <fo stays bounded; the perturbation eg will, in general, grow over a long time interval. If a > 0, then <fo grows exponentially; after a short time, the differences between u and UQ can be O(l) unless the perturbation is exponentially small. Asymptotic expansion. By solving linear problems, one can increase the power of e which multiplies the nonlinearity. Again, we illustrate this for the foregoing example. If 0o solves and we write the solution w of (5.3.2) in the form 170 Initial-Boundary Value Problems and the Navier-Stokes Equations then we obtain for w\ : Obviously, the nonlinearity is now multiplied by e2. Repeating this process, we introduce into (5.3.2). Equating terms of order e°, e',e2, ...,ep~' successivly, we obtain linear equations for <fo, ..., <t>p-\. If these are solved, the nonlinear equation for wp can be written down: where and Fp is also known in terms of <fo, ...,<£p_|. In this way, we have reduced the nonlinearity to order ep+l; the solution u has the asymptotic expansion Generalization. Without going into formal details, we want to indicate gener- alizations of the previous example. Consider a system where P is a differential operator of any of the forms treated in Sections 5.1, 5.2, and assume the coefficients of P are polynomials in u, for simplicity. Now suppose uo(x, t) is a known approximate solution in some time interval 0 < t < T; more precisely, if we substitute UQ into (5.3.4), the equations will be satisfied with small defects: (Formally, we can define the terms cG. eg by the above equations. The param- eter e is introduced artificially in such a way that G — O(l), g = O(\).) Then we can view the given system (5.3.4) as a pertubation of (5.3.5), which suggests an ansatz Nonlinear Systems in One Space Dimension 171 For w one obtains: Here PI and Pi are differential operators whose coefficients also depend on UQ; however, since UQ is assumed to be known, this is suppressed in the notation. The operator P\ is linear, and the nonlinear part is multiplied by e. If the equation for w is of the type discussed in Sections 5.1, 5.2, we can derive a priori estimates for u;. As in the example, the growth behaviour of the solution w of the linear problem is crucial. If the solutions of (5.3.6) satisfy then one can obtain a priori estimates for w of the form Therefore, if |e| is sufficiently small, the solution w - and thus u - will exist in 0 < t < T. Also, if 0o solves the linearized equation then u — UQ + e0o + O(e2). Higher order asymptotic expansions can be derived as in the example; the functions </>Oi . . . , 4>p-\ solve linear prob- lems. Remark on numerical calculations. The approximate solution u0 can also be obtained through interpolation of numerically computed data vh. It is possible to bound the defects eG, eg defined in (5.3.5) in terms of divided differences of vh; the latter can be computed. For this reason, if a numerical solution vh is computed in 0 < t < T, and sufficiently many divided differences stay within reasonable bounds, one has an indication that the partial differential equation has a solution in 0 < t < T. A better test is obtained if one repeats the computation with step-sizes 172 Initial-Boundary Value Problems and the Navier-Stokes Equations If the bounds for the divided differences do not grow for decreasing step-sizes, the defects of the interpolant UQ = UQ will finally be small enough to ensure existence of a solution of the p.d.e. On the other hand, if the divided differences do grow for decreasing h, one has an indication for blow-up at a time TB < T, 5.4. On Global Existence for Parabolic and Mixed Systems Global (in time) existence theorems for systems can only be derived under rather special assumptions. However, since we have a local existence and uniqueness result, we can always piece local (in time) solutions together and obtain a half-open maximal interval of existence 0 < t < TQ. Here TO is finite or TO = oo. If TO is finite, then cannot be bounded. Otherwise we could apply the cut-off technique described in Section 5.2 and use Theorem 5.1.3 to extend the solution beyond TO. For the systems (5.4.1) we thus have the alternative: Lemma 5.4.1. Either (i) a smooth solution u exists for 0 < t < oo, or (ii) there is a blow-up time TO < oo with Phrased concisely, the lemma states that "the solution exists as long as it stays bounded". We want to describe conditions which exclude possibility (ii); thus these conditions imply all-time existence. First consider a specific example: Suppose u = (v,p) is a smooth solution. The basic energy estimate follows from Nonlinear Systems in One Space Dimension 173 Integration from 0 to t yields and The coefficient matrix of the example (5.4.2) clearly satisfies an estimate of the form We will show in the following theorem that estimates like (5.4.3), (5.4.4), (5.4.5) exclude possibility (ii) of Lemma 5.4.1; thus they imply all-time existence. Theorem 5.4.2. Suppose that (5.4.5) holds, and a solution u of (5AA) is defined for 0 < t < T < oo and satisfies Then are finite; thus u can he extended beyond T. Furthermore, there is a bound where K* only depends on e, K2, Kj, and ||u(-,0)||//i, but does not depend on T. Proof. Differentiation of (5.4.1) yields 174 Initial-Boundary Value Problems and the Navier-Stokes Equations To estimate |u|oo, we use Lemma 3.2.5, and find that If we set then we have shown that hence To bound y(t), we fix a time 0 < t < T with y(t) > 1. Case 1. There is t0 < t with y(t0) =1. Let T0 denote the largest of these to-values, thus y(r) > 1 for T0 < r < t. Integration of (5.4.8) gives us and thus y(t) < 1 + (a + /3)/C3. Case 2. For 0 < r < t we have y(r) > 1. Integration of (5.4.8) yields and the theorem is proved. In the example (5.4.2), the estimates (5.4.6), (5.4.7) are valid a priori in any interval 0 < t < T where the solution exists. Therefore, according to the previous theorem and Lemma 5.4.1, the solution exists for all time. Furthermore, since the constants KI and Kj, do not depend on T, the bound K* derived in the theorem is uniform for 0 < t < oo. In particular, the values of the solution stay bounded for all time: Nonlinear Systems in One Space Dimension 175 A special mixed system. We shall now show all-time existence for the mixed hyperbolic-parabolic system Integration by parts gives us Therefore, by the Sobolev inequality, Lemma 3.2.5, For the first x-derivatives we obtain Thus, using (5.4.9), we obtain a bound for This process can be continued, and existence follows. Notes on Chapter 5 For local existence theorems there is no essential difference between linear and nonlinear equations. Global existence requires global bounds, however. One such result for parabolic systems is given in Theorem 5.4.2. Others can be found in Smoller (1983) and in Ladyzhenskaya, Solonnikov, and Uralceva (1968). Much less is known for the inviscid case e = 0. We cannot expect 176 Initial-Boundary Value Problems and the Navier-Slokes Equations global existence. Classical solutions cease to exist beyond a certain blow-up time; see John (1974), Lax (1964), Glimm and Lax (1970). If one can write the equations in conservation form then one can introduce again weak solutions. A (not necessarily smooth) func- tion v is a weak solution if for all </> e C0°. Weak solutions are not unique. Generalizing (4.4.25), Lax (1957) introduced entropy conditions for systems. Glimm (1965) proved that there is always a global weak solution which satisfies these conditions, provided the initial data are sufficiently close to a constant vector. However, it is not known whether the solution is unique. For later results we refer to DiPerna (1985), Liu (1983), and Majda (1984). An important example is the system of conservation laws for gasdynamics: Here p, u, p, e denote the density, velocity, pressure, and internal energy, re- spectively. Also, an equation of state which connects e. p, p has to be speci- fied. Many people have studied these equations for the last 150 years using less mathematical machinery. For an account see Courant and Friedrichs (1948), Whitham (1974), and Landau and Lifshitz (1959). Little is known about the behavior of the solutions of (5.4.1) as e — > 0. However, Gilbarg (1951) proved that the viscous gasdynamical equations allow traveling waves which converge for € — » 0 to a weak solution of the limit equation. For other results we refer toSmoller(1983). 6 The Cauchy Problem for Systems in Several Dimensions The results of Chapter 3 about linear parabolic, strongly hyperbolic, and mixed systems can mostly be generalized from one space dimension to an arbitrary number of space dimensions. These generalizations are carried out in the first three sections of this chapter. In particular, we show well-posedness of the Cauchy problem for the linearized compressible Navier-Stokes and Euler equa- tions. Section 6.4 treats short-time existence for nonlinear systems; we carry out the details for the symmetric hyperbolic case and sketch generalizations to systems of different type. For a special class of nonlinear parabolic systems in two space dimensions, we will prove all-time existence in Section 6.5. 6.1. Line Parabolic Systems We consider second-order parabolic systems of the general form Here s is the number of space dimensions and Dt is the operator Di — The coefficient matrices AIJ — Aij(x. t), Bi = Bi(x,t), C = C(x.t) and the forcing function F = F(x, t) are assumed to be C^-smooth, real, and 1-periodic 777 178 Initial-Boundary Value Problems and the Navier-Stokes Equations in each component Xj, i = I,..., s. The concept of parabolicity will be defined below. For the function w(x, t) an initial condition is given, where / takes values in Rn. As before, we seek a smooth solution u = u(x, t) taking values in Rn which is 1-periodic in each x*, i= 1,..., s. In Section 6.1.1 the concept of strong parabolicity will be defined; it leads to a simple derivation of solution-estimates. Existence of a solution can again be shown by proving the analogous estimates for a difference scheme. This will be sketched in Section 6.1.2. Existence, uniqueness, and estimates can be generalized from strongly parabolic systems to systems for which a pointwise definition of parabolicity is used. For our applications this generalization is of minor importance, however, and we only sketch the proof. 6.1.1. Estimates for Strongly Parabolic Systems The following definition generalizes our previous concept of strong parabolicity to systems in any number of space dimensions. Definition. The system (6.1.1) is called strongly parabolic in 0 < t < T if there exists a constant 6 > 0 such that the estimate holds for all x € R*, 0 < t < T. and all vectors y; 6 Rn, i = 1,..., s. The basic result about existence, uniqueness, and estimates is Theorem 6.1.1. If (6.1.1) is strongly parabolic in 0 < t < T, then the Cauchy problem (6.1.1), (6.1.2) has a unique classical solution u = u(x, 0 in 0 < t < T. The solution u is a C00-function. For each p = 0,1,2,..., there is a constant Kp with The constant Kp depends on 6 and the maximum norm of the coefficients of (6.1.1) and their derivatives of order < p, but not on F, /. The Cauchy Problem for Systems in Several Dimensions 179 Proof. We start with the estimates. Integration by parts yields Therefore, (6.1.3) follows for p = 0 if we apply Lemma 3.1.1. This estimate also shows the uniqueness of a classical solution. To estimate first derivatives, we differentiate the equation (6. 1 . 1 ) with respect to £n i = 1 , . . . , s, and obtain a system of similar type for the vector The coefficients of this system depend on Aij, Bj. C and the first derivatives of these matrices. The desired estimate follows as above. Estimates for higher derivatives are obtained by repeated differentiation. In the next section we sketch the proof of the existence of u. 6.1.2. Existence via Difference Approximations Let h = 1 IN denote a gridlength, N a natural number. For any multi-index let xv = (hi/i , . . . , hva) denote the corresponding meshpoint. We consider gridfunctions vv = v(xv) which are assumed to be 1 -periodic in each coordinate direction. By Ej we denote the translation operator in the j-th coordinate: ej = (0, . . . , 0. 1 , 0, . . . . 0) with 1 at position j. The powers of Ej are and E°J = I is the identity. The forward, backward, and centered difference operators in the j-th coordinate direction are 180 Initial-Boundary Value Problems and the Navier-Stokes Equations Using these notations, we approximate (6.1.1) by the system of ordinary differ- ential equations If we introduce a discrete scalar product and norm for gridfunctions by then all computational rules of Section 3.2.2 carry over in an obvious way to the multi-dimensional case. Therefore, we can estimate the solutions of (6. 1 .4) and their divided differences independently of h. We just mimic the continuous process. As in the one dimensional case, we can interpolate vv(t) = v£(t) with respect to the x-variables by Fourier polynomials wh(x, t). (See Appendix 2.) These are uniformly (with respect to h) smooth; in the limit h — * 0 we obtain a solution u = w(x, t) of the Cauchy problem. There is no essential difference from the one dimensional case. 6.1.3. General Linear Second-Order Parabolic Systems In Section 2.4.3 we have defined parabolicity for problems with constant coef- ficients. Applying this concept to all frozen-coefficient problems, we are lead to: Definition. The system (6.1.1) is called parabolic in 0 < t < T if there is a constant 8 > 0 such that for all x € R*, 0 < t < T, and all u> = (tj| , . . . , ws) e Ra, the eigenvalues K of the matrix satisfy he Cauchy Problem for Systems in Several Dimensions 181 One can prove Theorem 6.1.2. The statements of Theorem 6.1.1 are valid if the assumption of strong parabolicity is replaced by the assumption of parabolicity. We only sketch the proof of this result. The essential point is to obtain the a priori estimates (6.1.3); then existence of a solution follows as before by use of a difference scheme. To obtain the estimate (6.1.3) for p = 0, first fix a point (XQ, £Q) and consider the constant-coefficient parabolic operator If we proceed as in Section 2.7 - but replace Fourier transforms by Fourier expansions - we can construct an inner product with for all smooth functions u = u(x). Using the corresponding norms and a partition of unity argument as in Section 3.2.5, we obtain the basic energy estimate as before. Estimates for derivatives can be obtained by differentiating the differential equation. As noted in Section 3.2.6, a basic property of parabolic differential operators is their smoothing ability. This smoothing property is also valid for equations in more than one space dimension and can be proved as in the one dimensional case. 6.2. Linear Hyperbolic Systems A first-order system is called symmetric hyperbolic if P(x, t, dfdx) has the form 182 Initial-Boundary Value Problems and the Navier-Stokes Equations and Bj(x,i) = BJ(x,t) for all x,t. As before, the coefficients Bj, C, and the forcing function F are assumed to be real, C°° -smooth, and 1 -periodic in each X. To obtain existence for the Cauchy problem (6.2.1), (6.1.2), we consider first the parabolic problems The existence result of Theorem 6.1.1 applies, and the estimates are independent of e > 0. For e -* 0, one obtains a solution of the symmetric hyperbolic problem. A basic property of hyperbolic equations is the "finite speed of propagation". We formulate the result in the following theorem; the proof uses the convergence of an explicit difference scheme. Theorem 6.2.1. Define The solution-value U(XQ, to) depends only on those values of F(x, t) for (z, t) lying in the cone and only on those values of f(x)for x lying in the interval \XQ — x\ < at^. FIGURE 6.2.1. Domain of dependence and mesh. The Cauchy Problem for Systems in Several Dimensions 183 Proof. 1. As in Section 6.2.1, we introduce meshpoints #„, v e Z*. but discretize in time also. Let k > 0 denote the time step-size. We approximate (6.2.1) by the difference approximation Here denotes the usual centered difference operator. The three level difference scheme (6.2.2) is supplemented by the starting conditions We fix a relation k = h/(a + 77), rj > 0 arbitrary, and send h —> 0. The difference equations (6.2.2), (6.2.3) define an approximation v = vh for the exact solution u, and we will prove the convergence The domain of dependence of vh belongs to the cone Since rj > 0 is arbitrary and u is smooth, the assertion of the theorem follows. 2. It remains to show (6.2.4). First note that the operators DOJ are antisymmet- ric: Therefore, also the operators Q(t) defined in (6.2.2) are antisymmetric: 184 Initial-Boundary Value Problems and the Navier-Stokes Equations We give a simple bound for Q(t): The estimate implies that The smooth solution u = u(x, t) of the differential problem satisfies Thus we can consider u^t) as a solution of perturbed difference equations: Here ^v(i), gv are bounded independently of h and k. The difference w — u — v solves Taking the inner product with wv(t + fc) -f wv(t — fc) and summing over j/, we find that The Cauchy Problem for Systems in Several Dimensions 185 If one defines the auxiliary quantities then one obtains the recursive estimate It is important to observe that This implies a lower bound for L(t), and therefore the above recursive estimate gives us that Hence the auxiliary quantities L(t) also satisfy Initially, at t = k, we have and by a discrete analog of Gronwall's Lemma (Lemma 3.1.1) it follows that Since L(t) bounds ||u>(t)|j^ , the convergence (6.2.4) is proved. 186 Initial-Boundary Value Problems and the Navier-Stokes Equations The existence result mentioned above for symmetric hyperbolic systems can be extended to more general hyperbolic equations. Consider the first-order system (6.2. la) where the operator P(x,t,d/dx) has the form (6.2. Ib), but the matrices Bj(x, t) are not necessarily symmetric. We call the symbol of the differential operator P and make the Definition. The system (6.2.1) is called strongly hyperbolic if there exists a smooth 1 -periodic symmetrizer, i.e., there is a 1 -periodic positive definite Hermitian matrix function depending smoothly on all variables, with This definition generalizes the conditions discussed in Section 2.4.1 for the multi-dimensional constant-coefficient case and the conditions in Section 3.3. 1 for the one dimensional variable-coefficient case, but it requires in addition smooth dependence on (x, t, u). (In the constant-coefficient case smooth depen- dence on u> was not essential.) If we require that all frozen-coefficient problems obtained from (6.2.1) be strongly hyperbolic in the sense of Section 2.4.1, then we can use the matrices of Theorem 2.4.1 and obtain the symmetrizer H = S*5; compare Lemma 2.4.2. The only ext assumption in the above definition is the requirement of periodicity and s oothness of H — H(x1t1u)). The matrix S~l(x. t,u>) contains in its columns the eigenvectors of P{x. t. iu}). Thus, if the eigenvalues of P(.T, t, iu>) are all purely imaginary and if linearly independent eigenvectors can be chosen as smooth periodic functions of (x,t,uu), then the system is strongly hyperbolic. This is the case if the system is strictly hyperbolic, i.e., the eigenvalues of P(x, t, iu;) are purely imaginary and always distinct. One can prove Theorem 6.2.2. If the system (6.2.1) is strongly hyperbolic, then the Cauchy problem is well-posed. For every set of smooth data F = F(x,t). f = f(x) The Cauchy Problem for Systems in Several Dimensions there exists a unique smooth solution. Estimates as in the one dimensional case are valid; i.e., the solution is as smooth as the data. Sketch of the proof. We use the theory of pseudodifferential operators and refer to Eskin (1981, Chapter 5) or Nirenberg (1973). To conform with the usual framework, we do not assume that the data and coefficients are 1 -periodic. Instead, we assume that F, / € C°° and that the coefficients are constant outside a bounded set. Because of the finite speed of propagation, this is no restriction. Let 0j = 0j(x) denote a partition of unity where the support of $j has diameter dj and sup^ dj is sufficiently small. Also, at every point x at most p functions 0, are different from zero, p independent of x. We define the pseudodifferential operator H(t) by For sufficiently small dj defines a scalar product which is equivalent to the usual LI -scalar product. To prove this, we define and note that This ensures that can be estimated from below by const ||v||. Let 0 = 0(|u;|), 0 e Cx, denote a "cut-off function with We write (at each t)187 188 Initial-Boundary Value Problems and the Navier-Stokes Equations where With these notations, Again, HP 4- P*H* is a pseudodifferential operator. The calculus of pseudod- ifferential operators implies that its symbol has the form The part multiplied by 0 vanishes, and 5(x,t,u;) is bounded. Hence, HP + P*H* is a bounded operator. Because the factor 1 — 0(|u;|) in the definition of H cuts off high frequencies, the operator HP is also bounded. Therefore one obtains that and the basic energy estimate follows. 6.3. Mixed Hyperbolic-Parabolic Systems and the Linearized Navier-Stokes Equations There are no problems in obtaining existence and uniqueness results, and esti- mates of smooth solutions for mixed hyperbolic-parabolic systems of the form Here we assume that the uncoupled systems are second-order strongly parabolic and first-order symmetric hyperbolic, re- spectively. The coupling terms #12 and RI\ are general first-order operators. The Cauchy Problem for Systems in Several Dimensions 189 The proof proceeds as in the one dimensional case; we refer to Section 3.4.1. The solutions are as smooth as the data. One can also generalize this result to the case where the uncoupled systems (6.3.2) are second-order parabolic and first-order strongly hyperbolic, respec- tively. The proof for this more general case requires that one modify the L2- norm to obtain the estimates. We shall now discuss how these results apply to the linearized Navier-Stokes equations. The (nonlinear) viscous compressible system reads Suppose U(x, t), P(x, 0, #(x, t) are smooth functions with R > 0 and P = r(R); these functions are considered as an exact or approximate solution of (6.3.3). We substitute into (6.3.3) and neglect all terms which are quadratic in the corrections u', p', p'. If we drop the ' sign in our notation, we obtain the linear equations The inhomogeneous terms Gj, G2 are determined by U, P, R and vanish if U, P, R solve (6.3.3) exactly. Clearly, the equations (6.3.4) are of the general form (6.3.1) with u — u, v = p. For // > 0, // > 0, the second-order operator 3 190 Initial-Boundary Value Problems and the Navier-Stokes Equations is strongly parabolic since Here j/j e R3 is arbitrary, and y'l) is the i-th component of j/j. Thus the uncou- pled u-equation in (6.3.4) is strongly parabolic as long as R(\,t) is bounded. The uncoupled p-equation in (6.3.4) is symmetric hyperbolic since it is of first- order and scalar. We consider also the inviscid case /* = /*'= 0. Except for zero-order terms and forcing functions, one obtains the first-order system (2.4.2) where 17, V, W, R are now (known) functions of (x, t). The symbol PI(X, t , iu;) equals and the symmetrizer is Thus the linearized inviscid compressible equations form a strongly hyperbolic system if K = (dr/dp) (R) > 0. Again, as in the one dimensional case, the symmetrization of the system is equivalent to the introduction of a scaled density. 6.4. Short-Time Existence for Nonlinear Systems Local (in time) existence results for nonlinear strongly hyperbolic, parabolic, and mixed systems in s space dimensions can basically be derived in the same way as in the case of one space dimension. However, the proofs become technically more difficult because the Sobolev inequalities depend on the number s of space dimensions. We first treat nonlinear symmetric hyperbolic systems in The Cauchy Problem for Systems in Several Dimensions 191 detail. Then corresponding existence results for parabolic and mixed systems are stated. 6.4.1. Nonlinear Symmetric Hyperbolic Systems Consider a symmetric hyperbolic problem As in the one dimensional case, we assume uniform bounds for the (real) coef- ficients and their derivatives, i.e., for all arguments u G Rn. These global bounds can be replaced by local bounds in a neighborhood of the initial data if one applies the cut-off technique described in Section 5.2. We start with an a priori estimate of ||w||//s+2. As it turns out, if this norm is estimated in some time interval, then higher derivatives can be bounded in the same interval. In the proofs we repeatedly use the Sobolev inequality (see Theorem A.3.6) It implies that Here [s/2] is the largest integer < s/2, and c is a numerical constant independent of w, v. Lemma 6.4.1. Suppose that u solves (6.4.1) in 0 < t < T. Then where C depends only on the constants KQ, ..., A's+2 o/(6.4.3). Proof. Differentiation of (6.4.1) yields 192 Initial-Boundary Value Problems and the Navier-Stokes Equations where are multi-indices with a + a' = v and Therefore, the time derivative can be estimated in terms of together with The symmetry Bj = Bj and integration by parts yield and thus One fin for |i/| < s + 2, To estimate a term (6.4.5), we apply the chain rule and write Here each Ca consists of partial derivatives of the coefficient B} w.r.t u (and Ca acts as an r-linear map on D"1 u . . . D°ru.) According to assumption (6.4.3), the Cff are bounded. Consider first the case Then (by Sobolev's inequality) The Cauchy Problem for Systems in Several Dimensions 193 since j^l + [s/2] + 1 < s + 2. Therefore, Secondly, let thus (Observe that |a| + \/3\ = \v\ + 1 < 5 + 3.) Then and we find that If each \ai\ is < s + I - [s/2], then (6.4.6) applies; hence If, say, |<TI | > s + 1 - [s/2], then all other \a,\ satisfy \ai\ < s + 1 - [s/2], and therefore Summarizing, the estimate is shown, and the lemma is proved. The following result is an immediate implication of the previous lemma. Corollary 6.4.2. There is a time T > 0 depending on \\f\\H°+2' but not on higher derivatives of f, with the following property: If u solves (6.4.1). (6.4.2) inO<t<T, then 194 Initial-Boundary Value Problems and the Navier-Stokes Equations Proof. If / = 0, then u = 0. Thus let / ^ 0, and suppose that y(t) solves where C is determined as in Lemma 6.4.1. Then (by Lemma 4.1.2) for some T > 0. We shall now show that all higher derivatives of the solution can be estimated in the same time interval 0 < t < T. The existence of a smooth solution is assumed. Lemma 6.4.3. Let T = T(\\f\\H,+2) denote the time of Corollary 6.4.2. For every p = 0,1,2,..., there is a constant Mp, depending on \\f\\H'+2+"> with Proof. For p = 0 the result is shown. Assuming it is proven up to p - 1, we shall prove it for p > 0. To this end, let \v\ = s + 2 + p. As in the proof of Lemma 6.4.1, we can bound in terms of (6.4.4) and (6.4.5). Since we already have the bound we find that It remains to estimate By Sobolev's inequality we have that The Cauchy Problem for Systems in Several Dimensions 195 Case 1. Then Case 2. |Q| + [s/2] = s + 1+ p\ thus \/3\ + [s/2] + \<s + 2 + p. Then and it remains to estimate Case 2a. all \Vi\ < s + p - [s/2]. The estimate of (6.4.8) follows by (6.4.7). Case 2b. \a\ \ > s + p - [s/2]. Then a\ — a, r = 1 , and thus Case3. \a\ + [s/2] >s+ 1+p; thus \0\ -f- [s/2] + 1 < s + 1+ p. In this case (D^ujoo < const by (6.4.7). Case 3a. all 1^) < s + p - [s/2]. We have bounded the term (6.4.8) in Case 2a. Case 3b. \a\ \ > s + p - [s/2]. We cannot have |<72| > s + p — [s/2] since otherwise Thus l/^uloo, » ^ 1, is bounded by (6.4.7), and To summarize, we have shown that 196 Initial-Boundary Value Problems and the Navier-Stokes Equations If we use the notation then and the induction step can be completed. We shall now sketch the proof for the existence of solutions. To this end, we consider the sequence of functions uk = uk(x, t) defined by the iteration Note that each function uk is determined by a ///iiar equation. In the same way as in Section 4.1.4, we can show the analogues of Lemma 4.1.5 and Lemma 4.1.6. The techniques to obtain the estimates are the same as used in the proofs of Lemma 6.4.1 and Lemma 6.4.3. Lemma 6.4.4. For a suitable time T\ = Ti(||/||j/.+2) it holds that Lemma 6.4.5. For each j = s + 3, s + 4, . . . there exists Kj with where T\ is determined in the previous lemma. As in Section 4.1.4, we can use the uniform smoothness of the sequence uk in 0 < t < T\ and employ Gronwall's Lemma to show the existence of a C°° -solution in 0 < t < T\. Uniqueness of a solution follows as in the one dimensional case; see Lemma 5.1.1. Theorem 6.4.6. Consider the \-periodicCauchy problem (6 A. I), (6 A. 2). The problem has a unique C°° -solution u = w(x, 0, defined for 0 < t < T. The time T > 0 depends on ||/||//-+2. but not on higher derivatives of f. The Cauchy Problem for Systems in Several Dimensions 197 6.4.2. The Compressible Euler Equations In two space dimensions the equations without forcing read or in matrix form If r'(p) = ^i(p) > 0, then we can introduce a new density p by This transformation leads to a symmetric hyperbolic system. In 3D we can proceed in exactly the same way. 6.4.3. More General Nonlinear Systems The local existence result stated in Theorem 6.4.6 can be generalized without difficulties to Symmetric hyperbolic systems where BJ and F depend smoothly on all variables. One can also treat Strongly parabolic systems 198 Initial-Boundary Value Problems and the Navier-Stokes Equations and obtain an existence interval depending on € > 0. If the system becomes symmetric hyperbolic for e = 0, then an existence interval 0 < t < T, T independent of e > 0, can be established. For e — > 0 the solutions of the parabolic problem converge (along with all their derivatives) to the solution of the hyperbolic problem. Proofs of these results have been given in Chapter 5 for one space dimension, but all arguments generalize. Similarly, we obtain local existence for Mixed hyperbolic-parabolic systems Further generalizations to general parabolic systems, strongly hyperbolic sys- tems (with smooth symmetrizers) and mixed systems can be obtained by chang- ing the Z/2-norm. We refer to Sections 6.1.3 and 6.2.2. These results establish short-time existence for the compressible viscous or inviscid Navier-Stokes equations. In the inviscid case, the usual assumption of dr/dp > 0 for the equation of state p = r(p) is required to ensure hyperbolicity. 6.5. A Global Existence Theorem in 2D To begin with, we consider a parabolic system where the first-order terms are in so-called self-adjoint form: (In this section e is arbitrary but fixed.) As before, we assume an initial condition According to Section 6.4, there is a time T > 0 and a C°° -solution u defined for 0 < t < T. We use the notation and start with a simple a priori estimate. The Cauchy Problem for Systems in Several Dimensions 199 Lemma 6.5.1. Suppose that u is a smooth solution defined for 0 < t < T. Then there are constant K\ , K2 wit Proof. Integration by parts gives us Integrating w.r.t. t, we find that and the lemma is proved. Now consider a parabolic system where A, B £ C00 are not necessarily symmetric. We assume that the coeffi- cients grow at most linearly for large arguments: The following lemma contains the key result of our global existence theorem. Lemma 6.5.2. Suppose that u is a smooth solution of (6.5.5), (6.5.2) defined forQ<t<T. If we assume the bounds (6.5.3), (6.5.4), (6.5.6), then are finite. Proof. Using (6.5.5), we obtain 200 Initial-Boundary Value Problems and the Navier-Stokes Equations One shows easily that and therefore, In Appendix 3 (see (A.3.7)) we prove the special Sobolev inequality Together with assumption (6.5.3) we obtain that Let us now show the bound for J\(t). Clearly, and if we set then This yields and with assumption (6.5.4) we conclude that Thus we have proved a bound for J\ (t). The Cauchy Problem for Systems in Several Dimensions 201 If we use (6.5.7) again and observe that J\ < ||Au||2, then follows. The special Sobolev inequality implies that / lemma is proved. Let us assume that also the first derivatives of the coefficients in (6.5.5) grow at most linearly for large arguments: Then we can apply the same technique to the differentiated differential equation (6.5.5). As a result, one obtains Theorem 6.5.3. Suppose that u € Cx is a solution of (6.5.5), (6.5.2) defined for 0 < t < T. We assume the bounds (6.5.3), (6.5.4) for u = u(x,t) and the bounds (6.5.6), (6.5.8) for the coefficients of the differential equation. Then are finite, and consequently u can be extended as a smooth solution beyond T. If a priori bounds (6.5.3), (6.5.4) (with Kj = Kj(T)) hold for any finite time interval, then u exists for all time. Proof. Differentiate (6.5.5) w.r.t. x and w.r.t. y to obtain (To obtain the above inequality, we observe thatand the 202 Initial-Boundary Value Problems and the Navier-Stokes Equations The inequality for (d/dt) J\ yields and the desired bound follows from Lemma 6.5.2. This process can be con- tinued, and by induction we obtain bounds for all derivatives. The remaining statements can be shown by the general arguments given at the beginning of Section 5.4. Notes on Chapter 6 Petrovskii (1938) proved the existence of solutions of the Cauchy problem for strictly hyperbolic equations. The proof was simplified by Leray (1953), who constructed a symmetrizer. Kreiss (1963) generalized the construction to the case where the algebraic multiplicity of the eigenvalues of the symbol P(x, £, iw) does not change. In this case the symmetrizer can be chosen as a quotient of differential operators. The smoothness of the symmetrizer is a major problem when the multiplicity of the eigenvalues changes. Besides the trivial case - when the system is symmetric and H = / - no general theorem is known. Recently Clarke Hernquist (1988) was able to construct a smooth symmetrizer for a particular class of symbols with changing multiplicity. The class of cor- responding differential equations was introduced by John (1978). For the con- struction of the symmetrizer for parabolic systems see also Mizohata (1956) and Kreiss (1963). 7 Initial-Boundary Value Problems in One Space Dimension In applications the interesting phenomena frequently occur near the boundary, and consequently the formulation of boundary conditions plays an important role. In this chapter we treat problems in one space dimension with an interval as spatial domain. After discussion of the heat equation as a specific example, more general parabolic systems will be considered in Section 7.2. The energy method in its discrete and continuous form will be employed to show well-posedness under Dirichlet and Neumann boundary conditions. For more general boundary conditions, the Laplace transform method is the appropriate tool, and we present it in detail in Sections 7.4 and 7.5. If a determinant-condition is satisfied then the problem is strongly well-posed in the generalized sense. Important concepts of well-posedness for initial-boundary value problems are discussed in Section 7.3. For hyperbolic equations the characteristics play, of course, an eminent role in determining correct boundary conditions: Values for the ingoing character- istic variables must be provided. If this is the case and if the solution is not overspecified at the boundary, then the hyperbolic problem is well-posed. Also, we will derive boundary conditions for mixed hyperbolic-parabolic systems and will apply the results to the linearized (compressible) Navier-Stokes equations. 203 204 Initial-Boundary Value Problems and the Navier-Stokes Equations A unified view of all results which can be obtained by the energy method is given in Section 7.8: The energy method applies if the spatial differential op- erators are semibounded on the space of functions obeying the (homogeneous) boundary conditions. Throughout this chapter we restrict ourselves to linear problems, but - as in the periodic case - nonlinear equations could again be treated by an iterative process. The presence of boundary conditions does not lead to essentially new difficulties. We refer to Chapter 8 for some further remarks. 7.1. A Strip Problem for the Heat Equation Consider the heat equation in the strip 0 < x < 1, t > 0. As we did earlier, we prescribe an initial condition Here / is assumed to be real. In addition, we require the boundary conditions We want to obtain a solution u(x, 0 which is smooth, including at the "cor- ners" (x, t) = (0,0), (x,t) = (1,0) of the strip, and need compatibility condi- tions to be satisfied. Before discussing this further, let us try to find a solution using a series expansion. Solution in series form. The first step is to construct special solutions of the differential equation which satisfy the boundary conditions. These are functions in separated variables: Introducing into (7.1.1) and (7.1.3), we find that u(x) must be a solution of the eigenvalue problem Initial-Boundary Value Problems in One Space Dimension 05 FIGURE 7.1.1. Strip problem for the heat equation. The solutions of (7.1.5) are given by and therefore the special solutions (7. 1 .4) read Any finite sum of these functions clearly satisfies the differential equation and boundary conditions. If the given initial function f(x) can be expanded in a seres which converges sufficiently fast, then is a classical solution of the initial-boundary value problem (7.1.1M7.1.3). Uniqueness. For simplicity we consider only real-valued functions and define the Z-2-scalar product and norm by 206 Initial-Boundary Value Problems and the Navier-Stokes Equations The rule of integration by parts reads Suppose u and v are two solutions of the initial-boundary value problem (7.1.1)- (7. 1 .3). We obtain, for w = u — v, i.e., ||w(-,t)||2 < ||w(-,0)||2 = 0. Thus w = 0, and the solution is unique. Compatibility conditions, generalized solutions, smoothing. Suppose the strip problem (7.1.1M7.1.3) has a solution u(x,t) which is Cx -smooth in 0 < x < 1, t > 0, i.e., up to the boundary and the initial line. This implies that / e C30, and the initial data are compatible with the boundary conditions: Differentiating the boundary conditions with respect to t and using that we find that This yields necessary conditions for / if we want a solution which is C°°- smooth in 0 < x < 1, t > 0. The easiest way to comply with these conditions is to assume that / 6 C°° is identically zero in a neighborhood of x = 0 and x = 1. Let One can show: If / € 0^(0,1), then the formula (7.1.6) defines a solution u(x,t) of the strip problem, which is (7°°-smooth in 0 < x < 1, t > 0. An estimate as in the uniqueness argument given above shows that Initial-Boundary Value Problems in One Space Dimension 207 Since the set CQC(O, 1) is dense in 1/2, we obtain a generalized solution for all initial functions / € L2 by the usual extension of the solution operator. One can show that the generalized solution is Cx -smooth for 0 < x < 1, t > 0. Indeed, the formula (7.1.6) remains valid for / e 1/2, t > 0. For all / 6 Z/2 the solution u(x,t) depends even analytically on the argument (x.t) for t > 0. This follows from the exponential decay of the coefficients in (7. 1 .6). A priori estimates of derivatives. We want to give another existence proof using a difference approximation. First let us derive the a priori estimates for the solution. Suppose that u(x,t) solves the strip problem and is C^ -smooth in 0 < x < 1 , t > 0. We have seen that Differentiation of (7.1.1) and (7.1.3) with respect to t gives us for v = ut: Therefore, Repeating this process, we find estimates for all time derivatives and the space derivatives of even order. Sobolev inequalities imply bounds for the remaining space derivatives, and estimates for mixed derivatives follow from the differen- tial equation. The difference scheme. We shall now approximate the strip problem by an ordinary initial value problem. Let AT be a natural number, let h = \/N denote the gridsize, and let x,, = i/h. v = 0, . . . . N, denote the gridpoints. The gridf unction v(t) with components will approximate u(xv,t). We replace (7.1.1), (7.1.2) by 208 Initial-Boundary Value Problems and the Navier-Siokes Equations The boundary conditions (7.1.3) are replaced by The equations (7. 1 .8) can be used to eliminate VQ and v.\ from (7. 1 .7), and one obtains an ordinary (linear) initial value problem for v\, . . . , iw-i- Thus there is a unique solution v(t) = vh(t) of the above difference approximation. Estimates for vh(t) and the limit h — > 0. We shall estimate all difference- differential quotients of v = vh independently of h. To this end, let us introduce some notation. If v, w are (real) gridfunctions, their discrete L2 -scalar product and norm* are defined by In analogy to integration by parts we have Lemma 7.1.1. For all gridfunctions i>, w it holds that Proof. The formula follows from In the next lemma, we let v(t) = vh(t) denote the solution of the difference approximation defined above. Lemma 7.1.2. Assume that f e C£°(0, 1) and define f(x) = Ofor x £ (0; 1). For every q = 0, 1 , 2. . . ., there is an ho > 0 such that (Here fh denotes the initial function restricted to the proper grid.) '''Since the expressions (7.1.9) ignore the boundary data VQ, vt\, etc.. we obtain a scalar product and norm on the space of gridfunctions defined on the interior grid x\ , . . . , x\ _ | . Initial-Boundary Value Problems in One Space Dimension 209 Proof. Using the discrete boundary conditions and Lemma 7.1.1, we find that and therefore, This proves the statement for q = 0. Now let w — dv/dt. Differentiation of (7. 1 .7) and (7. 1 .8) with respect to t yields The initial condition for w reads Here tv(0) = /^, ^ = 1, . . . , AT — 1. Furthermore, using the boundary condi- tions to determine VQ, VN, we obtain that if h is small enough. This shows that and therefore, Thus we obtain, as before, This process can be continued and the lemma follows. The discrete norms ||(£>+D_)«/ft||A approach \\d^f/dx^\\ as /i -> 0, and thus they are bounded independently of /*. Therefore we have estimates for all time derivatives of vh(t) which are independent of h. Interpolation leads to a solution of the strip problem for h —»• 0: 210 Initial-Boundary Value Problems and the Navier-Stokes Equations Theorem 7.1.3. Assume that f € C^O, 1). Then the initial-boundary value problem (7.1.1 H7.1.3) has a unique solution Proof. Note that implies that In general, If we introduce the notation then Thus we have bounded the divided differences with respect to x. By the results of Appendix 2 we can interpolate the gridfunctions v£(t) with respect to x and obtain C00 -functions wh(x,t) which are uniformly (with respect to /i) smooth. In the same way as in the periodic case, we can send h — * 0 and apply the Arzela-Ascoli theorem to obtain a C00 -solution u of the strip problem. Smoothness of the generalized solution for t > 0 . Let / 6 1/2 . and consider a sequence /* € C£°(0, 1) with By definition, the corresponding solutions uk converge to the generalized solu- tion u belonging to the initial function /. For t > 0, we can derive estimates of the derivatives of uk which depend only on ||/fc|| but not on derivatives of fk. As before, this implies smoothness of the limit u for t > 0. To show the estimates, we first recall that Initial-Boundary Value Problems in One Space Dimension 211 i.e., Also, (Note that u*x(0,0 = uf(0,f) = 0.) Using (7.1.10), integration from 0 to t gives us that This process can be continued, and we can bound in terms of ||/fc||2. Then we can argue as in Section 3.2.6 and obtain Theorem 7.1.4. Let f e 1,2, and let u denote the generalized solution to the strip problem. The function u is a C00 -smooth in 0 < x < 1, t > 0; i.e., u is smooth up to the boundary for t > 0. 7.2. Strip Problems for Strongly Parabolic Systems In this section we consider parabolic systems in the strip 0 < x < 1, t > 0. At time t = 0 we give initial data As boundary conditions we require n linearly independent relations between the components of u and ux at each boundary point x = 0, x = 1; i.e., the boundary conditions have the form 212 Initial-Boundary Value Problems and the Navier-Stokes Equations with constant n x n matrices L,-n, Lj\. The n x 2n matrix (Ljo, Lj\) has rank n for j = 0 and j — 1 since the boundary conditions are linearly independent. The matrix coefficients A, B, C in (7.2.1) are assumed to be C°° -smooth. Furthermore, we require that for some 6 > 0; i.e., the system (7.2.1) is symmetric parabolic. For the initial function / we assume that All functions and matrices are taken as real, for simplicity. The constants ci , 02, etc. introduced below will depend on the time interval 0 < t < T, where T is arbitrary but fixed. Extensions. The reader can generalize all arguments from the real to the com- plex case and assume, instead of (7.2.4), This generalization (to a strongly parabolic system) essentially requires one to replace 2(u, Auxx) by (Auxx, u) + (u, Auxx) in the arguments given below. We refer to Section 3. 1 . Also, without difficulty, one can add a smooth forcing F = F(x, t) in (7.2.1). Furthermore, the coefficients in the boundary conditions could depend on t. This would make the proofs technically slightly more complicated, however. Solutions in series form. Boundary conditions of the general form (7.2.3) can be motivated as follows: If the coefficients A, B, C do not depend on time, then we can try to solve the strip problem by the same technique as in the last section and first construct special solutions in separated variables. The ansatz leads to the eigenvalue problem For the ordinary differential equation (7.2.5), the above boundary conditions are well established. In fact, under quite general additional assumptions, one can show that the eigenvalues form a sequence Afc with Re A* — > — oo, and that an arbitrary / € 1/2 can be expanded in terms of the functions which span the Initial-Boundary Value Problems in One Space Dimension 213 invariant eigenspaces. If this is the case, the strip problem can be solved as in the last section in series form. 7.2.1. Solution-Estimates under Various Boundary Conditions The basic energy estimate. If the coefficients A, B, C are time dependent, the above approach of series expansion does not easily apply. When can we expect a unique solution of the strip problem (7.2.1M7.2.3)? In order to derive sufficient requirements on the boundary conditions, let us suppose that u is a C°° -solution and let us try to derive the basic energy estimate. Recall the rule of integration by parts: Equation (7.2.1) gives us that In order to obtain the basic energy estimate, we arrive at the following condition, which we call the Requirement for an energy estimate. For all functions w = w(x). w 6 C00, which satisfy the boundary conditions the estimate holds.* Here c may be dependent on T but not on w. *The value 6/2 of the coefficient of ||ifx||2 can be changed to any other positive value without altering the requirement. This will follow from the considerations below. 214 Initial-Boundary Value Problems and the Navier- Stokes Equations Before we discuss this requirement further, we shall treat the important special cases of Dirichlet and Neumann boundary conditions. The above requirement, which leads to the basic energy estimate, does in fact imply the existence of a unique smooth solution u of the strip problem. We shall prove this below using a difference approximation. Dirichlet conditions. The simplest boundary conditions are the so-called Dirichlet conditions If they are imposed, then the boundary term in (7.2.7) is zero. Neumann conditions. These are of the form If w fulfills (7.2.6), then and we can use the Sobolev inequality (see Appendix 3) to show (7.2.7). Mixed conditions. Clearly, we can also require a Dirichlet condition at one boundary point and a Neumann condition on the other. If ,4(0, t), say, is di- agonal, we can use a Dirichlet condition for some components of u(0, t) and a Neumann condition for the others, If .1(0, 0 is not diagonal but has a complete set of eigenvectors, then we can introduce new dependent variables u so that the matrix .4(0. t) becomes diagonal. For u we can impose boundary conditions of the form described. The general case. To discuss the general case of boundary conditions (7.2.3), first note that the requirement imposed by such a condition remains the same if we multiply the conditions with a nonsingular matrix on the left. For example, if Lj\ is nonsingular, we can assume a Neumann condition at x — j\ and if LJI =0, we can assume a Dirichlet condition at x = j. If rank Lj\ = r,, 1 < r j ; < n — 1 , we can assume that Initial-Boundary Value Problems in One Space Dimension 215 Partitioning LJQ correspondingly, we "split" the boundary conditions into a Neumann and a Dirichlet part: The following result from linear algebra will be applied at each boundary point. Lemma 7.2.1. Let A G Rn •" be arbitrary and let L0,L, € Rn<" be of the form where rank L{ = r, rank (Lo, L\) = n. The following conditions are equivalent: (i) There exists a constant c > 0 such that for all w, wx € Rn which satisfy (ii) Ifa.be Rn are vectors with then Proof, (i) =» (ii): Let a, 6 satisfy (7.2.10), and choose wx e R" with Considering the vectors we find that Since (7.2.9) is fulfilled for any /3, there exists - by assumption - a bound of the right-hand side which is independent of 0. This implies (a, Ab} = 0. (ii) => (i): Let w, w x satisfy (7.2.9) and decompose* *ker L = {« | Ls — 0} is the kernel or nullspace of L. By 5X = {s\ (s, v) — 0 for all v € «S} we denote the orthogonal complement of a subspace S. 216 Initial-Boundary Value Problems and the Navier-Stokes Equations Clearly, L^w = 0, and therefore (ii) implies that If si, . . . , sn_r denotes a basis of kerZ/f , then w% fulfills the matrix equation The system-matrix is nonsingular; thus one can solve for w", and the estimate (7.2.8) follows. Remark. Condition (ii) states geometrically that the kernel (= nullspace) of LQ! is orthogonal to the /1-image of the kernel of L{. Since ker Ltf has dimension r and ker L{ has dimension n — r, and since A is nonsingular in our application, we obtain that the equality is an equivalent formulation of (ii). Formally, condition (ii) is also applicable for Dirichlet and Neumann conditions characterized by and respectively. If the boundary conditions are neither of Dirichlet nor of Neumann type, then (7.2.12) puts a severe restriction on A. We apply the previous lemma to show Lemma 7.2.2. There is an energy estimate (see (7.2.7)) if and only if implies that Proof. First assume that (7.2.14) holds for all vectors with (7.2.13). By the previous lemma, Initial-Boundary Value Problems in One Space Dimension 217 The proof of Lemma 7.2.1 shows that c(j, t) can be bounded uniformly for 0 < t < T. An application of Sobolev's inequality shows (7.2.7). Conversely, suppose that (7.2.7) holds for all functions w with (7.2.6). Since - for each fixed t - the vectors w(j,t), wx(j,t), j = 0.1, can be prescribed arbitrarily, it is necessary that an estimate holds. (The right-hand side of (7.2.7) cannot be used to bound a term \wx(j. t)\ .) Another application of Lemma 7.2.1 finishes the proof. For later purposes we show Lemma 7.2.3. Suppose the matrix function A satisfies (7.2.4) and suppose the requirement for an energy estimate to be valid. Then the n x n matrices are nonsingular. Proof. Otherwise there exists a vector aj ^ 0 with Condition (7.2.14) implies that in contradiction to (7.2.4). A priori estimates of the solution and its derivatives. Suppose that the bound- ary conditions are such that (7.2.7) holds; i.e., there is an energy estimate. Furthermore, assume that u is a C°° -solution of the strip problem. Clearly, and thus 218 Initial-Boundary Value Problems and the Navier-Stokes Equations We want to show that we can also estimate the derivatives of u; we set v = ut. Differentiation f (7.2.1), (7.2.3) yields Thus, except for the inhomogeneous term P/w, the time derivative v = ut satisfies the same differential equation and boundary conditions as u. By (7.2.1), Using the Sobolev inequality we obtain that i.e., As before, w* havfi 4lKady A t^und far ||u||, w; attain u estimate fcr v = ut. By (7,2,15) the space derivatives U T, uxi are also bounded, Repeated differentia- tion with respect to t gives us the desired estimates. We summarize the result in Lemma 7.2.4. Suppose that the boundary conditions are such that the re* quirement for an energy estimate is fulfilled. Given any nonnegative integers p, q and any time T > 0, there exists a constant K = /C(p, ?, T) such that in 0 < t < T. The constant K is independent of f 6 C£°(0, 1). Initial-Boundary Value Problems in One Space Dimension 219 7.2.2. Existence of a Solution via Difference Approximations The difference scheme. The notations for the gridsize h = 1 JN, the gridpoints XQ, . . . , x,v, etc., are the same as in the last section. We replace the strip problem (7.2.1H7.2.3)by The discretized boundary conditions can also be written in the form We assume the requirement for an energy estimate to be valid. Then it follows from Lemma 7.2.3 that the above equations can be solved for vo(t). vu(t) if h is sufficiently small. Thus we can eliminate VQ and VN from (7.2.16). Therefore the difference scheme has a unique solution v(t) = vh(t) for 0 < h < ho. The basic estimate. In the following we estimate the solution vh(t) of the difference scheme independently of h; we start with Lemma 7.2.5. There is a constant K and a step-size ho > 0 with Proof, a) We remind the reader of the notations the discrete 'integration-by-parts rule' 220 Initial-Boundary Value Problems and the Navier-Slokes Equations and the product formula b) The difference equations (7.2.16) yield where Since the pair v0(t), D+v0(t) satisfies the boundary conditions at x = 0, we obtain, from (7.2.8), Also, Finally. ||J%u|U i ll/>4-u|U. and Therefore we have the estimate Using the discrete Sobolev inequality (see Appendix 3, formula (A.3.15)) Initial-Boundary Value Problems in One Space Dimension 221 we obtain A similar estimate holds for \vN\2. Choosing e = 6/(16c4), we find that This proves the basic estimate for all sufficiently small h. Estimates of derivatives, existence, smoothing. In a similar way as in the last section, by considering the difference equations satisfied by w = dv/dt, we also can estimate derivatives. Lemma 7.2.6. Suppose that f e C^°(0, 1) and define f(x) = Qfor x $ (0, 1). As before, assume the requirement for an energy estimate to be valid. Given any nonnegative integer q and any T > 0, there is a constant K and a step-size ho > 0 with Proof. The function w = dv/dt satisfies the difference equations and the same boundary conditions as v. The initial data are Here ^(0) = f(xv}, v - 1, . . . , N - 1, by (7.2.17). Using (7.2.18) one obtains, for sufficiently small h, and thus We can proceed in exactly the same way as for the a priori estimate of ut and show that 222 Initial-Boundary Value Problems and the Navier-Stokes Equations Thus the extra term Qt(xv, t)vv(t) poses no problem, and we obtain the desired estimate for w = dv/dt. The lemma follows by repeated differentiation with respect to t. Clearly, we can use the time derivatives to estimate the divided differences independently of h. Therefore, as in the last section, we can use interpolation to obtain existence of a solution for h — > 0. We summarize the main result in Theorem 7.2.7. Suppose that f e Co°(0, 1), and assume that the boundary conditions are such that (7.2.7) is valid for all functions with (7.2.6); i.e., we have the basic energy estimate. Then the strip problem (7.2.1)-<7.2.3) has a unique solution u. The solution is C°° -smooth in 0 < x < 1, t > 0. The solution and its derivatives can be estimated in terms of f and its derivatives. For all initial data / € £2 there is a unique generalized solution. The smooth- ing properties of parabolic equations remain valid for the initial-boundary value problem. The proof proceeds in the same way as in the last section by showing an estimate of in terms of ||/||. The generalized solution is a C°° -function in 0 < x < 1, t >0. 7 A DigcuisSon of Concepts of In this section we give a somewhat informal discussion of different concepts of well-posedness for initial-boundary value problems. We do not restrict our- selves to the case of one space dimension and consider a system of differential equations with initial data and boundary conditions Initial-Boundary Value Problems in One Space Dimension 223 Here P is a linear spatial differential operator whose coefficients may depend smoothly on x and t, and L is a linear operator combining values of u and its derivatives at the boundary. In general, L may also involve time derivatives of u. The boundary of the domain ft C Rs is assumed to be smooth. We consider F, /, and g as the data of the problem, the operators P and L and the domain ft being fixed. Roughly speaking, the initial-boundary value problem (7.3.1)-(7.3.3) is called well-posed if for all smooth compatible data F, /, and g there is a unique smooth solution u, and in every finite time interval 0 < t < T the solution can be estimated in terms of the data. Clearly, to arrive at a definition, we must specify the norms which enter the estimates.* In the following we shall use the notation For example, in case of the strip problem in one space dimension we have The Cauchy problem. Let us recall the definition of well-posedness for the spatially periodic Cauchy problem or the pure Cauchy problem. In those cases well-posedness required an estimate of the form for 0 < t < T. Using Duhamel's principle, one can replace such an estimate by estimates for the case F = 0. We have used this for constant-coefficient opera- tors P in Chapter 2. Also, one could develop a theory of well-posed problems where the estimate (7.3.4) is weakened in the following way: First, without much restriction, one can assume that f(x) = 0 because the transformation *It is not essential to make the smoothness assumptions for F, /, g, and u more precise. The reader can always replace "smooth" by C°°. Once estimates are derived for this case, more general - less smooth - data can be treated by approximation as long as the norms for the data are defined. One obtains a generalized solution. 224 Initial-Boundary Value Problems and the Navier-Stokes Equations leads to zero initial data. Then, instead of (7.3.4), one could require an estimate Such a concept of well-posedness would be satisfactory in the sense that "no derivatives are lost". If F 6 CQ°, then all derivaties of u vanish on the initial line t — 0, and - by differentiating the differential equation (7.3.1) - one obtains equations like (7.3.1) for the derivatives of u. Only the lower order terms will have changed. Thus, as we have seen, in many cases one can derive estimates like (7.3.5) for the derivatives of u in terms of the derivatives of F. Clearly, (7.3.5) is - at least formally - a weaker requirement than (7.3.4). However, for the pure Cauchy problem or the periodic Cauchy problem not much seems to be gained by such a generalization of well-posedness. We do not know of any problem for which (7.3.5) holds but (7.3.4) is not valid. (If the coefficients of P are allowed to become singular for t = 0 it seems likely that (7.3.5) can indeed become a truly weaker requirement.) Strongly well-posed problems. Consider now the initial-boundary value prob- lem (7.3.1M7.3.3). Generalizing (7.3.4) we give Definition 1. The initial-boundary value problem (7.3.1H7.3.3) is called strongly well-posed if for all smooth compatible data F, /, g, there is a unique smooth solution u, and for every finite time interval 0 < t < T there is a constant KT such that in 0 < t < T. The constant KT may not depend on F, /, or g. The above estimate holds for parabolic equations with Neumann boundary conditions in any number of space dimensions (see Section 8.1) and for hyper- bolic equations in one space dimension. One can give examples to show that there are real difficulties in estimating the boundary term Initial-Boundary Value Problems in One Space Dimension 225 in more than one space dimension for hyperbolic problems. The failure to estimate this term is not due to the techniques employed. Homogeneous boundary conditions. We can always transform to homoge- neous boundary conditions by constructing a function ^ = ^(ar, 0 with Then v = u - ijj satisfies the homogeneous condition Lv = 0. This motivates Definition 2. Consider the initial-boundary value problem (7.3. 1 H7.3.3) with g = 0. We call the problem well-posed if for all smooth compatible data F and / there is a unique smooth solution u, and we have, instead of (7.3.6), There are technical difficulties in working with this concept of well-posedness. To illustrate these, assume we have obtained the basic estimate (7.3.7), for ex- ample by integration by parts. We would like to show similar estimates of the derivatives of the solution in terms of the derivatives of the data. To this end, we differentiate the given equations (7.3.1) and the boundary conditions (7.3.3) with respect to t and in the tangential directions to obtain equations of a similar type for the derivatives. In this process, however, inhomogeneous boundary terms will appear, in general, if the boundary operator L has variable coeffi- cients. These can be subtracted out by other functions ifr, as before. However, derivatives of ip appear as inhomogeneous terms in the differential equation, and in the resulting estimate we "loose" derivatives; i.e., we need higher derivatives of the data to bound lower derivatives of the solution. This is intolerable if one wants to go over to nonlinear problems, for example. To summarize, the above concept of well-posedness (with g = 0) leads to technical difficulties if the differentiation of the boundary conditions introduces inhomogeneous terms. An example. To illustrate the foregoing, consider the following simple example 226 Initial-Boundary Value Problems and the Navier-Stokes Equations The data are assumed to be compatible. We try to derive the basic estimate and proceed as in Section 7.1: The boundary term is Using Sobolev inequalities, the term originating from an inhomogeneous Neumann condition can be estimated prop- erly: However, the term originating from an inhomogeneous Dirichlet condition cannot be treated. We can only derive energy estimates if we first transform to homogeneous Dirichlet conditions. To this end, choose a function o> £ £*** with Then satisfies homogeneous Dirichlet conditions at x = 0. Therefore v can be es- timated in terms of the data. (The first derivative of go is needed but this is not important.) We can proceed as in Section 7.2 to estimate derivatives of v. No problems occur here since the coefficient which defines the homogeneous condition, Initial-Boundary Value Problems in One Space Dimension 227 is constant. Thus the relation remains homogeneous if we differentiate it to derive a condition for vt. Another example. Consider a parabolic strip problem Here Ljo(t), -Lji(t), j = 0, 1, are n x matrices, and rank (Ljo(t), Lj\(t)) = n. Neumann boundary conditions. If Lji(t),j = 0,1, are nonsingular for all t > 0, there are no difficulties in deriving an estimate of the form (7.3.6). We refer to Section 8.1.2 where we treat this situation in more than one space dimension. The above strip problem is strongly well-posed if the boundary conditions allow us to express ux(j, t) in terms of u(j, 0, 9j(t). Dirichlet conditions. Another extreme case is Lj\(t) = 0,j = 0,1. The boundary conditions read After transforming to homogeneous conditions as indicated above, we obtain the basic estimate and estimates of derivatives. Mixed conditions. If the boundary conditions contain a Dirichlet and a Neu- mann part, we must transform the Dirichlet part to become homogeneous with a constant coefficient matrix. Then, if we can derive the basic estimate, we can also estimate derivatives. Strongly well-posed problems in the generalized sense. Instead of transform- ing to homogeneous boundary conditions, it is also of interest to study the case / = 0. (This will become more apparent below, when we will use the Laplace transform in time; defining u(x, t) = 0 for t < 0, we obtain a continuous function u only if u(x,0) = f(x) = 0 .) Definition 3. Consider the problem (7.3.1M7.3.3) with / = 0. We call the problem strongly well-posed in the generalized sense if for all smooth compatible data F and g there is a unique smooth solution u and we have, instead of (7.3.6), 228 Initial-Boundary Value Problems and the Navier-Stokes Equations The theory of strongly well-posed problems in the generalized sense is well developed for parabolic and hyperbolic systems. In case of constant coefficients, there are necessary and sufficient conditions of an algebraic nature. For variable- oefficient problems the estimates are valid if all relevant frozen-coefficient equations are strongly well-posed in the generalized sense. As we shall show, initial-boundary value problems for strictly hyperbolic and symmetric hyperbolic systems which are strongly well-posed in the generalized sense are also strongly well-posed in the sense of Definition 1 . For parabolic systems such a result does not hold. As mentioned before, the assumption / = 0 does not pose technical problems if one wants to estimate derivatives. We need only to assume that F vanishes identically in a neighborhood of t = 0; then all derivatives of u are zero at t = 0. The set of all smooth F vanishing identically near t = 0 is still dense in Weakly well-posed problems. Finally one can weaken the required estimate further by assuming that both the initial function / and the boundary data g vanish. We give Definition 4. The problem (7.3.1H7.3.3) with / = 0, g = 0 is called weakly well-posed if for all F 6 CQ° there is a unique smooth solution u and, instead of (7.3.6), we have At present, there is no general theory available for weakly well-posed prob- lems. We anticipate that any such theory would be extemely complicated. 7.4. Half-Space Problems and the Laplace Transform One aim of this section is to indicate the limitations of the energy method in a discussion of well-posedness. We shall employ the Laplace transform in time to solve parabolic initial-boundary value problems and shall show that these problems are strongly well-posed in the generalized sense if and only if aL2 Initial-Boundary Value Problems in One Space Dimension 229 certain determinant condition is fulfilled. Using this technique, one can decide the question of well-posedness in the generalized sense. In contrast to this, if the boundary conditions are neither of Dirichlet nor of Neumann type, the energy method applies only in exceptional cases. 7.4.1. Parabolic Half-Space Problems*; Main Result Consider an equation in the domain 0 < x < oo, t > 0, under n linearly independent boundary conditions at x = 0, and an initial condition Assumptions. The n x n matrices A, LQ, L\ are assumed to be constant; results for variable coefficients are stated at the end of this section, without proof however. Throughout, the assumption of strong parabolicity is essential. Concerning the boundary conditions: If rankl/i = r, then we can assume, without restriction, that the matrices are of the form where LQ, L{ have size r x n, and the boundary conditions take the form (The cases of a Dirichlet condition, r = 0, and a Neumann condition, r = n, are formally included in our discussion. However, the reader may focus attention on the mixed case where 1 < r < n - 1; only in the latter case do the Laplace transform method and the energy method differ significantly.) As previously, it is not a severe restriction to put strong smoothness assump- tions on the data, namely '"Since we work in one space dimension, the half-space is actually only the half-line 0 < x < oo in this section. 230 Initial-Boundary Value Problems and the Navier-Stokes Equations Once solution-estimates are derived under these assumptions, more general data can be treated by approximation. Solution concept. We seek a solution for which all derivatives are 1/2 -functions of 0 < x < oo at each time t: This requirement can be thought of as a boundary condition at x = +00. The main result. Before giving further motivations, let us formulate the main result of this secti ; it will be proved below via Laplace transformation. Hence- forth, we assume that the matrix A has n distinct eigenvalues /**. We transform A to diagonal form introduce the notation and de the square matrix Theorem 7.4.1. If del CQ ^ 0, then the parabolic initial-boundary value problem (7.4. l)-(7.4.3) has a unique solution. For f = 0, the problem is strongly welhpascd in the generalized sense. The same result Is valid {with exactly the same determinant condition} for an equation with lower-order terms, 7.4.2. The Energy Method Assume that the problem (7.4.1)-<7.4.3) has a solution u and let g11 = 0. We try to obtain a solution-estimate by the energy method; thus we consider the time derivative of the energy: Initial-Boundary Value Problems in One Space Dimension 231 Since the value |wx(0, i)|2 cannot be bounded in terms of the Z/2-norms we must bring the boundary conditions into play. The proof of the following result, which slightly generalizes Lemma 7.2.1, is left to the reader. Lemma 7.4.2. Let A, Ln, L, e Cn'n, where LQ, L{ have size r x n, rankLJ = r, rank(Lo,£i) = «• The following conditions are equivalent: (i) There exists c> 0 JMC/I (ii) //"a, 6 G Cn are vectors with 232 Initial-Boundary Value Problems and the Navier-Stokes Equations If the matrices A, LQ, L\ meet the requirements of the previous lemma, then and we obtain an estimate of the solution in terms of the data. More precisely, in 0 < t < T. One can also show the existence of a smooth solution, and therefore the problem is strongly well-posed in the sense of Definition 1, Section 7.3. If the conditions of Lemma 7.4.2 are not met, then the energy method does not apply, and the question of well-posedness may be investigated by other means only. Example. To illustrate the differences, we apply Theorem 7.4.1 and the energy method to the following simple example: Here The matrix CQ reads with del CQ = a — 2; thus Theorem 7.4.1 applies whenever a •£ 1. Initial-Boundary Value Problems in One Space Dimension 233 When does the energy method apply? We check condition (ii) of Lemma 7.4.2 and find Thus the energy method applies only for a = -|. This example is typical for parabolic problems under mixed boundary condi- tions: The energy method applies only in exceptional cases whereas the Laplace transform method applies in most cases, and yields well-posedness in the gen- eralized sense. What can be said if detCo = 0, i.e., if Theorem 7.4.1 does not apply?- We will show in Section 7.5 that the problem indeed becomes ill-posed in the generalized sense; again, this result remains valid if lower-order terms Bux + Cu are added to the differential equation. 7.4.3. An Eigenvalue Problem Suppose that F = 0, g = 0 in (7.4.1H7.4.3) and substitute into the equations. One obtains a solution* if and only if (Note that all derivatives of u(-,0 are in 1,2 if / € LI\ this follows from sf = Afxx.) Considering s € C as a parameter, we have obtained an eigenvalue problem and give Definition 1. A number s G C is called an eigenvalue of (7.4.5) if - for the number s in question - there is a non-trivial solution / = /(x), / e C°° D LI. Using elementary means of linear algebra, we shall discuss the constant- coefficient problem (7.4.5) below. First let us indicate the relevance of eigen- values for the question of well- or ill-posedness. Lemma 7.4.3. If there is a sequence sv of eigenvalues with Re sv —> oo, then the problem (7.4.1)-(7.4.3) is ill-posed in any sense. Proof. If such a sequence s v exists, then there is no bound on the exponential growth rate (in time) of genuine solutions. *Recall that the r uirement u(-,t) (E LI is part of our solution-concept; consequently we exclude all solutions f of sf = Af xx which grow exponentially as x —> oo. 234 Initial-Boundary Value Problems and the Navier-Stokes Equations Now we solve the eigenvalue problem (7.4.5) explicitly; the reader is reminded of the notations (7.4.4). For any s € C, Re s > 0, the general solution of is given by* Clearly, If we introduce these expressions into the (homogeneous) boundary conditions (7.4.3), we obtain a linear system for the vector a 6 Cn. After division of the first r equations by — ,/s, the system takes the form Thus we have shown Lemma 7.4.4. A number s e C, Re s > 0, is an eigenvalue of (7.4.5) if and only if the determinant of the matrix (7.4.6) vanishes. These results motivate Theorem 7.4.1 (but they do not prove it): If Re a —r oo, then the matrices (7.4.6) converge to Co; consequently, if det Co ^ 0, there ore no eigenvalues with arbitrarily large real parts. If one had a converse of Lemma 7.4.3, then well-posedness would follow. Unfortunately, a complete proof of Theorem 7.4.1 is more elaborate. A simple conclusion can be drawn, however, from the results shown: Lemma 7.4.5. Let detC0 = 0, and assume that LQ = 0, i.e., that the Neumann part of the boundary conditions has no lower-order terms. Then the problem (7.4.1>-(7.4.3) i'j ill-posed. Proof. All numbers s £ C, Re s > 0, are eigenvalues. The result follows from Lemma 7.4.3. *For any complex number c with Re c > 0 let \fc denote the root with Re \/c > 0. Initial-Boundary Value Problems in One Space Dimension 235 7.4.4. The Laplace Transform; Elementary Properties In this section we introduce the Laplace transform u = u(s) of a function u = u(t) and prove some elementary results, which will be needed below. Essentially all the results follow quite easily from corresponding properties of the Fourier transform. To avoid confusion, we use here the following notation for the Fourier transform of v = v(t): Laplace transform vs. Fourier transform. Suppose u = u(t), 0 < t < oo, is a continuous function with values in Cn, which satisfies an estimate for some real constants C, a. The analytic function* is the Laplace transform of u. The Laplace and Fourier transform are closely related; to explain the relation, we set and note that each function has the Fourier transform Therefore we have the relation The Fourier inversion formula gives us *The integral is defined componentwise. 236 Initial-Boundary Value Problems and the Navier-Stokes Equations and thus we have shown the following inversion formula for the Laplace trans- form: Applications of Parseval's relation. For a function i> = v(t) and its Fourier transform TV = (.FvXf) it holds that In terms of the Laplace transform, the formula reads As a consequence one obtains - for each finite time T - an estimate of u in terms of its Laplace transform: In our applications below we consider continous functions Suppose that u satisfies an estimate then denotes the Laplace transform in time. We want to show Lemma 7.4.6. Under the above assumptions let u(-,t) € LI for each t > 0, and assume we have a bound TVien Initial-Boundary Value Problems in One Space Dimension 237 and consequently, (Under our assumptions the integrals exist and are finite.) Proof. The result follows if one integrates the relation with respect to x and interchanges the order of integration. Integrability of the function and Fubini's theorem justify the interchange. Laplace transform of time derivatives. Suppose that u — u(t). 0 < t < oo, is continuously differentiable and Then integration by parts gives us that Future does not affect past. We will need a result which states - roughly speaking - that we may alter data g(t) fort>T without changing the solution for t < T. For illustration, we consider an ordinary initial value problem where L e Cn n, L + L* < 2{3I, and g = g(t) is continuous with Laplace transformation gives us and thus 238 Initial-Boundary Value Problems and the Navier-Stokes Equations Using the inversion formula (7.4.7), we obtain an explicit expression for the solution u(t). Since g(s) depends on all values g(t), one might be tempted to conclude from this formula that each value u(to) depends on all values g(t), too. However, as is well-known for the above initial value problem, if we alter the data g(t) for t>T, the solution u(t) remains unchanged for 0 < t < T. A slight generalization of this result is contained in Lemma 7.4.7. Suppose that u — u(t), g = g(t), 0 < t < oo. are continuous functions bounded in norm by some exponential ceat, and suppose that their Laplace transforms satisfy an estimate for some constants c\, a\ > a. If g(t) = Ofor 0 < t < T, then u(t) = Ofor Proof. We fix a time T\ < T and use (7.4.9) for For 77 —> oo we obtai Since T\ < T is arbitrary and u is continuous, the result follows. Estimates of the Laplace transform. Let tion by pans gives usThen integra- Initial-Boundary Value Problems in One Space Dimension 239 This process can be repeated. Therefore, for any p = 1 , 2, . . . , there is a constant Kp = Kp(g) independent of s with In other words, as \s\ — » oo, Res > 0, the function |^(s)| decays faster than any power \s\~p. 7.4.5. Solution via Laplace Transform Consider the parabolic initial-boundary value problem (7.4.1)-(7.4.3) under the assumptions stated in Section 7.4.1. In addition, we assume first that / = 0 and LQ = 0; thus the Neumann part of the boundary condition at z = 0 has no lower-order terms. If the matrix Co introduced in (7.4.4c) is singular then, according to Lemma 7.4.5, the problem is ill-posed, and consequently we assume that detCo ^ 0. We shall construct a solution via Laplace transformation and derive solution-estimates which imply strong well-posedness in the generalized sense. Estimate of the Laplace transform. To begin with, assume that there is a solution u = u(x, t) which is "well-behaved" in the sense that all the following operations are permitted. The solution formula derived below can subsequently be used to justify this assumption. Laplace transformation gives us We prove the following estimate of u in terms of P and g: Lemma 7.4.8. There is a constant c independent of the data F = F(x, t) and g = g(t) with 240 Initial-Boundary Value Problems and the Navier-Stokes Equations for all s € C, Res > 0. (Here and in the following, the powers of \s\ have significance for large |s|, not for s « 0.) Proof, a) For brevity we set p = v/s, |arg p| < f . Introducing the vector we can write the second-order differential equation for u(-,s) as a first-order system: The boundary conditions become The system-matrix can be diagonalized; using the notation (7.4.4), we set and obtain (note that d>~'.4<I> = A~2) In the new variable the system is diagonal: Initial-Boundary Value Problems in One Space Dimension 241 The boundary condition at x = 0 reads where Thus the matrices BQ, B\ have size n x n, and is nonsingular since det Co ^ 0 by assumption. Therefore we obtain that b) With a number d > 1 , to be specified below, we set The differential equation (7.4.1 1) yields for each fixed 5, Re s > 0, Since all diagonal entries A^ of A have positive real part and since |argp| < there is 8\ independent of p with and therefore On the other hand, using integration by parts and the boundary condition (7.4.12), we find that 242 Initial-Boundary Value Problems and the Navier-Stokes Equations Here the constant c\ depends only on the matrices BQ, B\ in (7.4.12). If we choose d > 1 -f c{ and combine the two estimates, we obtain that Thus there is a constant c2 independent of the data and of 5 € C, Res > 0, with In terms of the original variables u, g1, g11, F, the desired estimate follows, and the lemma is proven. Estimate of the solution. The previous lemma gives us an accurate bound of the Laplace transform u of the solution u in terms of the Laplace transforms F, g1, g11 of the data F, g1, g11. We can now apply the results of Section 7.4.4 - which were themselves simple applications of Parseval's relation - to estimate u in terms of F and g1, g11. One obtains Lemma 7.4.9. First assume that g11 = 0. For each time T > 0 there is a constant KT, independent of the data F, g1, with If g11 is not necessarily zero then Proof. Assume that g11 = 0 and consider, for example, the integral of |wx(0, t)|2. We fix 77 > 1 and obtain, from (7.4.9), Initial-Boundary Value Problems in One Space Dimension 243 According to Lemma 7.4.7, the integral on the right-hand side can be bounded by Here by (7.4.10a). The integral of |<77|2 is estimated similarly using (7.4.8). In this way one obtains a bound by Now we change the data in such a way that they vanish for t > T + e, e > 0. Reasoning as in the proof of Lemma 7.4.7, we note that the solution remains unaffected for t < T; since e > 0 is arbitrary, the result follows. Existence of a solution. Thus far we have not shown the existence of a "well-behaved" solution for which the above computations are justified. On the Laplace-transform side, existence can be shown quite easily. All we need is the following simple existence result for scalar equations on the interval 0 < x < oo. Lemma 7.4.10. Consider a scalar equation where Re A > 0, q G C00. \\q\\ < oo. The problem has a unique solution v = v(x) with ||v|| < oo. Proof. All solutions of the differential equation have the form If we set then v(x) and therefore, 244 Initial-Boundary Value Problems and the Navier-Stokes Equations Integration in x yields For an initial value v(0) = v0 different from the one given above, the solution shows exponential growth, hence is not in LI. Now consider the diagonal system (7.4.11) with boundary condition (7.4.12) at x = 0; here s € C, Re s > 0, is fixed. Using the previous lemma, we find that the side condition ||u>(-,s)|| < oo determines a unique solution w(x,s). Then, reversing the transformations used in the proof of Lemma 7.4.7, we obtain u(x, s). The inverse Laplace transform (7.4.7) gives us By assumption, g e CQ°(() < t < oo). Hence g(s) is an analytic function of 5, Re s > 0, and the same is true for u(x, s). Also, u(x, s) is a smooth function of x, and \u(x, s)\ decays rapidly for \s\ —> oo. Therefore, This shows that u is a solution of the differential equation. Clearly, u satisfies the boundary condition. It remains to show that u(x, 0) = 0. By Residue Calculus we can choose any positive value for 77 without affecting the result u = u(x, t) of the inverse Laplace transform. Therefore, This completes the proof of Theorem 7.4.1 for / s 0 under the assumption that there are no lower-order terms, i.e., Lower-order terms. The case of nonzero lower-order terms can be treated in much the same way. For the vector Initial-Boundary Value Problems in One Space Dimension 245 one obtains a system where and A/o is the matrix introduced in the proof of Lemma 7.4.8. Since the eigen- values of MO are all different, one can diagonalize M(p) by a transformation The boundary conditions for take the form with Thus, if |p| is sufficiently large, the estimates for w follow as before; the crucial condition remains unchanged. In this way, we obtain well-posedness in the generalized sense for problems with lower-order terms. Thus far we have restricted ourselves to homogeneous initial data / = 0, however. Inhomogeneous initial data. If / ^ 0, we can introduce the function which satisfies homogeneous initial conditions at t = 0. Unfortunately the derivative fxx enters the forcing term of the differential equation for v, and therefore, if we apply Lemma 7.4.8 in a straightforward way, we need to estimate the solution u. 246 Initial-Boundary Value Problems and the Navier-Stokes Equations It is possible, however, to derive an estimate in terms of ||/||2 only. For simplicity we restrict ourselves to an equation and assume boundary conditions (7.4.3) with g = 0, L!Q — 0. (Again, lower order terms could be treated by a perturbation argument.) Laplace transformation of (7.4.13) yields thus the function /(x) enters the discussion in the same way as the function F(x, s) did previously. If det Co ^ 0 we find, as in Lemma 7.4.7, Using (7.4.9) for fixed rj > 0, T > 0, we obtain an estimate of the solution on the boundary x = 0, The time integral of ||u(-,£)||2 can be treated in the same way using (7.4.106). To summarize, we have shown Lemma 7.4.11. // det CQ ^ O7 F = O, g = 0, ihen the solution u — M(OS, t) of the initial-boundary value problem satisfies 7.4.6. Extensions We want to sketch some extensions of the previous results without giving the proofs in detail. Strip problems. The results can be extended to strongly parabolic systems in a strip Initial-Boundary Value Problems in One Space Dimension 247 with boundary conditions of the form (7.4.3) on either side of the strip, i.e., at x — 0 and x = 1 : To check for well-posedness, one considers now the two matrices instead of the single matrix (7.4.4c); one can show that the strip problem is well-posed in the generalized sense if and detC,^0. To sketch a proof of this result, we note that the general solution of is given by The vectors ao, o\ have to be determined by the boundary conditions. If we substitute the above function u(x.s) into the boundary condition at x = 0, we note that a\ contributes a term which is exponentially small for large Re s. Similarly, the contribution of GQ is exponentially small at x = 1. For this reason, if Res is sufficiently large, the vectors OQ = <TO(S), CTI = ai(5) are uniquely determined. To derive the proper solution-estimate, we can basically proceed as in the proof of Lemma 7.4.7. Only the scaling-matrix D = D(x) must be chosen more carefully: Here d(0), d(l) are positive smooth functions for which the ratios are sufficiently large (depending on the boundary conditions). 248 Initial-Boundary Value Problems and the Navier-Stokes Equations Variable coefficients. The previous results can be generalized to parabolic sys- tems with variable coefficients A = A(x, t), etc. Also, the coefficients of the boundary conditions are allowed to depend smoothly on time, Assuming that A(x, i) can be diagonalized at x = 0, x = 1, we obtain matrices Cj(0> j' — 0> I- If t"e determinant condition is fulfilled, then the initial-boundary value problem is well-posed in the gener- alized sense. (See also the remarks at the end of Chapter 8.) 7.5. Mildly Ill-Posed Half-Space Problems Consider a parabolic problem in 0 < x < oo, t > 0, with initial condition and boundary conditions at x = 0, All matrices are constant (in general, complex), A, B, C have size n x n, LQ . L\ have size r x n, and LQ] has size (n-r) x n. We make the reasonable assumption that L\ and LQ* have full rank in order to have n linearly independent boundary conditions. Furthermore, we assume for simplicity that the eigenvalues of A are distinct. First we want to use the ideas of the previous section to derive a formal solution fo ula. Second, we will show that the problem is not strongly well-posed in the generalized sense (see Definition 3 of Section 7.3) unless Here Initial-Boundary Value Problems in One Space Dimension 249 Hence there are two different ways in which a problem (7.5.1M7.5.3) can become ill-posed: First, there can be a sequence of eigenvalues s^ with Re $„ — > oo; see Lemma 7.4.3. Second, no such sequence exists, but det Co = 0. In the second case the ill-posedness is of a milder nature: One can still estimate the solution if one uses derivatives of the data. In contrast to the weakly ill-posed Cauchy problems of Chapter 2, a perturbation by lower-order terms still does not lead to arbitrarily fast exponential growth. 7.5.1. Formal Solution Laplace transformation of (7.5.1)-(7.5.3) gives us For each fixed s with sufficiently large real part, we want to solve this constant- coefficient problem. The general solution of the differential equation (7.5.4) with side-conditon (7.5.6) reads Here Kk = Kk(s) are the roots of and 0fc = 0fc(s) G C" are corresponding eigenvectors, (It is assumed that Res > 0 is so large that the 2n roots K of the above characteristic equation are distinct and exactly n of them have negative real parts.) Perturbation arguments show that and one can normalize the eigenvectors 0jt(s) so that 250 Initial-Boundary Value Problems and the Navier-Stokes Equations If we introduce the notations then we can write for short and the boundary conditions (7.5.5) take the form There are two possibilities: (i) There is a sequence «„ 6 C with Re «„ — » oo for which the above system- matrix is singular. Then we can argue as in Section 7.4.3 and show that there is no bound on the exponential growth rate (in time) of genuine solutions. Therefore the initial-boundary value problem is ill-posed in any sense. (ii) If Re * is sufficiently large then the matrix in (7,5,8) is nonsmgular, In this case let denote the solution of (7.5.8). The function u(x,s) defined in (7.5.7) solves (7.5.4)-(7.5.6); inverting the Laplace transform, we have formally solved the half-space problem. There arc no difficulties in extending the formal solution process to inhomo- geneous differential equations and inhomogeneous initial conditions. If all data are sufficiently "well-behaved" then - by the same arguments as before - the formal solution is a genuine C°° -solution. Nevertheless, as we will prove in the next section, the problem is ill-posed in the generalized sense if det CQ = 0. 7.5.2. Necessity of the Determinant Condition Using the previous notations, we show Theorem 7.5.1. The half-space problem (7.5.1M7.5.3) is not well-posed in the se f Definition 3, Section 7.3, // detQ) = 0. Initial-Boundary Value Problems in One Space Dimension 251 Proof, a) W ivide the I-part system (7.5.8) by -p = -v"s and obtain By assumption, Co is singular; furthermore, taking Res sufficiently large, we may assume that (Otherwise we are in case (i) mentioned in the previous n.) From the rank-assumption for Lf , L^1, it follows that there exists a vector b) Let us outline the rest of the proof. We will construct a sequence of scalar functions and define data by For these data we solve (7.5.9) and obtain vectors (r(t/)(ri + i£) with According to (7.5.7), we obtain solutions u(t/)(x, rj + iO of (7.5.4H7.5.6), and the inverse Laplace transform gives us functions u(j/)(x, t). We will show that whereas the data satisfy c) The lower bound (7.5.10) is a consequence of the following perturbation result from linear algebra. Lemma 7.5.2. Suppose that CQ is a singular n x n matrix and q £ range CQ. Let 252 Initial-Boundary Value Problems and the Navier-Stokes Equations and solve Then Proof. There is a nonsingular matrix P such that where A) has full rank, say rank DQ = m. There is a component j with since otherwise Pg e range (PCo). If 60 > 0 with (7.5.1 1) did not exist then for some sequence e -»• 0, and the j»-th equation of would yield a contradiction. d) To continue the proof of Theorem 7.5.1, we define a sequence of functions Then the Z/2-norm of 7(l/) equals 1, and therefore According to (7.4.9), the inverse Laplace transforms g("\t) have bounded Z/2- norms over 0 < t < 1, say. Now invert the Laplace transform of data Initial-Boundary Value Problems in One Space Dimension 253 and obtain Thus one finds If one observes the lower bound (7.5.10) and notes that the supports of a(v) are confined to small intervals for large z/, then one obtains a similar result for Also, for each fixed v we can apply arbitrarily small changes to obey compati- bility conditions at t = 0, and the theorem is proved. 7.6. Initial-Boundary Value Problems for Hyperbolic Equations 7.6.1. The Method of Ch teristics In this section we consider hyperbolic systems in the strip 0 < x < 1, t > 0 with initial data At each boundary point x = 0, x — 1 we prescribe (inhomogeneous) linear relations for u\ i.e., we consider boundary conditions of the general form 254 Initial-Boundary Value Problems and the Navier-Stokes Equations Specific assumptions about the matrices LO, L\ will be derived below. The coefficients and data functions are assumed to be C°° -smooth with respect to all variables. Hyperbolicity of the system (7.6.1) requires the eigenvalues Aj(z, t) of B(x, t) to be real and assumes the existence of a smooth transformation 5 = S(x, t) such that In the interior 0 < x < 1 the eigenvalues may change sign. However, we assume that have a constant sign as a function of time; i.e., each function (7.6.4) is either > 0 for all t, =0 for all t, or < 0 for all t. If we introduce new variables (so-called characteristic variables) then the system (7.6.1) transforms to an equation where B is replaced by A. To simplify notation, we assume that the given system is written in characteristic variables already, thus B = A in (7.6.1). The case of n scalar equations. To discuss the system, we use the method of characteristics and start with the case C = F = 0. The differential system separates into n scalar problems Thus uj{x,t} ia constant along the cnoracteriitici (i{ft,ti defined by The case of constant \j. Assume first that Then the characteristics are straight lines, and thus Let M+, Tio, M- consist of the variables Uj corresponding to indices j with Aj > 0, \j = 0, \j < 0, respectively. Clearly, UQ is determined by the initial data Initial-Boundary Value Problems in One Space Dimension 255 FIGURE 7.6.1. Strip with characteristics. but we need boundary conditions to determine u+ and u_. Acceptable boundary conditions are i.e., we prescribe the ingoing characteristic variables at each boundary. If the initial function f(x) and the boundary data </o(0, g\(t) are not compat- ible, then the solution will have discontinuities along the characteristics which start at the corners (x.f) = (0,0). (x.t) = (1,0). Thus, in contrast to the parabolic case, there is no smoothing during time evolution here. To avoid dif- ficulties connected with nonsmoothness, we will assume that the data /. g0. q} vanish near the corners. As usual, once solution estimates are derived for this case, the more general situation can be treated by a limiting process. The boundary conditions (7.6.6) can immediately be generalized to One says that the ingoing characteristic variables are described in terms of the outgoing ones. Boundary conditions for the characteristic variables UQ belonging to speeds Aj = 0 are neither necessary nor allowed. The case \j = \j(x, t). For simplicity we assume that all eigenvalues (7.6.4) are different from zero on the boundary; one says that the boundary is not characteristic. If an eigenvalue \j(x,t) changes sign as a function of x, then possibly the variable HJ belongs to a positive characteristic speed at x = 0 and to a negative speed at x = 1. Nevertheless, we use the notation u+ and u_ to assemble the variables iij with Aj > 0 and Aj < 0, respectively, at each 256 Initial-Boundary Value Problems and the Navier-Stokes Equations boundary point. As in the case of constant \j, we obtain a unique solution u(x, t) if the boundary conditions have the form (7.6.7). Also, as follows from Section 3.3.2, we can add a diagonal term and a forcing function F(x. t). The assumption for the boundary to be nonchar- acteristic is not essential. By solving scalar equations, one obtains Theorem 7.6.1. Assume that /, F, go, ond g\ vanish in a neighborhood of the corners (x, £) = (0, 0), (x, t) - ( 1 , 0). The system with initial and boundary conditions (7.6.2), (7.6.7) has a unique smooth solu- tion. The solution vanishes near the corners. 7.6.2. Solution Estimates If the coefficient C = C(x, t) is not diagonal we want to use the iteration to show the existence of a solution. To start with, we show the following basic estimate! Lemma 7.6.2. Assume that the boundary is not characteristic. For every finite time interval 0 < t < T there is a constant KT independent of f, F, <?0) 9\ with the following property: Ifu solves and satisfies (7.6.2), (7.6.7) then Initial-Boundary Value Problems in One Space Dimension 257 Proof. 1 ) To begin with, let us scale the variables u+ , u_ such that the matrices 5b(0, SKO become "small". Let and introduce new variables v = Du. Equation (7.6.9) transforms to and thus only lower-order terms change. The boundary conditions (7.6.7) be- come where D_(0. /) is a diagonal submatrix of D(0,t), etc. Clearly, the matrices Sj(t) become as small as we please by choosing D(x, t) appropriately. To sim- plify notation, we assume |5o| + |5j | to be sufficiently small from the beginning. 2) From (7.6.9) we obtain that Integration by parts gives us and thus 3) Consider, for example, the boundary point x = 1 and note that 258 Initial-Boundary Value Problems and the Navier-Stokes Equations Using the boundary condition at x = 1, we find that Here we have assumed S\ to be so small that A similar consideration applies at x — 0, and we obtain that The lemma follows by integration. We show next how to estimate derivatives of the solution of (7.6.9), (7.6.2), (7.6.7) if the solution is smooth; we let v = ux. Differentiation of (7.6.9) with respect to x gives us To obtain boundary conditions for v, we first differentiate (7.6.7) with respect to t, and similarly at x = 1. Then we use (7.6.9) to replace ut by u£: where Here Thus we have equations for v = ux which are of the same type as the equations for u. By Lemma 7.6.2 we can estimate in terms of <?o, g\ , /, F and their first derivatives. Initial-Boundary Value Problems in One Space Dimension 259 Differentiation of (7.6.9) with respect to t gives us, for w = ut, Therefore we can also estimate w = ut in terms of 50, 9\, /, F and their first derivatives. This process can be continued, and we obtain Lemma 7.6.3. Assume that the initial-boundary value problem (7.6.9), (7.6.2). (7.6.7) has a smooth solution and that the boundary is not characteristic. The derivatives of the solution satisfy estimates of type (7.6.10). 7.6.3. Existence of a Smooth Solution Assume that the compatibility conditions of Theorem 7.6.1 are satisfied and that the boundary is not characteristic. We consider the iteration (7.6.8). By Theorem 7.6.1 each function of the sequence is C°°-smooth. Also, and corresponding estimates for ^-derivatives hold. Thus, by Gronwall's Lemma 3.1.1 and Picard's Lemma 3.3.4, the sequence uk is uniformly smooth in any finite time interval. Furthermore, As before, Gronwall's Lemma and Picard's Lemma imply convergence of the sequence uk to a Cx -limit u. The limit u solves the initial-boundary value problem and satisfies the estimates of Lemma 7.6.3. We summarize: Theorem 7.6.4. Assume that the boundary is not characteristic and that the data go, g\, /, F are compatible at t = 0. The hyperbolic initial-boundary value problem (7.6.9), (7.6.2), (7.6.7) has a unique solution. The solution is a C°° -function, which satisfies the estimate (7.6.10). Similar estimates hold for all derivatives. If the initial and boundary data do not satisfy compatibility conditions, then we can approximate the data and go to the limit. In this way we can introduce generalized solutions. As the coupled case shows, these will - in general - 260 Initial-Boundary Value Problems and the Navier-Stokes Equations have discontinuities travelling along the characteristics which start in the corners of the strip. 7.6.4. The Strip Problem vs. the Half-Space and Pure Cauchy Problems Suppose that the coefficients B = A, C and the data F, / are defined for — oo < x < oo, t > 0. The finite speed of propagation for hyperbolic systems makes it possible to split the strip problem into two half-space problems and a pure Cauchy problem. To this end, let </>i 6 C°° denote a monotone function with and define First we consider the following three strip problems (here the index j = 1,2,3 numbers the problerrh not a component)! Clearly, solves the given system. The finite speed of propagation implies that there is a time interval 0 < t < T\, T\ > 0, where ui(x,t) = 0 for 5/6 < x < 1. Therefore, we can consider ui as the solution of the right half-space problem Initial-Boundary Value Problems in One Space Dimension 261 at least for 0 < t < T). Similarly, u2 and u^ can be considered locally as solutions of a left half-space and a pure Cauchy problem, respectively. These considerations are valid in some finite time interval 0 < t < TO; at time TO o has obtained a new initial function and can restart. Thus, in principle, we can solve the strip problem by solving half-space and pure Cauchy problems. The strip problem is well-posed if the corresponding half-space and pure Cauchy problems are well-posed. As in Sections 7.4, 7.5, one can employ the Laplace transform in time to discuss boundary conditions which are much more general than the conditions (7.6.7). We return to this technique in Chapter 8 to treat problems in more than one space dimension. 7.6.5. Equations Not in Characteristic Variables In many applications the differential equation (7.6.1) is not given in characteristic variables, i.e., the matrix B is not diagonal. To test for well-posedness under boundary conditions, one can try to derive energy estimates directly without diagonalizing the system. If B = B* then where Thus the boundary conditions have to provide bounds for Consider, for example, the boundary point x = 1 and assume that 5(1,0 has r positive and n — r negative eigenvalues; hence there are r ingoing and n-r outgoing characteristics. This suggests specifying r independent boundary conditions at x = 1 , for example, Using the boundary conditions, one can eliminate r variables Uj(l,0 in the quadratic term 262 Initial-Boundary Value Problems and the Navier-Stokes Equations Similar considerations apply at x = 0. If the remaining quadratic terms have the correct signs, then an energy estimate can be derived. In this way one has avoided the transformation to characteristic variables. A difficulty of this approach is that it provides only sufficient conditions for well-posedness. It might well happen that quadratic terms with the wrong sign appear, the problem being well-posed nevertheless. For the case where we first transform to characteristic variables, this is reflected in our additional assumption that |50(t)| + \S\(t)\ is sufficiently small; compare the proof of the basic estimate of Lemma 7.6.2. In applications hyperbolic differential equations often appear as second-order equations. By introduction of auxiliary variables, these can be written as first- order systems, however. We refer to Sections 2.1.4 and 3.3.3 for two simple examples. 7.7. Boundary Conditions for Hyperbolic-Parabolic Pr lems In this section we treat mixed systems in the strip 0 < x < 1 , t > 0, under initial and boundary conditions. For the uncoupled systems we assume a form as described in Section 7.2 (parabolic case) and Section 7.6 (hyperbolic case). Then we allow certain coupling terms in the differential equation and in the boundary conditions. The resulting systems are shown to be well-posed. We give an application to the linearized compressible Navier-Stokes equations. 7.7.1. The Basic Estimate for Mixed Systems Consider a parabolic system in the strip 0 T with boundary conditions For each fixed t we assume the conditions formulated in Theorem 7.2.7 to be fulfilled (see also Lemma 7.2.1). We allow the matrices Lj,, etc. to depend smoothly on t but assume that the rank T} of LJ, is constant. Initial-Boundary Value Problems in One Space Dimension 263 Consider further a hyperbolic system in the same domain with boundary conditions i.e., the ingoing characteristic variables are expressed at each boundary point in terms of the outgoing ones. (See Section 7.6.1 for notations.) We want to discuss the coupled system with boundary conditions and initial conditions The coefficients Bn = BU(X, 0, etc. and all inhomogeneous terms are assumed to be C°° -smooth. We first assume the existence of a solution and show Lemma 7.7.1. Suppose that the boundary is not characteristic for the hyper- bolic system; i.e., A(0, 0 and A(l, 0 are nonsingular. In any finite time interval T we can estimate in terms of Proof, a) First note that one can easily extend Lemma 7.2.1 to inhomogeneous equations as follows: If condition (ii) of the lemma is met and 264 Initial-Boundary Value Problems and the Navier-Stokes Equations then for some constant c. b) As explained in the proof of Lemma 7.6.2, we can assume without loss of generality that |S0| + |Si| is sufficiently small. We let c) First consider the w-equation (7.7.1), Integration by parts gives us that and that Using the Sobolev inequality we obtain that Initial-Boundary Value Problems in One Space Dimension 265 d) Second, the v-equation (7.7.2) yields Integration by parts and the boundary conditions imply e) We take a > 0 sufficiently large, and the above estimates give us that Another application of (7.7.8) and integration with respect to t finishes off the proof of the lemma. 7.7.2. Estimates of Derivatives and Existence of a Solution If we differentiate the u-equation (7.7.1) with respect to t we obtain Here uxx can be expressed by ut, etc. using (7.7.1) again. Thus uxx and ux can be treated like zero-order terms as in the proof of Lemma 7.2.4. The v-equation (7.7.2) is differentiated with respect to t and with respect to x. One obtains a system Boundary conditions for ut and vt follow by differentiation of the given bound- ary conditions with respect to t; then boundary conditions for vx are obtained if we replace vt by vx using the differential equation vt = Avx + — Thus 266 Initial-Boundary Value Problems and the Navier-Stokes Equations we have a system for (ut, vt, vx) which has the same structu as the given system for (u, v). Consequently, estimates for wt, ux, uxx, v(, x are derived. The process can be continued, and all derivatives can be estimated. If we assume that all data vanish in a neighborhood of the comers of the strip, then existence of a smooth solution can be shown by using the iteration The boundary conditions for u*+1 are the same as those for u except that v is replaced by vk in (7.7.3). Similarly, u is replaced by w* in (7.7.5), (7.7.6). This proves Theorem 7.7.2. Consider the hyperbolic-parabolic strip problem (7.7.1)- (7.7.7). Assume that the boundary is not characteristic for the hyperbolic part; i.e., A(0, 0 <*nd A(l, 0 are nonsingular. If the compatibility conditions are sat- isfied, then the problem has a unique smooth solution (u(x, t), v(x, t)). We can estimate the solution and its derivatives in terms of the data and their derivatives. 7.73. The Case of a Characteristic Boundary We now allow one or both of the boundaries x = 0, x = 1 to be characteristic for the v-equation The boundary conditions (7.7.5), (7.7.6) for v remain unchanged, hence none of the characteristic variables VQ appear in the boundary conditions for v. For the variable u we restrict ourselves to boundary conditions of Dirichlet type (If the boundary x — 0, say, is not characteristic, we can allow a boundary condition of the more general form (7.7.3), (7.7.4) at x — 0.) To show the basic estimate, we proceed as in the proof of Lemma 7.7. 1 . From integration by parts we obtain the boundary terms The first two of these terms are zero, the third term is treated as before: Initial-Boundary Value Problems in One Space Dimensi 267 Th we obtain an estimate of in terms of the data. Bounds for derivatives can be derived as before. To summarize: Theorem 7.7.3. The result formulated in Theorem 7.7.2 remains valid if one or both of the boundaries x = 0, x = 1 are characteristic provided that the boundary condition for u at a characteristic boundary is ofDirichlet type. With some restrictions, we found that coupled hyperbolic-parabolic systems are well-posed if the uncoupled systems have this property. It is possible to relax the assumptions of Theorem 7.7.3 further; however, there can be difficulties if the hyperbolic part is characteristic at the boundary and the parabolic part has a non-Dirichlet boundary condition. (The difficulties arise from the boundary term (u,B\2v). Actually, in one space dimension, the difficulties can be overcome since the characteristic variables satisfy ordinary differential equations along the boundary. This technique does not generalize to more than one space dimension, however.) In any concrete case, one can try to use the above tools to derive an energy estimate and estimates for derivatives. Also, the method of Laplace transform is applicable for discussing much more general boundary conditions. 7.7.4. Application to the Linearized Navier-Stokes Equations in ID Upon neglecting zero-order terms and forcing functions, the equations for w', p' derived in Section 3.5 read We assume that p.\ > 0. Clearly, the uncoupled equations are parabolic and hyperbolic, respectively. The base flow = U(x, t) is as- sumed to be known in the strip 8 Initial-Boundary Value Problems and the Navier-Stokes Equations Consider the boundary x — 0, for definiteness. We distinguish among three cases: 1) inflow at x = 0 : t/(0, t) > 0; 2) outflow at x = 0 : l/(0, t) < 0; 3) a wall at x = 0 : C/(0, t ) = 0. Our results do not cover a switch between inflow and outflow in time, since we assumed in Section 7.6 that the eigenvalues A,,(0, t) have constant sign. The general theory allows boundary conditions of the form given below. Case 1. (7(0, t) > 0. Here p' is an ingoing characteristic variable. We need two boundary conditions: and Case 2. U(0, t) < 0. In this case p' is an outgoing characteristic variable and need not be specified. For u'(0, t) we can take condition (7.7.9) or (7.7.10). Case 3. t/(0, t) = 0. The variable p' need not be specified. For u' we can take (7.7.9). Similar considerations apply at x = 1. By Theorem 7.7.3 the resulting strip problem for the coupled (u", ^-system is well-posed. Instead of the omoge- neous condition (7.7.9), one could also prescribe a smooth function The inhomogeneous case can be reduced to the homogeneous one by subtracting a suitable function from w'(x, t). 7.8. Semibounded Operators In the previous sections we have considered differential equations Initial-Boundary Value Problems in One Space Dimension 269 in the strip 0 < x < 1 , t > 0, together with an initial condition and boundary conditions. Restricting ourselves to the homogeneous time inde- pendent case, we write the boundary conditions in the form Here we assume that L, is a linear operator which combines values of u and its spatial derivatives evaluated at (x, t) — (j, £), j = 0, 1. In most cases our proofs of an energy estimate followed from an inequality of the type Thus the critical question is whether we can show an estimate if u satisfies the boundary conditions. We shall now formalize the procedure to some extent. For every fixed t the differential expression defines an operator Pi if we specify its domain of definition D. Let us define i.e., D consists of all C°°-functions which satisfy the boundary conditions. The following definition formalizes our requirement: Definition 1. We call the operators Pi , t > 0, semibounded on D if there exists a constant a such that for all * > 0 and all w 6 D. Clearly, if the operators P* are semibounded on D and if the strip problem has a solution u with u(-, t) € D for each t, then our considerations show Thus uniqueness and the basic energy estimate follow. 270 Initial-Boundary Value Problems and the Navier-Stokes Equations One might be tempted to believe that the existenc of a solution u with can also be shown once the operators P*, t > 0, are semibounded on D. How- ever, such a result cannot be valid without further assumptions: If we change the domain D to D\ c D by requiring additional boundary conditions, then the corresponding operators P{ are still semibounded; any solution u with would satisfy the additional boundary conditions. Obviously, this cannot be expected. In other words, a general existence result of the above type cannot be valid since one might have overspecified the solution at the boundary. This motivates the following: Definition 2. Suppose that P*, t > 0, is semibounded on a domain D of the form (7.8.1) and that D is determined by q linearly independent boundary conditions. Suppose further that q is minimal; i.e., if we specify a domain D\ by less than q boundary conditions, then the corresponding operators P\ are not semibounded. In this case, one calls P* maximal semibounded on D. (This concept is motivated by the next theorem. It is not sufficient to request that Pi lose semiboundedness on all domains D\ strictly larger than D. For example, suppose that integration by parts leads to a boundary term u(ux +uxx). If one requests the two boundary conditions ux — uxx — 0, then one cannot drop any condition. The correct condition is u = 0, however.) In the examples which we discussed previously the operators P* were indeed maximal semibounded; there was no overspecification and - under suitable compatibility conditions - we could show the existence of a solution, One can prove the following general result: Theorem 7.8.1. Consider a system of the form with boundary conditions Suppose that Ap(x, t) is nonsingular for all a?, t and that all coefficients are C°*- smooth. If the associated operators f*, t > 0, are maximal semibounded on the Initial-Boundary Value Problems in One Space Dimension 271 corresponding domain D, then the initial-boundary value problem is well-posed. Especially, for initial data f € C^CO, 1), there is a unique solution with «(-,<)€ Dfort > 0. Let us apply the concept of maximal semiboundedness to some simple exam- ples. Example 1. Consider the equation ut = ux under boundary conditions Here An estimate by 2a||«;||2 is possible only if the boundary conditions imply w(\) = 0. At x = 0 no boundary condition is allowed if the operator F = d/dx is to be maximal semibounded. Example 2. Consider a system where A is constant for simplicity. Here In order to obtain the desired estimate, we need boundary conditions of the form where |5o| -H l^i) must be sufficiently small. If the boundary conditions have the above form, but |Sn| + l-Si) is not small, then the operator P = Ad/dx is not semibounded on the corresponding domain. Nevertheless, one can derive an energy estimate as we have shown in Lemma 7.6.2. Thus semiboundedness is sufficient but not necessary for an energy estimate. Example 3. Consider the linearized Korteweg-de Vries equation 272 Initial-Boundary Value Problems and the Navier-Siokes Equations and assume the functions are real. Here If we want an estimate by 2a||u>||2 then the boundary conditions must imply At x = 1 we need one boundary condition, for example, w(l) = 0. Another possibility is For the left-hand side of (7.8.2) one obtains the quadratic form Condition (7.8.2) is satisfied if and only if both eigenvalues of the above 2x2 matrix are < 0. This yields a restriction for the coefficients a, b in the boundary conditions (7.8.4). At x = 0 we need two boundary conditions. One possibility is We can also require nnd ("7.8.31) bccomci equivalent to 1 «** 18d — 6C1 > fl It we impoic any of ihcie boundary conditions to the KdV equation, then the strip problems are well-posed according to Theorem 7.8.1. Notes on Chapter 7 The Laplace transformation and expansions into eigenfuncttons have been used for a long time in applied mathematics. See, for example, Carrier and Pearson (1976). Our aim was mainly to show that the energy method is rather restrictive with respect to the admissable boundary conditions, though it is very powerful when it applies. Initial-Boundary Value Problems in One Space Dimension 273 The proof of Theorem 7.8.1 is conceptually rather simple. We obtain im- mediately the basic energy estimate. Then, in the same way as for parabolic equations, we can estimate the time derivatives. This provides bounds for the z-derivatives. Corresponding estimates for suitable difference approximations (Kreiss (I960)) can be obtained in a similar way, and the existence of a solution follows as in the text. This page intentionally left blank In Chapters 3 to 6 we have assumed spatial periodicity, and Chapter 7 treated the case of one space variable 0<x<lorO<x<oo. Here we want to combine the two situations and consider a space variable under periodicity assumptions in #2, ...,£«. These special x-domains are fairly representative in the sense that problems posed in other domains (with smooth boundary) can be broken up into subproblems which can be transformed to the special cases. 8.1. Linear Strongly Parabolic Systems 8.1.1. General Assumptions We consider second-order systems in the strip. 2758 Initial-Boundary Value Problems in Serveral Space Dimension 276 Initial-Boundary Value Problems and the Navier-Stokes Equations At time t = 0 we give initial data and at the surfaces we prescribe linear (inhomogeneous) boundary conditions combining u and D\u. Introducing the notation we can write the boundary conditions in the general form where LJQ = Ljo(y,t),Lji = Lj\(y,t),j = 0,1, are n x n matrices. The coefficients Aij = Aij(x, t), etc., the forcing F = F(x, t), the initial function / = /(x), the inhomogeneous boundary terms QJ = g}(y, t), and the coefficients Ljt, = Lj,,(y, t) of the boundary conditions are all assumed to be C^-smooth, 1 -periodic with respect to X2,... , xs, and real, for simplicity. The differential equation (8.1.1) is assumed to be strongly parabolic;* i.e., there exists 6 — ST > 0 such that for all vectors 6 , . . . , & e Rn and all (x, t) in the strip (8. 1.2) with 0 < t < T. The n x 2n matrices are assumed to have rank n for all arguments, but funher assumptions about the boundary conditions are needed for well-posedness. We try to determine a solution u = u(x,t) with values in Rn which is Cx -smooth and 1 -periodic in each variable a?2, . . . , a?8. As before, if we want u to be smooth in the strip (8.1.2) including the boundaries of the domain at t = 0, we need compatibility conditions to be satisfied. The data *The definition given here coincides with our terminology in Section 6. 1 since the expressions j D^Dj-u and D^ Atj DJU differ only by a lower-order term. Initial-Boundary Value Problems in Several Space Dimensions 277 fulfill sufficient compatibility conditions if they vanish identically in some neigh- borhood of all points Notation. The inner product and norm in this section are given by For boundary integrals we use the notation where Y = [0, 1]*"1. Furthermore, Integration by parts. Let w = w(x), v = v(x) be smooth functions of x, 1-periodic in X2, . . . , xs. Then 8.1.2. Neumann Boundary Conditions For illustration, we will concentrate on so-called Neumann boundary conditions, which we write in the form We first show the basic energy estimate. 278 Initial-Boundary Value Problems and the Navier-Stokes Equations Lemma 8.1.1. Suppose that the smooth function u solves the parabolic prob- lem (8.1.1) with initial condition (8.1.3) and boundary conditions (8. 1 .8). There is a constant KT independent of f, F, go, g\ with Proof. From the differential equation (8.1.1), integration by parts (8. 1 .6), (8. 1 .7), and the parabolicity assumption (8.1.5) we obtain that We can use the boundary conditions (8.1.8) to replace D\ u in the above boundary term by w, go, g\. The one-dimensional Sobolev inequality can be integrated with respect to y to yield Therefore we obtain that nother application of (8.1.9) and integration with respect to t proves the lemma. We show next how to estimate derivatives; we write Dt = d/dt for time differentiation. We apply one of the operators Initial-Boundary Value Problems in Several Space Dimensions to (8.1.1) and obtain279 k = t. k = 2. . . . , s. (Since we do not get boundary conditions for D\u, we do not apply the operator D\ to (8.1.1).) Introducing the vector we can write the above system (8.1.10) as where A_ij is the block-diagonal matrix The terms not written out in (8.1.10) contain Noting that A\\ is nonsingular (see (8.1.5)) and using the given differential equation (8.1.1), we can express D\u by Therefore, D\u and D\u can be estimated in terms of F. u and the first derivatives of u. A key observation is that we can easily derive boundary conditions for u. An application of one of the operators Dt, D^.-.-D^lo (8.1.8) yields at each boundary x\ = j — 0, X[ = j = 1. With the same notation as before, Thus we have boundary conditions for u of the same structure as those for u. It is important to note that the inhomogeneous data 280 Initial-Boundary Value Problems and the Navier-Stokes Equations can be estimated i terms of the first derivatives of the given boundary data gj(y, t) and the u-values on the boundary. The latter have already been estimated in Lemma 8.1.1. Thus we have a parabolic system (8. 1 . 1 1 ) for u under Neumann boundary conditions (8.1.12). Lemma 8.1.1 applied to this problem gives us estimates for u, i.e., for To bound D\u, we take the inner product of (8.1.1) with u and integrate by parts. This yields Thus we obtain bounds for ||£>iu||2 also. This process can be continued, and inductively one obtains mma 8.1.2. Suppose thai the smooth function u solves the initial -boundary value problem (8. 1.1), (8. 1.3), (8. 1.8). In every finite time interval we can esti- mate u and its derivatives in terms of the data /, F, 50, g\ and their derivatives. Assume that compatibility conditions for the data /, F, go. g\ are satisfied. To show the existence of a smooth solution u, we can again use a difference scheme. Proceeding as in the continuous case, we can prove estimates for the solution v — vh of the difference scheme independently of the step-size h > 0. There are no essential differences from the case of one space dimension treated in Section 7.2. The periodic variables 13, . . . ,xa are discreiized as in Section 3.2. One obtains Theorem 8.1.3. Suppose that the data /, F, <fo, g\ are compatible. The parabolic problem (8. 1.1), (8. 1.3) under Neumann boundary conditions (8.1.8) has a unique smooth solution. One can estimate the solution and its derivatives in terms of the data and their derivatives. 8.1.3. Other Boundary Conditions If the matrix Lj\ — Lj\(y, t) in (8.1.4) is singular but of constant rank r}, we can write the boundary conditions as Initial-Boundary Value Problems in Several Space Dimensions 281 after we transform the Dirichlet part to become homogeneous. To derive an energy estimate, we can proceed as in one space dimension, see Section 7.2. If either Lj\ =0 (the pure Dirichlet case) or if at each boundary point alge- braic conditions as in Lemma 7.2.1 are met, then the problem is well-posed. Also, more general boundary conditions can be treated via localization, Laplace transform in time, and Fourier expansion in £2, .... x*. We refer to Section 8.4 where we will illustrate this technique for hyperbolic problems. 8.1.4. General Domains We want to indicate how our results can be extended to more general domains. Consider the example with a boundary condition As usual, w(-,0) = / is given at t = 0. Let (p = (p\(r) denote a monotone C00 -function with If we introduce the notations v? and we rite (8.1.13) as a system for MI, i FIGURE 8.1.1. Circle and annulus.then 282 Initial-Boundary Value Problems and the Navier-Scokes Equations To motivate the following, we first neglect lower-order terms and write Vj instead of Uj. Then we obtain the two separate equations Since u\ = 0 for r < 1 - 26, we consider the equation for v \ in the annulus 1 — 26 < r < 1 under boundary conditions This problem is of the type which we have treated, because we can map the annulus onto a periodic strip by the introduction of polar coordinates. Similarly, u2 = 0 for r > 1 - £, which motivates the consideration of a pure Cauchy problem for vz with "boundary condition" H^O, 0|| < oc- To treat the full problem, we define the iteration Here the functions u""*"1 are determined in the annulus subject to the boundary conditions of v\i the pure Cauchy problem is solved to determine MJ~H- The initial conditions are w"+I( ,0) = Vj/. In a similar way as for the iterations dis- cussed before (see, for example, Section 4.1.4), one can use Gronwall's Lemma 3.1.1 and Picard's Lemma 3.3.4 to show uniform smoothness and convergence as n —> oo. The function u = u\ 4- u2, Uj = lim u", solves the given problem. A similar procedure can be used to treat pr s in rather general domains with smooth boundaries, FIGURE 8.1.2. Domain and boundary strip. Initial-Boundary Value Problems in Several Space Dimensions 283 8.2. Symmetric Hyperbolic Systems in Several Space Dimensions In this section we consider first-order systems in the strip under initial conditions and boundary conditions at x\ = 0, x\ = 1. As before, all quantities are assumed to be real, for simplicity, C°° -smooth, and 1 -periodic in Z2,...,zs. This assumption applies also to the boundary conditions specified below. An essential assumption is the symmetry i.e., we have a symmetric hyperbolic system. We will also discuss such systems in the half-space and in more general domains. Our main emphasis is to show that the problem becomes strongly well-posed under certain boundary conditions. In the next section we apply our results to the linearized inviscid compressible Navier- Stokes equations. If one of the boundaries x\ = 0 or x\ = I is a wall, then naturally the u-component of the base flow vanishes there. In this case, as we will see, the corresponding boundary is characteristic. A complete discussion of this situation would be very complicated in several space dimensions. For a characteristic boundary we will restrict ourselves to the treatment of some special cases. 8.2.1. The Basic Estimate and Bounds for Derivatives After transforming the independent variables u, if necessary, we can assume that 284 Initial-Boundary Value Problems and the Navier-Stokes Equations is real and diagonal. First let us assume that B\ = A is nonsingular for all arguments. Then, without restriction, We partition the variables u correspondingly into As motivated by the discussion in one space dimension (see Section 7,6), we consider boundary conditions at a?, = 0,#i = 1 of the form Here Sb» Si are matrices of appropriate dimensions. Difference between one and more space dimensions. There is a significant difference between the cases of one and of more space dimensions: In one dimension, we could force the matrices SG, S\ to become "small" by applying a suitable transformation to ut if necessary; see the proof of Lemma 7.6.2. If we wer to apply such a transformation here, the matrices 82,...,Bs would lose their symmetry, in general. The symmetry is essential, however, for deriving the estimates for well-posedness by the energy method. For this reason, the assumption which we need below, constitutes a real restriction on the boundary condi- tions (8.2.3). The question remains whether or not it is possible to show well- posedness by other means if |50| + \S\ \ is large. We will employ the method of Laplace transform in Section 8.4 to discuss this question. It turns out that well- posedness can indeed be lost if |S0| + |S| | is not small. In one space dimension this is not possible. The basic estimate. We want to show first Lemma 8.2.1. Suppose that u(x,t) is a (real) smooth solution of (8,2.1 )- (8.2.3) which is \-periodic in xi, •..,x Initial-Boundary Value Problems in Several Space Dimensions 285 is sufficiently small for all arguments, then for any finite time interval 0 < t < T there is a constant KT with for 0 < t < T. The constant KT does not depend on f, F, or g. Proof. The differential equation (8.2.1) gives us that For i — 2,..., s, and thus For i = 1 we obtain an additional boundary term. Therefore Since |5o| + \S\ | is assumed small, we can treat the boundary term in exactly the same way as in one space dimension (compare the proof of Lemma 7.6.2) and obtain that Integration of the resulting inequality with respect to t proves the lemma. We will show below the existence of a smooth solution if the data are compat- ible. Then the estimate of the previous lemma states strong well-posedness of the hyperbolic initial-boundary value problem. In obtaining the basic estimate, we have not used that B\ = A is nonsingular. In the case that A has zero eigenvalues at the boundary, we can obtain a similar estimate to (8.2.4) if we replace the boundary term on the left side by 286 Initial-Boundary Value Problems and the Navier-Stokes Equations Zero eigenvalues of B\ — A cause difficulties, however, if we want to estimate derivatives of the solution. Estimates of derivatives. If we apply one of the operators to (8.2.1) we obtain k = t, k = 2, . . . , s. Introducing the vector we can write this system as where Using (8.2.1) again and observing the nonsingularity of B\ = A, we can express D\u in terms of u, u, and F. Boundary conditions for u are obtained if we apply Dt,Di,.,.,Da to (8.2.3). Since we have already estimates for \\u\\ and for ||«||p, we obtain estimates for u in terms of the data /, JT, g and their first derivatives. Clearly, we also have an estimate for D\u from (8.2.1). This process of estimating derivatives can be continued inductively. Initial-Boundary Value Problems in Several Space Dimensions 287 8.2.2. Existence of a Solution To show the existence of a C°° -solution, we assume compatible data; for ex- ample, it is sufficient that the functions vanish identically in a neighborhood of all points (zi = 0, y, * = 0) and (z, = l,y,t = 0) where y = (x2,...,ar,). We choose a gridlength h = I/AT, N a natural number. The space gridpoints are and we leave time continuous. The differential equation (8.2. 1 ) is replaced by a system of ordinary differen- tial equations for the gridfunction v(xl/, t) = vh(xv, t) of the form Here Q\ is the difference operator* The coefficients are to be evaluated at (xv, t). The boundary conditions (8.2.3) are replaced by where (0, y), (l,y) are points of the spatial grid. There are no difficulties in discretizing the initial condition (8.2.2) by The differential equation (8.2.1) is discretized only at the inter gridpoints of the strip. If we consider the equation (8.2.6) at a point xv = (h, y) close to the left boundary of the strip, say, then the difference operator *In the terminology of numerical analysis, we use "upwinding" to approximate AD) and apply a central difference formula in the periodic variables 2, . . . , x.s. 288 Initial-Boundary Value Problems and the Navier-Stokes Equations volves the value v_(0, y, t). We can use the discretized boundary condition .2.8) to eliminate this part of a v-value. Using periodicity with respect to the variables #2,...,xs, we have obtained a finite dimensional (linear) ordinary differential system for v(t) = vh(i) under initial conditions. Thus vh(t) is uniquely determined. As previously, by mimicking the estimates for the differential equation, one can bound all difference-differential quotients of vh independently of h. Simple error formulae (in numerical analysis terminology, the consistency of the difference approximation) show that vh converges to a smooth solution u(x, t) of the initial-boundary value problem for some sequence h —> 0. Some of the details of the estimates of vh are provided in the next section. We summarize the result in Theorem 8.2.2. Given the symmetric hyperbolic initial-boundary value prob- lem (8.2.1)-(8.2.3) where B\ = A is nonsingular and the matrices SQ and S\ are sufficiently small in norm. (For example. So = 0, 5j = 0.) // the data are compatible, there is a unique smooth solution. The problem is strongly well-posed. The smallness-assumption for (Sol + \S\ | can be quantified as in the proof of Lemma 7.6.2. What we actually need is the estimate (8.2.5) for all functions which satisfy the boundary conditions. 8.2 J. The Finite Speed of Propagation We shall prove that the principle of finite speed of propagation is valid for the hyperbolic initial-boundary value problem (8.2.1)-(8.2.3). As in the case of tne spatially periodic Cauchy problem (see Theorem 6.2.1), we construct a difference approximation whose solutions converge to the solution u(x, i) of the continuous problem. For the discrete systems the speed of propagation will be finite and bounded independently of the step-size. The spatial meshsize h-l/N and the spatial gridpoints are denoted as in the last section. We choose a time step where SQ will be assumed sufficiently small, but independent of h. Discretizing in time and space, we determine a grid function Initial-Boundary Value Problems in Several Space Dimensions 289 by Here Qi is defined in (8.2.7) and The gridfunction v is subject to the boundary and initial conditions (8.2.8), (8.2.9); as explained in the previous section, these conditions determine v uniquely. Clearly, the ratio bounds the speed of propagation in the discrete system (see Section 6.2); there- fore, once we have proven convergence of vh(xv,t) to u(x.t) as h — » 0 (in the sense of (6.2.4) ), the number SQ{ also bounds the propagation-speed for the continuous problem. It will be sufficient to show that we can bound vh uniformly in terms of the data. Then convergence follows, since vh - u satisfies the above difference equations with data which tend to zero for h — » 0. To prove these bounds for v = vh, we introduce a discrete scalar product by Also, we use the notation where the sum extends over and i/i = j is fixed. In (vague) analogy to integration by parts with respect to x\, we have the rules: Lemma 8.2.3. 290 Initial-Boundary Value Problems and the Navier-Stokes Equations where Proof. Since summat ver applies to all terms, it suffices to prove the result for s = 1. Then we have (Here we have used the index notation vv f (xv\ etc.) This proves the first relation, and the second is shown similarly. Recursive estimate for v = vn. The difference equation (8,2,10) gives us The inner products and norms on the right-hand side apply tQ the ghdfunctions v{ , f), etc,, and we have used estimates like To bound the term we apply Lemma 8.2.3 to treat the contribution of Initial-Boundary Value Problems in Several Space Dimensions 291 Since A_ and A+ are bounded away from zero, we have, with some 6 > 0, Using the discrete boundary conditions (8.2.8) and the assumption that is small, we can obtain an estimate for the boundary terms in terms of the boundary data g0 and g\. Therefore, The difference operator Q2 acts only in the periodic directions 12, . . . , xs\ thus summation by parts gives us Finally, it is not difficult to show that Summarizing, we have shown that 292 Initial-Boundary Value Problems and the Navier-Stoltes Equations Here c5/c2 = c^khso. If SQ is chosen so small that then we have the recursive estimate In any finite time interval we obtain the desired estimate of v in terms of the data. Thus we have proven Theorem 8.2.4. Under the assumptions of Theorem 8.2.2 the principle of finite speed of propagation is valid for the symmetric hyperbolic initial-boundary value problem (8.2.1H8.2.3). 8.2.4. The Strip Problem vs. the Half-Space and Periodic Cauchy Problems Thus far we have treated symmetric hyperbolic systems (8.2.1): a) under the side-condition of spatial periodicity; b) in a strip 0 < x\ < 1 with boundary conditions at x\ — 0, x\ = 1 and periodicity assumptions with respect to z2,..., av Only in the latter case did we need that B\ is nonsingular. In this section we will show how to use the strip problem to solve the hyper- bolic system (8.2.1) in the half-space with boundary conditions at x\ = 0. Conversely, the strip problem can be reduced to left half-space, right half-space and periodic Cauchy problems. Es- sentially these results follow quite easily from the finite speed of propagation. The reduction of the strip problem to half-space and periodic Cauchy problems will be useful in the next section, where we remove the assumption that B\ is nonsingular. Solution of half-space problems. Suppose the coefficients and data in (8.2.1), (8.2.2) are defined for all x € Ra, are 1-periodic with respect to a?2,..., xs, and are uniformly smooth. Furthermore, assume that the data f(x) and F(x, t) have compact support with respect to x\\ i.e., there is a constant M with We always assume symmetry £* — B*, and also require in this section that B\ is nonsingular. Our first result is Initial-Boundary Value Problems in Several Space Dimensions 293 Theorem 8.2.5. Under the above assumptions, consider the half-space prob- lem s < oo, t > 0, with the initial condition and the boundary condition at x\ = 0, Assume the data are compatible and |So| is sufficiently small. There is a unique smooth solution which vanishes for 0 < t < T if the argument x\ > c(T) is sufficiently large. Proof. We fix a time T > 0 and consider the above problem in the strip with the additional boundary condition at x\ = aM. According to Section 8.2.2, there is a unique smooth solution u(x,t) of this strip problem. The finite speed of propagation implies that u is identically zero in a neigh- borhood of the points if a is chosen sufficiently large. Setting u = 0 for x\ > aM, 0 < t < T, we have solved the half-space problem for 0 < t < T. Uniqueness follows as previously by an energy estimate. Also, the solution constructed in 0 < t < T does not change if we choose another Solution of strip problems. Consider the strip problem (8.2.1H8.2.3) and assume, as we did previously, that the coefficients and data are defined for all x. We want to show that we can express the solution of the strip problem as where w(1) and w(2) solve left and right half-space problems, respectively, whereas u(3) solves a periodic Cauchy problem. Such a representation is valid in a suf- ficiently small time interval 294 Initial-Boundary Value Problems and the Navicr-Stokes Equations after time 6 one can restart the process. We choose a number* 0 < r < 1/2 and a monotone function with and define (1) The right half-space problem is (2) The left half-space problem is (3) The periodic Cauchy problem is To be precise, the last problem is not periodic in x\ originally. However, using the assumption (8.2.11), we can alter the coefficients and data outside a sufficiently large interval and make them periodic. The finite speed of propagation ensures that we do not change w(3) in *The freedom in the choice of r will be useful in the next section. Initial-Boundary Value Problems in Several Space Dimensions 295 where 6 > 0 is sufficiently small. Clearly, the sum solves the given strip problem. The function u is nothing but the C°° -solution of the strip problem which we obtained already in Section 8.2.2. We have just described another representation. Using the values u( • , 6) as starting data at t = 6, it is clear that there are no compatibility problems if we continue this process. This shows Theorem 8.2.6. The solution u(x, t) of the strip problem (8.2. 1H8.2.3) can be obtained by solving half-space and periodic Cauchy problems in successive time intervals The time 6 > 0 depends on the finite speed of propagation. 8.2.5. Cases Where B\(x,t) Becomes Singular For the strip problem (8.2.1M8.2.3) we want to remove the assumption that B\(x, t) is nonsingular for all arguments. This is particularly easy if B\ becomes singular only in the interior 0 < x\ < 1. For any fixed finite time T, there is a number 0 < r < 1/2 such that tff'CM) exists for 0 < xi < r, 1 - r < x\ < 1, 0 < t < T. Then we split the strip problem into two half-space problems and a Cauchy problem as described in the last section. For definiteness, consider the right half-space problem. We change the coefficient B\(x,t) to For 0 < x\ < r it holds that B\ = B\\ furthermore, it is not difficult to check that B\ is nonsingular throughout if r > 0 is sufficiently small. Thus the changed right half-space problem (with B\) has a solution u(l). In some small time interval the function M(I) is zero for x\ > T, thus w(l) = w(l) solves the unchanged right half-space problem. Similar considerations apply to the left half-space problem. For the periodic Cauchy problem it was not important that the coefficient B\ can become singular. Adding the three solutions, we obtain a smooth solution 296 Initial-Boundary Value Problems and the Navier-Stokes Equations of the strip problem in a small time interval 0 < t < 6. We take t structed function u( •, 6) as initial data at t — 6; in this way, no compatibility problems arise when we restart the process at t = 8, As the proof of Lem a 8.2.1 showed, the basic energy estimate is valid also for singular B\; hence uniqueness of the solution is ensured. We have Theorem 8.2.7. The result of Theorem 8.2.2 remains valid if B\ becomes singular in the interior 0 < xi < 1 of the strip. Cases where B\ is singular on the boundary. For simplicity we restrict our- selves to right half-space problems (thus x\ > 0) and consider coefficient ma- trices B\ = B\ (x, t) of the following three forms. Case 1. where A_ < -77, A+ > -yl, 7 > 0, and AO is nonsingular. Case 2. where Case 3. where Here A_, AO, A+ are always real diagonal matrices. In Case 3, the matrices have no negative eigenvalues and (in contrast to Case 1) the matrix AO is allowed to become singular. The above representations are required for 0 < x\ < T, i.e., in some neighbor- hood of the boundary x\ =0. The differential equation (8,2,1) is supplemented by an initial condition (8.2.2) and a left boundary condition. Initial-Boundary Value Problems in Several Space Dimensions 297 In Case 3 no boundary condition is required. We want to show Theorem 8.2.8. Assume that B\(x,t) is singular on the boundary x\ = 0 and has either of the three forms described above. If\S\ is sufficiently small, the right half-space problem has a unique smooth solution. Proof. There are no problems in obtaining the basic energy estimate through integration by parts; compare the proof of Lemma 8.2.1. The boundary term of the left side of (8.2.4) is here Also, if we want to estimate first derivatives, we apply the operators to the differential equation and the boundary conditions. In the resulting system we have to express in terms of Dtu, DIU, .... Dau using the differential equation (8.2.1). In Case 1 we have, for Observing that remains bounded for x\ — * 0, no difficulties arise. In Case 2 the situation is even easier, since the "dangerous" part of the terms (8.2.12) vanishes identically near x\ =0. In Case 3 we do not need any boundary conditions. Thus we apply all the operators 298 I al-Boundary Value Proble nd the Navier-Stokes Equations to the differential equation ( .1) and obtain the a priori estimates for first derivatives. This process can continued, and one obtains bounds for all the derivatives in Case 1 and 2; in the third case all derivatives can be estimated. We shall now estimate the x\ -derivatives in the first two cases. To this end, we write the differential equation in the partitioned form Using the differential equation, we can estimate D\u-, D\u+ and their y, t- derivatives. Therefore we obtain, for v — D\UQ, For H and its y, t-derivatives we have already established a bound. The ma- trices Bj* are symmetric and Bf2(0, y,t) = 0 by assumption. Therefore, we can estimate VQ and all its y, ^-derivatives. Then we can obtain bounds for D\u-, D\u+ and the y, i-derivatives of these functions. In turn, this gives us estimates for D\UQ. This process can be continued. To show existence of a solution, we replace B\ by B\ -fa/, a > 0. If a > 0 is sufficiently small, there exists a solution depending on a. The estimates are dependent of a, and the desired result follows by a limit argument. All these considerations apply to a small neighborhood near x\ = 0. With the splitting technique at Section fc 1 a we obtain the result for the general caie. 8.2.6. Results for More General Domains For simplicity we restrict ourselves to two space dimensions and consider a founded domain « c R2 with a smooth boundary curve 1, Suppose we are given a symmetric hyperbolic system Along the boundary, we can write the differential equation in the form where d/dn and d/ds denote the derivatives in the inward normal and tangential directions, respectively. Here Initial-Boundary Value Problems in Several Space Dimensions 299 FIGURE 8.2.1. Domain Q with boundary F. where a denotes the angle between the z-axis and the inward normal to the boundary. We want to prove Theorem 8.2.9. Assume that Bn is nowhere singular on F and has exactly r negative eigenvalues. Suppose further that the boundary conditions consist ofr linear relations at each boundary point. The matrix L and the function g may vary smoothly along F. If at each boundary point we have an estimate with 6 > 0, c > 0, then the problem is well-posed. Proof. Integration by parts gives us the basic energy estimate 300 Initial-Boundary Value Problems and the Navier-Stokes Equations Here we have used the pointwise assumption (8.2.13) to treat the boundary integral. The basic estimate follows if we integrate with respect to t. To show the existence of a solution, we reduce the given problem to Cauchy tion of unity and let fy denote the support of <j>j. We split the data functions / = /(# , y\ F = F(x, y, t)t g = g(x, y, t) into FIGURE 8.2.2. Partition of unity and transformation. If the support fZj of 0, lies in the interior of fl then QJ = 0, and we solve the Cauchy problem with the data /,, F3 ignoring the boundary conditions. For a sufficiently small time interval, the solution fulfills the homogeneous boundary conditions automatically. If fy contains a part of the boundary F, then we transform a neighborhood of fij in such a way that the local piece of T is transformed into a piece of the new £j*axis and the normal directions are transformed to directions parallel to the x-axis. The new problem can be extended to a half-plane problem of the type dis- cussed earlier. Transforming back and adding the solutions, we have solved the problem in a small time interval. Thereafter, we can restart the proces 8.2.7. An Example in Two Dimensions Consider the systemproblems and half-plane problems, To this end, let denote a (finite parti Initial-Boundary Value Problems in Several Space Dimensions 301 in the half-plane under our usual periodicity assumption with respect to y. The general theory suggests imposing one boundary condition at x = 0 which expresses u- in terms of u+. We consider a condition of the general form* with smooth functions a and g. To obtain an energy estimate, we proceed as before: Now we replace w using the boundary condition. It is apparent that the desired estimate can be obtained as long as the boundary coefficient a(y) satisfies for all y. This quantifies our previous assumption of "smallness" of the coeffi- cient matrices in the boundary condition. (Compare Lemma 8.2.1, for example.) Now suppose that lQ(y)| 5: 1 for all y, a(yo) = 1 for some yo- Then our estimate breaks down since the above boundary integrand is *We could also treat the case a = a(y, t). 302 Initial-Boundary Value Problems and the Navier-Stokes Equations We still can derive a basic energy estimate if we first transform to homogeneous boundary conditions: Let The differential equation becomes and the boundary condition reads Proceeding as before, the boundary term becomes and thus we can estimate ||u||. In this way we do not obtain a bound for u on the boundary x = 0, however. Difficulties arise if we want to estimate derivatives. To obtain boundary conditions for uy, for example, we differentiate (8.2.14) with respect to y: The equation has the inhomogeneous term ayv. The difficulties are avoided if we first make a unitary transformation of the dependent variable such that the boundary con ions have constant coefficients. 8.3. The Linearized Compressible Euler Equations We linearize the inviscid compressible N-S equations at a constant state with As discussed in Section 2.4.2, w btain the stron hyperbolic system Initial-Boundary Value Problems in Several Space Dimensions 303 (If the ambient flow U — t/(x, t), etc. is not constant, the linearized equations read as above after neglection of zero-order terms and forcing functions. Com- pare Section 6.4.) The above system can be symmetrized by introduction of the scaled density p = cp/R; one obtains the symmetric hyperbolic system etc. We want to discuss this symmetric system in the half-space under the usual periodicity requirement with respect to y and z, and we want to derive boundary conditions at x = 0 which lead to a well-posed problem. (The matrix B\ is not diagonal; however, after a simple transformation the results of Section 8.2 apply.) The eigenvalues of B\ are Following the general theory, we will specify as many boundary conditions at x = 0 as there are negative eigenvalues of B\. Accordingly, we distinguish a number of different cases. 304 Initial-Boundary Value Problems and the Navier-Stokes Equations Case 1. U < 0 (outflow of the ambient flow). In the supersonic or transonic case |U\ > c, the matrix B\ has no negative eigenvalue and no boundary condition needs to be specified. In the subsonic case \U\ < c, there is exactly one negative eigenvalue, K4 = -U — c. Thus one boundary condition has to be specified. (This is necessary but not sufficient for well-posedness.) In order to obtain an energy estimate, we consider, as above, the expression and use integration by parts with respect to x, y, and z. Assuming functions which vanish at x = +00, the critical boundary term to be estimated is where the index F0 indicates integration over y and z at the boundary x = 0. We try to obtain an estimate analogous to (8.2.5) and specify a boundary condition for example. The boundary term (8.3.1) becomes Since U < 0, it suffices to require that This can be viewed as a requirement for the coefficient a in the boundary con- dition. With such a boundary condition the desired energy estimate is obtained, including on estimate of on the boundary x = 0. We can also require that or we can derive sufficient conditions for more general boundary conditions Once the basic energy estimate can be obtained with boundary nditions of the above form, we can also bound all derivatives of the soluti n up to the boundary. (Note that in the homogeneous case (8.3.2), the coefficient in the boundary ondition is constant.) Initial-Boundary Value Problems in Several Space Dimensions 305 Case 2. U > 0 (inflow of the ambient flow). In the supersonic case the matrix B\ has four negative eigenvalues, and consequently all four variables u, t>, w, p need to be specified. The boundary term (8.3.1) being given, there are no difficulties in obtaining the desired estimates. In the subsonic case there are three negative eigenvalues, and one eigenvalue is positive. We specify three conditions, for example: Substituting the expression for p into (8.3.1), we see that an energy estimate can be obtained if A possible choice is a — — 1. In the transonic case the matrix B\ has three negative eigenvalues, and the eigenvalue zero. As before, we need three boundary conditions to obtain an energy estimate. If the state at which we linearize is constant, then the conditions of Theorem 8.2.8 are au- tomatically fulfilled, and we obtain a well-posed problem. (After diagonalizing B\, we have Case 2 of the theorem.) If the state is not constant, we need conditions as required in Case 1 or 3 of Theorem 8.2.8. Case 3. U = 0 (a wall). In this case B\ is singular and has exactly one negative eigenvalue; hence we have to specify one boundary condition. For example, if the boundary condition takes the form then the integrand in (8.3.1) becomes and we obtain an energy estimate if a < 0. Also, we can use the conditions 306 Initial-Boundary Value Problems and the Navier-Stokes Equations or (Note that p is a density correction; thus also the latter condition might be physically reasonable.) In all cases we obtain the basic energy estimate, i.e., an estimate for If the conditions of Theorem 8.2.8 are not met, there are difficulties in estimating derivatives up to the boundary. 8.4. The Laplace Transform Method for Hyperbolic Systems Consider a strongly hyperbolic system in the half-space under an initial condition at t = 0 and boundary conditions at x = 0. The energy method employed in Section 8.2 provides a tool to derive sufficient conditions for well-posedness, but if the energy method does not apply then one does not know whether or not the problem is well-posed. The technique described here can be used - in principle - to decide the question of well-posedness. Employing Fourier transformation in y (or Fourier expansion in the periodic case) and Laplace transformation in t, one obtains a family of ordinary boundary value problems on the half-line 0 < x < oo, which can be discussed explicitly. In the case of one space dimension, we have proceeded in a similar way in Section 7.4; however, the presence of the dual variable uj to the space variable y brings a new element into the discussion here. There are no difficulties in generalizing from two space dimensions to three dimensions or more. A generalization to variable coefficients A = A(x, y, 0. etc. is more involved. Roughly speaking, it turns out that the frozen-coefficient principle is valid; i.e., if all frozen-coefficient problems are well-posed then the variable-coefficient problem is also well-posed. We will formulate a correspond- ing result without proof. Initial-Boundary Value Problems in Several Space Dimensions 307 8.4.1. An Example To illustrate the general theory, we will discuss a specific example, namely the symmetric hyperbolic system in the half-space with initial condition At x = 0 we prescribe a boundary condition where a € C is a given constant. In contrast to the previous section, we do not request periodicity in y here. Instead, u, /, <?, F, and all derivatives of these functions are assumed to lie in LI for each t > 0. Also, we assume the data to be compatible at t — 0, and require that F and g have compact support in 0 < t < oo. A family of eigenvalue problems. Let F = 0, g — 0. Substitution of a function with parameters into the differential equation (8.4.1) leads to If we add the boundary conditions we have obtained a family of eigenvalue problems. Definition 1. Let u; € R be fixed. A number s e C is called an eigenvalue of (8.4.5) for the parameter u; if there is a nontrivial solution 0 € C00 (8.4.5). 308 Initial-Boundary Value Problems and the Navier-Stokes Equations It is not difficult to show Lemma 8.4.1. If for some UQ G R there is an eigenvalue SQ € C with Re SQ > 0, then the initial-boundary value problem is ill-posed in any sense. Proof. The parameter a > 0 in (8.4.4) can be chosen arbitrarily large, and consequently there is no bound on the rate of exponential growth in time. (For- mally, the function u given in (8.4.4) is not a solution since u(x, -,0 £ Z/2- However, we can multiply on the Fourier transform side by a function i^(uj) having a peak at VQ and integrate w.r.t u/. See the proof of Theorem 2.2.2.) Discussion of the eigenvalue condition. The previous lemma naturally leads to the following question: What is the requirement on the parameter a 6 C appearing in the boundary condition such that there is no eigenvalue s e C with Re s > 0? This requirement on a is necessary for well-posedness, as follows from the previous lemma. An explicit discussion shows Lemma 8.4.2. The following conditions are equivalent: (i) The number a € C satisfies (ii) For all a; € R, all eigenvalues s € C of (8.4.5) satisfy Res < 0. Proof. Equation (8.4.5a) is equivalent to If Re s > 0 then the above matrix has the two eigenvalues with The eigenvector to /ci reads and therefore the general Z/2-solution of (8.4.5a) is given by Initial-Boundary Value Problems in Several Space Dimensions The boundary condition at x = 0 requires that309 and consequently the number s e C, Re s > 0, is an eigenvalue to the parameter u) if and only if For <jj = 0, equation (8.4.6) has no solution s with Re s > 0. Thus let u; 7^ 0; dividing (8.4.6) by |u;|, we obtain: The function maps the line Re z = 0 onto the boundary of the domain ft in Figure 8.4.1, and Re z > 0 is mapped onto Therefore the equation has no solution z with Re 2 > 0 if and only if i.e., a € R or lal < 1. FIGURE 8.4.1. Image of ^(z), Rez> 0. 310 Initial-Boundary Value Problems and the Navier-Stokes Equations 8.4.2. Example Continued: Formal Solution If there is no eigenvalue s E C with Re s > 0, then we can solve the problem formally. (This means, the algebraic equations and ordinary boundary value problems that appear below can be solved, but the question of convergence of integrals requires further study. As a rule, if the data are sufficiently "well- behaved", a formal solution is a genuine C°° -solution. However, the problem is well-posed only if one can derive the proper estimates of the solution in terms of the data.) To illustrate this, we consider (8.4.1H8.4.3) with / = 0, F = 0 and first apply Fourier transformation in y. Thus we write and obtain: Thus, for any fixed u>, we have to solve an initial-boundary value problem in one space dimension 0 < x < oo. Denoting the Laplace transforms in t by the equations transform to and the requirement is naturally added as a boundary condition at or = oo. The equations for v(ar,u>, s) can be solved for each pair Initial-Boundary Value Problems in Several Space Dimensions 311 separately. Using the eigenvalue- and eigenvector-results of the proof of Lemma 8.4.2, we find Here a = cr(u;, s) is determined by the boundary condition at x = 0, By assumption, there is no eigenvalue s with Re s > 0, and therefore we can solve for cr, Thus we can compute the function v(x. u;, s), and if there are no convergence problems, we can transform back and obtain the solution u(x, y, t) of the initial- boundary value problem. 8.4.3. Example Continued: Estimate of the Solution on the Boundary To investigate well-posedness, we try to estimate the solution in terms of the data. The most critical question is, as it turns out, whether one can estimate the solution on the boundary x = 0. To study this question, consider the example (8.4.1H8.4.3) with / = 0 and assume that Thus there is no eigenvalue s with Res > 0, and we can solve the problem formally. Using the notations of Section 8.4.2, we have that Let us denote the factor multiplying |/i(u;, s)|2 by p2(uj, s); thus, for u; =^ 0, and p2(0. s) = 1 for u; = 0. Re s > 0. There are two fundamentally different cases: 312 Initial-Boundary Value Problems and the Navier-Stokes Equations Case 1. There is a constant CQ with /^(w, s) < CQ for all u € R, Re s > 0. Case 2. The function p*(u>. s) is not bounded in the domain u> e R, Re s > 0. For the example under consideration, it is not difficult to show that Case 1 prevails if and only if |a| < 1, and therefore Case 2 prevails if and only if (Recall the assumption (8.4.7).) Concerning the question of well-posedness, the two cases of behaviour of p2(cu, s) - bounded or unbounded - lead to different answers. Only in the first case can one derive estimates of the solution on the boundary which express strong well-posedness in the generalized sense. (We will prove even more below, namely strong well-posedness in the usual sense.) In Case 2 such estimates are not possible, and the problem is not strongly well- posed in the generalized sense. Nevertheless, the explicit solution formula can be used in Case 2 also, and one can derive weaker estimates. We concentrate here on Case 1 and prove Lemma 8.4.3, Suppose that f>2(u>, s) < c$for all u> € R and all s € C with Re s > 0. Then, for each time T, there is a constant KT independent of the boundary data g with The constants KT can be chosen uniformly for 0 < T < T0; i.e., KT = A'(T0). In the above formula denotes the Li-norm of the boundary data at time t, and similarly ||u(0. , 0||r is the L2-norm of the solution on the boundary x = 0 at time t. Proof. First we recall the relation between u and its Laplace transform v. For fixed u; e R and 17 > 0 we apply (7.4.9) and find that Initial-Boundary Value Problems in Several Space Dimensions 313 The second estimate clearly follows from our assumption that p2(u>,s) < CQ. Since /i(u;, •) is the Laplace transform of g(u, •), equation (7.4.8) yields The resulting inequality is integrated w.r.t. w. After interchanging the order of integration, an application of Parseval's relation yields By an argument as given in the proof of Lemma 7.4.6, we can change the data g(y. t) for t > T without affecting the solution u(x,y,t) for t < T, and the result follows. 8.4.4. Example Continued: Strong Well-Posedness in the Generalized Sense We consider now the inhomogeneous equation with homogeneous initial data u(x, y, 0) = 0 and the same boundary condition as before, After Fourier and Laplace transformation, we obtain: We assume that there is no eigenvalue s with Re s > 0. Then, for every fixed s, u, the ordinary boundary value problem (8.4.9) for v(-.u>, s) can be solved, and we obtain a formal solution of the given initial-boundary value problem by 314 Initial-Boundary Value Problems and the Navier-Stokes Equations inverting the Laplace and Fourier transform. The formal solution is a genuine solution if one can derive sufficiently strong estimates. To derive estimates, we take the scalar product of (8.4.9) with v, integrate over 0 < x < oo and consider the real part: First, assume that |a| < 1. Using the boundary condition for v, we obtain that Therefore, (8.4.10) yields that Here K\ is independent of u;, s, /i, and /f . By Parseval's relation, we obtain estimates of u in terms of the data F and g; see Section 7.4, in particular (7.4.11). The estimates imply that the given problem is strongly well-posed in the generalized sense. The for ution is indeed a genuine solution. Second, assume that |a| = l and g = 0. In this c and we obtain, instead of (8.4.1 1), Thus the problem is weakly well-posed (see Definition 4 in Section 7.3); we do not obtain an estimate of the solution u on the boundary x = 0. Let us summarize the results for the example. The method of Laplace trans- formation shows that the condition |a| < 1 is necessary and sufficient for strong well-poscdncss in the generalized sense. In Section 8,2,7 we have treated the example already by the energy method, and could derive energy estimates un- Initial-Boundary Value Problems in Several Space Dimensions 315 der the same condition, namely |a| < 1. This might be misleading, because the method of Laplace transformation has generally a wider range of applicability. We have seen this already in the parabolic case. For hyperbolic systems, the energy method requires symmetric hyperbolicity, whereas the Laplace transform method can be applied to strictly hyperbolic systems, too. On the other hand, an application of the energy method is preferred, whenever possible, since intricate algebraic discussions are usually avoided. 8.4.5. Generalizations Consider a strongly hyperbolic initial-boundary value problem Here A, B, and R are constant matrices and The unknown vector-function u is partitioned accordingly, If / = 0, then we apply Fourier transformation in y, Laplace transformation in t, and obtain, with the same notations as before, The eigenvalue condition. Suppose that for some u; € R and some s € C with Re s > 0 there is a nontrivial solution </>(z) of 316 Initial-Boundary Value Problems and the Navier-Stokes Equations Then the problem (8.4.12) is ill-posed, because the functions are solutions for F = 0, g = 0, and hence there is no bound on the exponential growth rate in time. To discuss the above eigenvalue condition, we note that the general solution of can be written in the form Here KJ = KJ(U», 5) are the roots of and $j(x) = $j(ar, u>,s) are the corresponding vector-functions; these are poly- nomials in x of degree < mj — 1 where mj is the algebraic multiplicity of KJ. For a further discussion, the result of the next lemma is important. Recall first the assumption of hyperbolicity, which implies that all matrices have only purely imaginary eigenvalues. Lemma 8.4.4. Let u e R and s € C with Re s > 0 be given. The character- istic equation (8.4.15) has exactly r roots KJ with negative real parts and exactly n — T roots with positive real pans. (The roots are counted according to their algebraic multiplicity.) Proof, There is no purely imaginary root n\ = iu\ since otherwise in contradiction to the hyperbolicity assumption. Now fix uj and let The roots m of Initial-Boundary Value Problems in Several Space Dimensions 317 are Hence the assertion follows, since the KJ depend continuously on s, and no KJ According to the lemma, the sum in (8.4.14) consists of r terms. In particular, with a matrix $(u;, s) of size n x r. Introducing this representation into the boundary condition, we obtain Lemma 8.4.5. A number s e C w/7/j Re s > 0 is an eigenvalue for a given u> 1/f/ie r x r matrix /5 singular. Here $' consists of the first r and 3>ir of the last n — r rows of$. Formal solution and estimates. Suppose there are no eigenvalues s with pos- itive real parts; i.e., the matrices $7 — R$n are all nonsingular. As in the example, we can solve (8.4.13) and obtain a formal solution of (8.4.12) (with / = 0) by inverting the Fourier-Laplace transform. To derive estimates, consider first the case F = 0, / = 0. We have H = 0 in (8.4.13), and therefore One can derive an estimate for 0 < t < T, with KT independent of g, if and only if the function is bounded. Inhomogeneous differential equations and the symmetrizer. Now consider the case with an inhomogeneous term F in the differential equation (8.4.12). As before, we assume that the function (8.4.16) is defined and bounded. The simple process of Section 8.4.4 for deriving estimates of u in terms of g and F 318 Initial-Boundary Value Problems and the Navier-Stokes Equations does not work, in general. However, if the system (8,4.12) is strictly hyperbolic, one can construct a symmetrizer, which allows us to derive the estimates. Let us define Then (8.4.13) can be written as follows: One can show Lemma 8.4.6. Suppose that (8.4.12) is strictly hyperbolic and the function (8.4.16) is defined and bounded. For every fixed rj > here is a matrix function S = S(u/,£') which is defined for the arguments —1 < u/, £' < 1 and /kw the following properties: 1) The function S(u/,f) w C°°-smooth. 2) for a// arguments u/, f' fta matrix §A is Hermitian. 3) 7/y e Cn, /i e Cr are vectors w/7/i y7 = fly77 + Here d\ > 0 and C are constants independent ofy, /i, u;', f', r; > 0. 4) With a constant 62 independent ofu', £', 77 > 0, // holds that For a proof we refer the reader to Kreiss (1970). Let us assume here that the symmetrizer is constructed. Then one can derive estimates of the solution v = v(x,u>,s) of (8.4.13) as follows: We multiply (8.4.17) with 5, take the inner product with v, and integrate over 0 < x < oo. As a result we obtain Also, using integration by parts and the properties 2. and 3. of S, we find that Now we take the real part in (8.4.18), use property 4. of S, and multiply with 2(|s|2 + u;2)1/2. Then it follows that Initial-Boundary Value Problems in Several Space Dimensions 319 This gives us the desired estimate for the transformed function y. By Parseval's relation e obtain Theor 8.4.7. Suppose that the system ut = Aux + Buy is strictly hyper- bolic. Then the initial-boundary value problem (8.4.12) (with f = 0) is strongly well-posed in the generalized sense if and only if the function (8.4.16) is defined and bounded. Remark. The functions $j(x) = $,•(#, a;, s) in (8.4.14), and consequently the matrix 4>(u;. s), are not unique. However, if the <£/ are suitably normalized, then - for each fixed u> - the matrix function $(u;,s) can be extended continuously to Res = 0 in such a way that rank4>(u;.s) = r for all uj G R and all s e C with Res > 0. With such an extension, the expression (8.4.16) is defined and bounded if and only if the matrix is nonsingular for all uj and all Res > 0. This observation is helpful for a discussion of the boundedness of (8.4.16). If a matrix (8.4.19) is singular at ">o, so = s'€o. then SQ is called a generalized eigenvalue for ^Q. The correspond- ing roots of (8.4.15) might be purely imaginary, and the corresponding nontrivial solution <j>(x) is generally not in Z/2- The example (8.4.1), which we discussed in the previous sections of this chapter, is even strongly well-posed in the sense of Definition 1, Section 7.3, provided that |a| < 1. Thus, under the assumption that |a| < 1, we can also estimate the solution for inhomogeneous initial data. This has been shown already in Section 8.2.7 by a direct energy estimate. For the general case discussed here, one cannot show an energy estimate directly. Nevertheless, one can prove Theorem 8.4.8. Suppose that the system ut = Au£ + Buv is strictly hyper- bolic. If the initial-boundary value problem (8.4.12) is strongly well-posed in the generalized sense, then it is also strongly well-posed in the sense of Definition 1, Section 7.3. A proof of this result is contained in Rauch (1972 a,b, 1973). The arguments are rather involved. 320 Initial-Boundary Value Problems and the Navier-Stokes Equations All results generalize to half-space problems in more than two space dimen- sions. We Fourier transform in all space variables tangential to the boundary x = 0, and obtain a dual variable u which is a vector instead of a scalar. The other considerations remain the same. Variable coefficients. Consider a strictly hyperbolic problem (8.4.12) with smoothly varying coefficients We assume A to be diagonal* for all arguments, and nonsingular at x = 0. The (constant) block-structure of A(0, y, t) with A7 < 0, A7/ > 0, and the corresponding partitioning of u into u1, u11 are the same as in the case of constant coefficients. In particular, R = R(y, t) has constant size r x (n - r). For each boundary point (0, yo, to) we obtain a problem with frozen coefficients, For these problems we can decide the question of strong well-posedness by Theorems 8.4.7 and 8.4.8. For the variable-coefficient problem we state Theorem 8.4.9. The variable-coefficient problem is strongly well-posed if the differential equation is strictly hyperbolic and all frozen-coefficient problems (8.4.20) are strongly well-posed. Using the theory of pseudodifferential operators (see, for example, Eskin (1973) or Nierenberg (1970)) and the symmetrizer 5, we want to outline the proof of strong well-posedness in the generalized sense. As before, the main problem is the estimate. Let 77 > 0 be fixed, and transform the differential equation to We construct the symmetrizer 5 = S(x, y, t,uj',s') pointwise: at each point (or, y, 0 the properties 1., 2., and 4. of Lemma 8.4.6 hold, and at each boundary point (x = 0. y. t) property 3. also holds. (One can construct S as a smooth function of all arguments.) The symmetrizer S is used as the symbol of a *If A is not diagonal, we apply a transformation u = 5~'u, 5 = 5(x, y, t), such that S~' AS is diagonal. This introduces lower-order terms, but these do not influence well-posedness. Initial-Boundary Value Problems in Several Space Dimensions 321 pseudodifferential operator 5. For sufficiently rge 77 we obtain the desired estimate from Here the data functions are set to zero for negative t. The above estimate yields strong well-posedness in the generalized sense. Again, the results Rauch (1972 a,b, 1973) show that the problem is even strongly well-posed in the sense of Definition 1, Section 7.3. Strictly hyperbolic systems are not common in applications. For example, the linearized compressible Euler equations in 3D are not strictly hyperbolic. In Agranovich (1972) it is shown that the previous theorem remains valid for strongly hyperbolic systems if there is a transformation S(x.y, J.u/'i.u^) which is analytic in u>\, u>2 and smooth in x, y. t, and which diagonalizes the symbol In other words, for all arguments we have A corresponding result holds for strongly hyperbolic half-space problems in any number of space dimensions. One can show that a transformation S with the above properties exists for the linearized compressible Euler equations; there- fore, the theory outlined above does apply. In Oliger and Sundstrom (1978) a rather complete discussion of boundary conditions for the Euler equations is given. The existence of the transformation 5 seems to be typical for hyperbolic systems which appear in applications. Let us note further that problems in more general domains (with smooth boundaries) can be reduced to half-space problems and pure Cauchy problems by partition of unity arguments and local transformations. For this reason the study of half-space problems is most important. The main restriction we made throughout Section 8.4 is the assumption of a noncharacteristic boundary. Gen- eral results for a characteristic boundary are not known. However, Case 2 of Theorem 8.2.7 has been treated in Majda and Osher (1975) in the Fourier- Laplace transform framework. 322 Initial-Boundary Value Problems and the Navier-Stokes Equations 8.5. Remarks on Mixed Systems and Nonlinear Problems Mixed systems. As in the one-dimensional case, we can extend our results to mixed hyperbolic-parabolic systems. Initial-boundary value problems for parabolic systems were discussed in Section 8.1, for hyperbolic systems in Sec- tion 8.2. Now couple two such systems - in the same way as in the Cauchy problem - by adding lower-order terms. (See Section 6.3.1 or Theorem 2.5.1.) The initial-boundary value problem for the coupled system is well-posed with the same boundary conditions provided that the boundary is noncharacrerislic for the hyperbolic part. If the boundary is characteristic and the conditions of Section 8.2.5 are met, then the problem is still well-posed if we use a Diric t condition for the parabolic part. Much more general boundary conditions can be discussed in the framework of Laplace-Fourier transforms. Here we refer to Strikwerda (1977). For the treatment of the N-S equations, one is also interested in the behavior of solutions near the boundary when the viscosity converges to zero. We refer to Gustafsson and Sundstrom (1978) and to Michelson (1988). Nonlinear problems. No new difficulties arise for short-time existence of so- lutions of quasilinear problems. Consider, for example, a half-plane problem for a quasilinear symmetric hyperbolic system of the form with initial and boundary conditions We assume that all coefficients and data are Cx-smooth and that A, A\, B, B\ are Hermitian. The matrix .4(0, y, t) is assumed to be nonsingular and diagonal. The variables u = (u7, u11) are partitioned according to the signs of the diagonal entries of this matrix; see Section 8.2.1. One can show a result of the following type: Theorem. Assume that the linear problem which is obtained for (. = 0 is strongly well-posed. Then -for sufficiently small e - the nonlinear problem has Initial-Boundary Value Problems in Several Space Dimensions 323 a unique smooth solution in a time interval 0 < t < Tf. Here Te -» oof or c — > 0. As in Section 5.3, one can also derive asymptotic expansions of the solution. Clearly, nonlinear systems usually do not occur directly in the form described above. Nevertheless, local existence results can frequently be derived by a theorem of the above type. To illustrate the principle, we consider a quasilinear system together with quasilinear boundary conditions. Now x can be a vector of any dimension. Then we solve the linear problem i.e., we evaluate the nonlinear coefficients at the initial data. The same process applies to the boundary conditions for w. We substitute into the original equations. For the new variables u we obtain a system which has the above structure in a time interval of length 0 < t < Ke, i.e., the nonlinear terms are multiplied by e. Results of the type sketched above are valid for parabolic, for hyperbolic, and for mixed hyperbolic-parabolic problems. The systems need not be symmetric. Estimates can be derived which yield strong well-posedness in the generalized sense. We refer to Strikwerda (1977) and to Michelson (1988). Notes on Chapter 8 Energy estimates for initial-boundary value problems have a long history. Some references are Friedrichs (1958), Ladyzhenskaya (1984), and Lions (1961). For systems du/dt = P(d/dx)u with constant coefficients, Hersch (1963) derived necessary and sufficient conditions for weak well-posedness of half-space prob- lems. As in Section 8.5, after Fourier and Laplace transform, he obtained an eigenvalue problem. The half-space problem is weakly well-posed if and only if the eigenvalues s satisfy Re s < const. For a homogenous operator P(d/dx) this condition becomes: There are no eigenvalues s with Re s > 0. For parabolic systems the corresponding condition leads to weakly well-posed problems in the sense of Chapter 2, even for problems with variable coefficients. 324 Initial-Boundary Value Problems and the Navier-Stokes Equations This was proved by Agranovich and Vishik (1964), Eidelman (1964), and Solon- nikov (1965) using generalizations of Levy's parametrix. For hyperbolic first-order systems with variable coefficients we required the stronger condition: there arc no eigenvalues or generalized eigenvalues s with Re s > 0. A first result in this direction is due to Agmon (1962). The general theory for first-order systems was developed by Kreiss (1970) with extensions by Ralston (1971), Rauch (1972a,b, 1973) and Agranovich (1971, 1972). For single higher-order equations the theory is due to Sakamoto (1970, 1982). There is no genera] theory for equations with variable coefficients which satisfy the weaker eigenvalue condition (Lopatinskii condition) pointwise. For more details we refer to Eskin (1983) and Volevich and Gindinkin (1980). For difference approximations a similar theory has been developed. See Kreiss (1968), Osher (1972), and Gustafsson, Kreiss, and Sundstrom (1972) for results in one dimension, and Michelson (1983, 1987) for results in more dimensions. 9 The Incompressible Navier-Stokes Equations: The patially Periodic Case In the next two chapters we apply the general theory to the Navier-Stokes equations. For references and deeper insight we refer the reader to von Wahl (1985), Temam (1984, 1983), and Ladyzhenskaya (1969). 9.1. The Spatially Periodic Case in Two Dimensions Without further mentioning, all functions in this section are assumed to be real, C*00 -smooth and 27r-periodic in x and y. We consider the viscous incompressible N-S equations and give initial conditions Clearly, consistency of (9. 1 .2) and (9. 1 .3) requires us to assume that 325 326 Initial-Boundary Value Problems and the Navier-Stokes Equations Notations. We define an inner product and a norm for scalar functions by With 1 we denote the function which is identically one. Without changing the terms px, pv in (9.1.1), an arbitrary function po(0 could be added to the pressure. Thus p is not uniquely determined by the incompressible equations. We use the normalization* Our aim is to prove Theorem 9.1.1. Assume the compatibility condition (9.1.4) to be satisfied. The spatially periodic Cauchy problem (9.1.1 M9. 1 .3) with side-condition (9. 1 .5) has a unique solution (u, v, p). The solution is C°° -smooth and exists for all time 9.1.1. Outline of the Proof; Vorticity Equation A simplifying assumption. Suppose that (M, v,p) is a solution. The first mo- mentum equation (9.1.1) gives us, for the spatial average of u, Therefore is known in terms of the data. If we introduce then (!,M) = (1, v) = 0 and *The ambiguity disappears for the compressible equations and the normalization (9.1.5) can be justified by a limit process. The Incompressible Navier-Stokes Equations: The Spatially Periodic Case 327 and similarly for v. Since the coefficients (l.w), (l,v) can be considered as known, the study of the system for (u, v) is very similar to that of the system (9.1.1H9.1.3) with Henceforth we assume (9.1.6), for simplicity, and leave the extensions to the reader. The average flow quantities vanish: Vorticity. The expression is called the vorticity. Differentiation of the first momentum equation (9.1.1) with respect to y, of the second with respect to z, and subtraction gives us the vorticity equation Clearly, £ satisfies the initial condition If u. v were known, then the vorticity could be determined by the linear parabolic problem (9.1.7), (9.1.8). Conversely, as we will show, if £ is known, then at each time t the velocities are uniquely determined by the inhomogeneous Cauchy-Riemann system with side-condition This suggests the iteration 328 Initial-Boundary Value Problems and the Navier Stokes Equations We will use this iteration to prove global existence. Once u, v are determined, the pressure p can be obtained from 9.1.2. Some Results for Elliptic Equations We need some elementary results about elliptic equations, which we prove by Fourier expansions. Recall that all functions are assumed to be C°° -smooth and 27r-periodic in x and y. Lemma 9.1.2. If F = F(x, y) is a function with (1 , F) = 0, then has a unique solution w. There is a constant K independent of F with Thus we "gain" two derivatives. Proof. We can write The sums extend over all wave- vectors k ^ 0 with integer components k\ k2 In terms of Fourier coefficients, the equation Aw> = F is equivalent to This shows existence, uniqueness, and the estimate. Lemma 9.13. Let F, G denote functions with There is a unique solution (p of This function (p is also the unique solution of The Incompressible Navier-Stokes Equations: The Spatially Periodic Case 329 Proof. As in the proof of the previous lemma, we write F, G, and (f> as Fourier series. By assumption, The equations (£>x = F, ^y = G are equivalent to If <£(k) satisfies these equations then Conversely, if <£(k) is defined by (9.1.12), then (9.1.11) follows from (9.1.10). The remaining statements are shown in the same way. Lemma 9.1.4. Suppose that (1 , F) = (1 , G) = 0. Then the system has a unique solution (u,v). For each j = 1,2,..., there is a constant Kj independent of F, G with If F and G dependent smoothly on t, then so do u and v. Proof. If the system has a solution then and therefore Conversely, if we define u and v by the above formulae, the system is solved and the estimates follow from Smooth dependence on t holds, because we can differentiate the Fourier series ter ise. 330 Initial-Boundary Value Problems and the Navicr-Stokcs Equations 9.1 J. Vorticity Formulation of t Navier-Stokes Equa ns The Navier-Stokes equations for the primitive variables u, v, p are equivalent to the vorticity equation together with the inhomogeneous Cauchy-Riemann system for u, v and the elliptic pressure equation More precisely, Theorem 9.1.5. Suppose that fx + gv = 0, (I,/) = (1, g) = 0. If u, v, p solve ij defined as £ = vx — uv, then and p satisfies (9.1.13). Conversely, ifu, v, ^ solve (9.1.15), and p is defined as the solution of (9. 1.1 3), then u, v, p $o/v* the first set of equations. Proof. First suppose that u, v, p solve the equations (9.1.14). The system (9.1.15) has been derived in Section 9.1.1. Also, differentiation of the first momentum equation with respect to x and of the second with respect to y yields, for the pressure, This proves the first part of the equivalence. Conversely, assume that u, v, £ solve the equations (9.1.15). We define F and G by and obtain The Incompressible Navier-Stokes Equations: The Spatially Periodic Case 331 By Lemma 9. 1 .3 there is a unique solution (p of Clearly, if we set p = -(p, the momentum equations are satisfied by w, v, p. Also, and therefore — (p = p is the solution of (9.1.13). The initial conditions u — /, v = g at t = 0 follow from the uniqueness of the solution w, v of and the initial condition for £, £ = ^x — fy at t = 0. 9.1.4. Uniqueness and Existence To begin with, we show uniqueness; we assume that w_,, Vj, PJ, j = 1,2, solve (9.1.14). Subtraction yields for the differences u — u\ - U2, etc. Thus, using integration by parts, and therefore the initial conditions imply that Furthermore, we obtain from (9.1.13) that Ap = 0, (l,p) = 0, and thus p = 0. This shows the uniqueness of the solution of (9.1.14). To prove existence we use the iteration (9.1.9) starting with u° = v° = 0. For every fixed n we have linear equations with C°° -coefficients; thus wn, vn, £n exist for all time. (See Lemma 9.1.4 for the determination of un+l, vn+l in terms of £"+'.) We first show the following estimate for the sequence fn. Lemma 9.1.6. Let h = gx - fv denote the initial data of £n. There is a constant K depending on v and \\h\\H\ , but independent ofn and t with 332 Initial-Boundary Value Problems and the Navier-Stokes Equations Proof, a) We write f = £n+l, u = un, v = vn; thus Therefore. Here and thus A similar computation for (^, v£ Hence we have, for all n, b) Integration by parts and (9.1.17) imply that Here Adding a similar estimate for (d/di)\\£y\\2, one finds that The Incompressible Navier-Stokes Equations: The Spatially Periodic Case 333 and therefore c) A Sobolev inequality (Theorem A. 3.6) and Lemma 9.1.4 give us that d) To summarize, if we add and (9.1.19), we obtain that Here c is a constant depending only on v. By (9.1.18), Therefore, is bounded independently of n and t. Next we show that all space derivatives of £n can be estimated independently of n in any given time interval. It is convenient to note that mixed space derivatives can always be estimated by pure space derivatives. More precisely, for all functions w, where 7^ is a numerical constant independent of w. This can be shown by Fourier expansion. (See formula (A.3.2) in Appendix 3.) Lemma 9.1.7. Suppose that the sequence £n is defined by the iteration (9.1.9), and T > 0 is arbitrary but fixed. For each j — 1,2,..., there exists Kj with The constant Kj depends on f, T, and \\h\\fj,, but is independent ofn. Proof. We abbreviate, as before, f = £n+I, u = un, v = vn, and let j > 2. Using induction on j, we can assume the above estimate to hold up to j — 1; thus 334 Initial-Boundary Value Problems and the Navier-Stokes Applying the operator D\ to (9.1.17), we obtain that Therefore, The differentiated products in the last formula are evaluated by Leibniz' rule. One obtains an estimate in terms of Consider the u-terms for definiteness, and For k = j - 1: and A similar estimate applies to the v-terms in (9.1.21). Thus we have show Clearly, the term can be bounded in the same way. Introducing the functions The Incompressible Navier-Stokes Equations: The Spatially Periodic Case 335 and observing (9.1.20), we have that Here c does not depend on n. Therefore, by Gronwall's Lemma 3.1.1, in 0 < t < T. By Picard's Lemma 3.3.4 we obtain the desired bound independent of n. This completes the proof of the lemma. The above considerations (and Sobolev inequalities) show that we have bounds independent of n for all derivatives of the functions in 0 < t < T. We can now argue in exactly the same way as in the proof of local existence for Burgers' equation; see Theorem 4.1.6. Thus - in any time interval 0 < t < T - the sequence un, vn. £n converges along with all its derivatives to the solution w, v, f of (9.1.15). The Equivalence Theorem 9.1.5 shows existence of a solution u. i>, p of the N-S equations. Theorem 9.1.8. Consider the system of equations (9.1.15) with initial data h = gx — fy There is a unique Cx -solution. The solutions exists for all time. 9.1.5. Maximum Principle and Smallest Scale It is an important property of the vorticity (in 2D) that it obeys a maximum principle. We formulate this next. Lemma 9.1.9. For all t > 0 the vorticity £ satisfies Proof. Let a > 0 be a constant and set 336 Initial-Boundary Value Problems and the Navier-Stokes Equations The vorticity equation yields Suppose I attains its maximum over at (#o5 2/0? *o)'» this contradicts the differential equation for £. Thus we have to = 0; i.e., a positive maximum can only be attained at t = 0. Similarly, a negative minimum can only occur at t = 0. If we send a — > 0, the estimate for £ follows. The maximum principle allows one to improve the estimates of the previous section considerably. In Henshaw, Kreiss, and Reyna (1988) it is proved that can essentially be bounded in terms of if denotes the Fourier expansion of the vorticity, these results imply that provided that |f( ,0)|oo = W). Thus, there is very little energy in the high wave-numbers |k| ^ i>~'/2, and therefore the smallest spatial length scale is ~ i/1/2. We have restricted ourselves to smooth initial data. However, in the same way as for Burgers' equation, we can show a smoothing principle for the vorticity: We can estimate in terms of )(£(•, 0)||2. Therefore we can define generalized solutions for nons- mooth initial data. The Incompressible Navier-Stokes Equations: The Spatially Periodic Case 337 9.2. The Spatially Periodic Case in Three Dimensions As in the previous section, we restrict ourselves for simplicity to the case with zero forcing. The equations have the form Without further mentioning, all functions in this section are assumed to be real, C00-smooth and 27r-periodic in x, y. z . We require that and obtain, as in the 2D case, as long as the solution exists. 9.2.1. Vorticity Formulation; Local Existence In three space dimensions the vorticity is defined as the vector If one applies the curl-operator to the momentum equations and observes that div u = 0, then one finds after some computation the vorticity equation Here B(u) is the Jacobian matrix of the velocity field, To show a local existence result, we want to use an iteration like (9.1.9) in the 2D case. To this end, we first note that as follows directly from the definition (9.2.2). Also, integration by parts implies the vanishing of the spatial averages 338 Initial-Boundary Value Problems and the Navier-Stokes Equations This allows us to determine u in terms of £ by a generalized Cauchy-Riemann system. We show by Fourier expansion: Lemma 9.2.1. Suppose that £ = £(x) satisfies Then the system with side-condition has a unique solution u. For each j = 1 , 2, . . . , there is a constant dent of £ withj indepen- Proof. Suppose that the system has a solution. We can combine the equations to obtain These equations allow us to express the Fourier coefficients of u, v, w in terms of the Fourier coefficients of £1, 6, 6- For example. This shows uniqueness and the estimates. Conversely, defining u, v, w in terms of its Fourier coefficients by formulae as above, we obtain existence. To prove local existence for the N-S equations, we consider the sequence defined by The Incompressible Navier-Stokes Equations: The Spatially Periodic Case 339 starting with u° = 0. In contrast to the 2D case, one obtains here an estimate °f llf HH' on'y m a fin»te (small) time interval 0 < t < T, where T depends on the /fi-norm of the initial vorticity, i.e., on ||curlf||//i. Then one can show - as in the 2D case - that the functions un, £n are uniformly smooth in this interval 0 < t < T. As before, it follows that the functions un. £n converge along with all their derivatives to C°° -functions u, f which solve the vorticity equation (9.2.3) and the generalized Cauchy-Riemann system (9.2.4) at each time t , 0 < * < T. To obtain a solution u, p of the N-S equations, it remains to determine the pressure. We take the divergence of the momentum equations to find If we impose the side-condition (l,p) = 0, the pressure is uniquely determined. (Note that (1, J) = 0.) Except for the analog of Lemma 9.1.6 (which is replaced by a local result), these details are more or less the same as in the 2D case. Also, the uniquenes of a solution follows as in Section 9.1.4. We summarize the result in Theorem 9.2.2. The spatially periodic Cauchy problem for the 3D viscous incompressible Navier-Stokes equations with C°°-data has a unique C°° -solution existing in some time interval 0 < t < T. The time T > 0 depends only on the viscosity v and on ||h||H, . Here h = curl f is the vorticity at time t = 0. 9.2.2. Extension of the Solution Interval for Bounded Velocities It is not known whether the solutions u, p of the viscous incompressible N-S equations (9.2.1) exist for all time 0 < t < oo or whether it is possible (for some initial data) that the solution ceases to exist due to the development of a singularity in finite time. If one assumes a solution in a time interval one can ask for conditions which allow the extension of the solution beyond T. For example, the Local Theorem 9.2.2 implies: If 340 Initial-Boundary Value Problems and the Navier-Stokes Equations is finite, then the solution can be extended beyond T. In other words, if the solution ceases to exist at t — T, then the function must have a singularity at t = T. The boundedness of ||f !!„, implies bounded- ness of the velocity: In this section we want to show that boundedness of the velocity is also sufficient to extend the solution beyond T. One can prove even more, namely estimates of all derivatives of the solution in terms of the initial derivatives and the maximal velocity. Notations. For any j ; = 0, 1 , . . . , let We assume throughout that u, p is a smooth solution of (9.2.1) defined in 0 < t < T; we start with the basic energy estimate. This estimate is valid without any assumption on jujoo . Lemma 9.2 3. The energy-change satisfies and therefore Proof. From (9.2. la), Here S consists of nine terms. The Incompressible Navier-Stokes Equations: The Spatially Periodic Case 341 Integration by parts yields that and similarly Therefore, Furthermore, and the lemma is proved. We now proceed to estimate the first space derivaties of u and assume the velocity to be bounded, Since the dependence of the estimates on the viscosity v is of some interest, we will make it explicit below. The constants c, c\, 02, etc. are numbers independent of v, T and the solution u under consideration. Lemma 9.2.4. For the function J\(t) measuring the first space derivatives ofu we have that The right-hand side also is a bound for Proof. Application of any of the operators to the momentum equations yields 342 Initial-Boundary Value Problems and the Navier-Stokes Equations Therefore, Summing these three estimates for D — D\, DI, DT,, we find that Integration of the inequality - together with the estimate of / J2 dt by Lemma 9.2.3 - proves the result. We want to show now a similar estimate for t/2(0; more precisely, Then we can proceed exactly as in the proof of the previous lemma and obtain an estimate of J2(t) in terms of To ghow (9.2.6), we apply any of the operators to the momentum equations: Therefore, Now Leibniz' rule is applied to the term for example, The Incompressible Navier-Stokes Equations: The Spatially Periodic Case 343 Typical terms which need to be estimated are There is no difficulty in bounding the third term by (Actually, the three terms of this form add up to zero since div u = 0). Also, we can use integration by parts to remove the first derivative from uy and from Dv in the first and in the second term, respectively. This shows that (9.2.6) holds. To summarize, if the velocity is bounded in 0 < t < T, then J|(0, -/z(0 are also bounded. Therefore, is bounded, and our previous considerations show that the existence interval can be extended beyond T. Theorem 9.2.5. If the solution u to (9.2.1) satisfies a bound ; then u can be extended to a solution in an interval 0 < t < T + AT. AT > 0. Remark. Instead of (9.2.7) one only needs to assume Then the solution can be extended beyond T; see von Wahl (1985). This page intentionally left blank 10 The Incompressible Navier-Stokes Equations under Initial and Boundary Conditions In two space dimensions the nonlinear equations have the form We consider the equations in the region and prescribe initial conditions and boundary conditions at x = 0, x = 1. All data are assumed to be Cx-smooth and 27r-periodic in y; we seek a C30-solution which is 27r-periodic in y. 10.1. The Linearized Equations in 2D In this section we start our discussion of the linearized equations under boundary conditions. To this end, let 345 346 Initial-Boundary Value Problems and the Navier-Stokes Equations denote smooth functions of x, y, J which are 27r-periodic in y, and substitute into the N-S equations. Neglecting terms quadratic in the corrections u', etc. and dropping the prime ' in our notation, we find the linear equations where The forcing function F in (10.1.2) is the defect of (V,V,P) in (10.1.1). It is convenient to study (10.1.2) for a general smooth matrix function A(x.y.t) which is 27r-periodic in y. Our plan is to approximate (10.1.2) by the evolutionary system We shall choose boundary conditions for (u. v. p) in such a way that our theory for mixed parabolic-hyperbolic equations applies for every fixed e > 0. Then, if we can estimate the solutions of (10.1.3) and their derivatives independently of e, we obtain a smooth solution of (10.1.2) in the limit e -* 0. We start with the basic estimate. Lemma 10.1.1. Suppose that u,p solve (10.1.3). There is a constant c > 0, depending on the coefficients U, V, in the differential equation (10.1.3) but independent of e and F, with (Here we use the notation for boundary terms.) The Incompressible Navier-Stokes Equations under Initial and Boundary Conditions 347 Proof. The differential equations (10.1.3) yield The result follows from integration by parts and estimates of zero-order terms. The boundary contributions appear from integration by parts in the ^-variable. Boundary conditions providing the basic estimate. The boundary conditions - together with the term on the right-hand side of (10.1.4) - have to provide bounds for the boundary term We consider different possibilities. Case 1. The lines x = 0, x — 1 represent walls. In this case it is reasonable to prescribe the boundary velocities for the nonlinear system (10.1.1). (This is obvious for u and confirmed by observation for v .) Thus one chooses a base flow with and obtains for the linearized system. Clearly, the boundary term (10.1.5) vanishes, and the basic estimate follows. Case 2. Inflow at x = 0, outflow at x = 1. Now one chooses Again, we can use the Dirichlet conditions (10.1.6). Also, the following condi- tions of Neumann type may be used: 348 Initial-Boundary Value Problems and the Navier-Stokes Equations Here QJ , /3j , QJ , hj are given smooth functions which are 27r-periodic in y. For the boundary term (10.1.5) we obtain Here etc. Using a (one-dimensional) Sobolev inequality for each fixed y and inte- grating over y, we have Thus the basic estimate follows Lemma 10.1.1. Remarks. 1. The above boundary conditions are of the general form discussed for mixed parabolic-hyperbolic systems in Section 8.5. Therefore, if smooth initial data are given for u, v, p and provided that suitable compatibility conditions are fulfilled at t = 0, the equations (10.1.3) determine a unique smooth solution satisfying these boundary conditions. 2. In many applications the viscosity constant i/ > 0 is very small. Then, if estimate (10.1.8) is employed, rapid exponential growth of the energy is allowed. However, if the coefficients QJ , fy in the boundary conditions (10.1.7) are suitably chosen, one obtains an estimate of the boundary term (10.1.5) by c{||0||2 + ||/i||r}- In this case, inequality (10.1.8) need not be employed, and the rate of exponential growth becomes independent of i/. The same is true, of course, for Dirichlet boundary conditions. 3. It is no restriction to assume the Dirichlet conditions (10.1.6) to be ho- mogeneous. If inhomogeneous data are given, we can introduce new variables u — u — 0 which satisfy the homogeneous condition, and if we differentiate with respect to y and t, the condition remains homogeneous. In contrast to this, consider (10.1.7). Of course we could also transform to the homogeneous case, but if we want to estimate y- and £ -derivatives, new inhomogeneous terms are introduced, and nothing is gained. The Incompressible Navier-Stokes Equations under Initial and Boundary Conditions 349 4. We have assumed conditions of the same type at x = 0 and x = 1, either of Dirichlet or of Neumann type. Clearly, this assumption is not necessary. In applications one often specifies u = v = 0 at inflow and (10.1.7) at outflow. If i/ > 0 is small and a. /3 are suitably chosen (see Remark 2), such a Neumann condition at outflow leads to a "smoother" solution than the simple requirement u = v = 0 at outflow. 10.2. Auxiliary Results for Poisson's Equation In this section we prove some elementary estimates for solutions of Laplace's equation and Poisson's equation in the strip As in the last section, all functions are assumed to be C^-smooth and 2?r- periodic in y. We apply Fourier expansion in y-direction and use the notation Then Parseval's relation reads and integration over x gives us Lemma 10.2.1. Suppose that u e C°° solves and u is 2ir-periodic in y. There is a constant K\ endent of u with If the spatial average ofu vanishes, i.e., 350 then with K2 independent of u.Initial-Boundary Value Problems and the Navier-Stokes Equations Proof. Since Aw = 0 it follows that Thus we have a representation Using (10.2.1) for uv, we find that Similarly, Elementary computation shows that (for k ^ 0) Using we find that the estimate ||u for all integers k ^ 0. The Incompressible Navier-Stokes Equations under Initial and Boundary Conditions 351 To show the estimate for \\u\\2, we first observe that and therefore Thus (10.2.1) yields This proves the lemma. An easy generalization of the last lemma is Lemma 10.2.2. Suppose that u e Cx solves and u is 2'K-periodic in y. For every j > 1 there is a constant Kj independent of u such that Proof. The equation Aw = 0 allows us to express even order ^-derivatives in terms of y-derivatives: Thus, any derivative term appearing in |w|//j reduces to The estimate of the last lemma finishes the proof. Next we want to show estimates of u in terms of boundary data on the lines x = 0, x = 1. 352 Initial-Boundary Value Problems and the Navier-Stokes Equations Lemma 10.23. Let 2it-periodic C00-functions go(y), g\(y) be given. The boundary value problem has a unique C™ -solution w(x, j/) which is 2it-periodic in y. For any j — 0, 1,... there is K3 independent of g with Here denotes the Hi-norm of the boundary data. Proof, First assume that there is a smooth solution u. We use (10.2.1) and the representation (10.2.2) to obtain On the other hand, The formula (10.2.2) gives us the relations Thus we find that d the bound ||ii||2 < /i'ollpllr follows. Similarly. The Incompressible Navier-Stokes Equations under Initial and Boundary Conditions 353 If we apply this estimate to instead of u, and use Lemma 10.2.2, then we obtain the desired bound of \\u\\2HJ . The bound clearly implies uniqueness. Existence of a solution follows , because we can define u in terms of its Fourier expansion. Now consider the inhomogeneous differential equation where F € C00 is 2n -periodic in y. We ask for estimates of a C°° -solution with homogeneous boundary data and require, as before, that u is 27r-periodic in y. We show Lemma 10.2.4. The above strip problem has a unique solution u. For any j ; = 0, 1 , . . ., there is a constant Kj independent of F with In this sense, we "gain" two derivatives. Proof. First assume that the problem has a solution u. Fourier expansion of (10.2.3), (10.2.4) yields Thus each function is determined by an ordinary boundary value problem. By we denote the £2-norm of such a function of x, and obtain through integration by parts (for k ^ 0) 354 Initial-Boundary Value Problems and the Navier-Stokes Equations Thus and with (10.2.5a) For k = 0 an explicit integration of the boundary value problem (10.2.5) shows that Thus we have obtained that and Parseval's relation (10.2.1) yields This shows the desired bound for j = 0. There are no difficulties in estimating higher t/-derivatives of because (10.2.3), (10.2.4) can be differentiated with respect to y. If we want to estimate higher i-derivativcs, we use 00.2.3) and replace them by y-derivatives. e.g., Thus we obtain the estimate of the lemma. Existence of a C3* -solution follows as before since we can define u by its Fourier expansion. If both - the boundary data and the differential equation - are inhomogeneous, i.e, then we decompose the problem in an obvious way, and obtain the solution as a sum, The Incompressible Navier-Stokes Equations under Initial and Boundary Conditions 355 Here Lemma 10.2.3 applies to w(1) and Lemma 10.2.4 applies to w(2). One obtains the estimate 10.3. The Linearized Navier-Stokes Equations under Boundary Conditions The linearized system (10.1.2) is not of the standard form of an evolution equa- tion since we do not have an equation for pt. To show the existence of a so- lution of (10.1.2) under initial and boundary conditions, we go over to (10.1.3) for e > 0 and send e — > 0. 10.3.1. Convergence Theorem We consider the system (10.1.3) and restrict ourselves, for simplicity, to the no-slip condition (10.3.1) and the initial condition As previously, the forcing function F and the coefficients U, V, A in (10.1.3) are assumed to be (7°° -smooth and 27r-periodic in y. Also, we require that This ensures compatibility of the data at t = 0. For each e > 0 the initial- boundary value problem (10.1.3), (10.3.1), (10.3.2) has a unique C°°-solution which is 27r-periodic in y. To begin with, note that at t = all derivatives of u, p are = 0, as follows from (10.3.3). Our aim is to show that all derivatives of u, p can be bounded independently of e in any interval 0 < t < T. Then, as e — >• 0, we obtain a smooth solution of the linearized Navier-Stokes equations. As remarked earlier, the solution is only unique if one fixes a (time-dependent) constant to determine the pressure, e.g., 356 Initial-Boundary Value Problems and the Navicr-Stokes Equations We will show Theorem 103.1. Consider the linearized viscous incompressible equations (10. 1.2) in with boundary conditions u = v = Qatx = Q, x — 1 , initial condition u = v = 0 at t — 0, and side -condition (10.3.4). Assume that the data and coefficients are C°°-smooth, 2^-periodic in y, and fulfill (10.3.3). Then there is a unique C°° -solution which is 2n -periodic in y. In any interval 0 < t < T, the solution is the limit of the corresponding solutions of (10. 1.3) as c — * 0. The assumptions of the theorem can be relaxed: One can allow, for example, certain inhomogeneous initial conditions and different boundary conditions; see Section 10.3.2. The proof of the estimates of all derivatives of u = u£ , p = pf proceeds in several steps. We fix a time interval 0 < t < T. For brevity, we will call a function u = u>{(#, y, t) estimated, if we have established a bound for which is uniform in 0 < e < 1 . As previously, the main difficulty is estimating ^-derivatives, because one cannot differentiate the boundary conditions in x- direction. Step 1. The basic estimate (see Lemma 10.1.1) gives us bounds for u, tp. We can differentiate the differential equation and the boundary conditions arbitrarily often w.r.t. y and t and obtain systems of the same form where the lower-order terms are already estimated. Thus we get bounds for all y, i-derivatives of u, cp. In particular, we have bounded and the basic estimate (10.1.4) gives us a bound for ux. From the differentiated systems we obtain, with the same argument, bounds for all t/, < -derivatives of ux. To summarize, we have estimates for and all j/, ^-derivatives of these functions. Step 2. We differentiate the equation ept + ux -\-vv =Q w.r.t. x and replace uxx using the first equation (10. 1.3a). Then we obtain The Incompressible Navier-Stokes Equations under Initial and Boundary Conditions 357 where the function H\ is already estimated. Integrating the above ordinary differential equation for px(x, y, t) in time and observing that we have bounded px. Also, since all y, ^-derivatives of H\ are estimated, we have bounds for all y, ^-derivatives of px. Step 3. Differentiating the u-equation (10.1.3a) w.r.t. x, the v-equation w.r.t. y, and summing the two results, we obtain that where L(u,ux,uy) is a linear expression in u,ux,uy. We replace ux + vy by — ept and find that The function #2 is already estimated, and using the same argument as in Step 2, we have established a bound for the function and all its y, t-derivatives. At each time t > 0 the pressure p can be written as a sum where and From Lemma 10.2.4 we obtain bounds for p2- 7>2x. P2xx and all y, ^-derivatives of these functions. In Step 2 we have bounded the function px and all its y, t- derivatives; thus we can estimate the difference p\x and all its y, t-derivatives. 358 Initial-Boundary Value Problems and the Navier-Stokes Equations Step 5. We want to derive a bound for p\ itself. To this end note that as follows from tpt + ux + vv =0 and the boundary conditions. Therefore, the function has spatial average zero and, using Lemma 10.2.1, we can bound p\. (Note that Pi* = pix-) Since ps is already bounded, we obtain an estimate for p\ itself. In the same way we obtain also a bound for each time derivative pit, put, etc., and Lemma 10.2.2 yields bounds for all other derivatives of p\. Thus we have established estimates for all derivatives of p\ . Step 6. We have shown estimates for p, px, pxx, u, ux and all y, ^-derivatives of these functions. It remains to bound higher ^-derivatives (and their y, t- derivatives); this can now be done recursively as follows: The equations ( 10. 1 .3a) can be solved for uxx, vix, and therefore these functions are also bounded. Thus all first derivatives of HI are estimated, and (10.3.5) gives us bounds for all third derivatives of p. Then (10.1.3a) yields bounds for uIXI, uIIIX. A simple induction argument finishes the proof. 103.2. Remark on Initialization In the last section we have assumed that the initial conditions are homoge- neous. Let us explain the reason for this. Suppose that we would give general nonhomogeneous initial data For every fixed e > 0 the corresponding initial-boundary value problem (10.1.3) has a solution. However, we need bounds of the derivatives which are inde- pendent of e. Our estimates of the previous section show these bounds if the y, ^-derivatives are bounded independently of 6 at £ = 0. For the y-derivatives we have no difficulties, provided the boundary conditions are compatible with the initial conditions. Therefore, the crucial question is // one can bound the time derivatives independently oftatt = 0. By (10.1.36) the first time derivatives are bounded independently of e at ( = 0 if and only if 360 Initial-Boundary Value Problems and the Navier-Stokes Equations For example, the functions may satisfy inhomogeneous boundary and initial conditions. We substitute into (10.1.1) and obtain equivalent (nonlinear) equations for the corrections u', v', p'. If we drop the prime ' in our notations, the equations read Here F is a new forcing which is determined by {/, V. P. This system can be approximated by (10.4.1) together with If we have boundary conditions (10.3.1), (10.3.2) and assume (10.3.3), then we can proceed in the same way as above for the linearized equation. As e — > 0, we obtain convergence to a solution of the nonlinear incompressible equations. The process which has been chosen here to obtain a solution of the incom- pressible equations does not describe the passage of the compressible to the incompressible N-S equations. If one wants to study this limit, one should replace (10.4.2) by (We recall the continuity equation pt + div (pu) = 0 and formally substitute p - 1 + ep.) In both cases, (10.4.1) with (10.4.2) or (10.4.1) with (10.4.3), we have a coupled hyperbolic-parabolic system. To illustrate the differences between (10.4.2) and (10.4.3), we assume inflow (U > 0) at x = 0 and require the boundary condition u = u = 0atx = 0. Whereas the boundary x =. 0 is characteristic for the equation (10.4.2), the variable p is an ingoing charac- teristic variable for (10.4.3). Therefore, no boundary condition for p is needed with (10.4.2), but (10.4.3) requires a boundary condition. For example, one can prescribe For the limit-equations (€ = 0), no boundary condition for p is allowed. In Kreiss, Lorenz, and Naughton (1988) it is shown that convergence for c. — > 0 is also obtained with (10.4.3), at least away from a boundary layer at x = 0. APPENDIX 1 Notations and Results from Linear Algebra Vectors and matrices. Let C denote the field of complex numbers and let C" be the vector space of column vectors We define an inner product and a norm by Here Hj is the complex conjugate of Uj. If A = (aj/t) € Cn-n is a complex n by n matrix, then denotes the complex conjugate transpose of A It holds that The spectral-norm of A is 361 362 Initial-Boundary Value Problems and Die Navier-Siokes Equations One can show that where p (B) denotes the spectral radius of a matrix B, i.e., the largest absolute value of all eigenvalues of B. A matrix U 6 C" " is called unitary if If U is unitary then, for any u E C", and therefore A matrix U with columns u1 , . . . , un e Cn is unitary if and only if i.e., the columns form an orthonomnal system. An important result from linear algebra is Schur's Theorem. Let A e C"'" denote a matrix with eigenvalues \\,...,\ in any prescribed order. There is a unitary matrix U such that is upper triangular with diagonal entries Proof. We use induction on n; the case n = 1 is trivial 1. One can choose an orthonormal basis of C" starting with u1. The matrix U\ with columns «',...,«" is unitary, and The matrix ^4 has the eigenvalues such that where RZ is upper triangular and AS, . . . , An appear on the diagonal. If we setThere is a unitarymatrix U2 Notations and Results from Linear Algebra 363 then and the induction is completed. This page intentionally left blank APPENDIX 2 Interpolation Fourier interpolation in ID. Let m e {1,2,...}, h = (2m + 1) ', xv = vh, v = 0, ±1, ±2,— Suppose that v(x) is a 1-periodic function, v(x) = v(x + 1), defined at all gridpoints x = xv. We want to interpolate v(x) by a Fourier polynomial at all grid points: Theorem A.2.1. 7"/ie interpolation problem (A.2.2) has a unique solution Proof. If we introduce the inner product then the gridfunctions 365 366 Initial-Boundary Value Problems and the Navier-Stokes Equations form an orthonormal system, i.e., The above result is obvious for k = I and follows from the geometric-sum formula otherwise. Consequently, if w(x) is an interpolant of the form (A.2.1), then Thus the coefficients a/ are uniquely determined by the data function v(x). Also, we can consider the interpolation-condition as a linear system of 2m + 1 equations for the 2m -I- 1 unknown a^. Since the solution vector is unique (if it exists), the system-matrix is nonsingular, and existence follows. Fourier interpolation is very useful since the interpolant w(x) inherits "smooth- ness" from the data v(x), with estimates independent of h. More precisely, if we introduce norms by then the following result holds: Theorem A.2.2. Ifw(x) denotes the Fourier interpolant of the data v(x) then and // is important that the factor (ir/2)2p does not depend on h. Proof. The first equation is nothing but Parseval's relation; it applies since the functions Interpolation 367 are orthonormal w.r.t. the continuous and discrete inner products. Therefore, differentiation of w(x) gives us Since we find that Here and the result follows. Fourier interpolation in more space dimensions. The generalization to s space dimensions is straightforward. Suppose v = v(x), x = (x|....,x.s), is a func- tion of s variables which is 1 -periodic in each variable xi and which is defined at all gridpoints (v\h, i^h, .... i/.,/i), v\ = 0, ±1, ±2, — Then there is a unique Fourier polynomial which interpolates v at all gridpoints, As in Theorem A.2.2, derivatives of w can be estimated in terms of correspond- ing divided differences of v\ the constants in the estimates are independent of h. Interpolation of nonperiodic functions in ID. s Consider a discrete function / defined on a grid x,, = vh, h = l/N, v = 0, 1,2, ...,7V . We want to interpolate / so that the derivatives of the interpolant can be estimated in terms of the divided differences of /, with constants independent of h. There are many 368 Initial-Boundary Value Problems and the Navier-Stokes Equations ways to do this. Here we first extend / to a periodic gridfunction so that bounds of divided differences are preserved. Then we use Fourier interpolation to obtain the desired interpolant. For transparency, we describe the corresponding extension of smooth functions first. Consider a function / e Cp[0, 1], and let 0 < 6 < | be given. We will extend the domain of definition of/to-oc<ar<oo. The extended function, again denoted by /, will have its support in —6 < x < I + 6. Furthermore, Here C depends on 8 and p but not on /, and || - ||E , || • || denote the Z-2-norrns over the whole line and the original interval, respectively. To begin with, let us preliminarily define f(x) for 1 < x < 3/2 as the solution of the differential equation with initial conditions The solution is (Note that the p-th derivative of the extended function jumps at x — 1, in general. This is not essential, however.) For x < 0 we use the corresponding construction. Now we modify the function so that its support becomes compact. To this end, let <^> € Cx denote a cut-off function with Then the desired extension is given by <^/. By the Sobolev inequality stated in Lemma A.3.10, we can estimate |£>>/(1)|, |£^/(0)|, j = 0, 1, 2, ... ,p - 1, in terms of ||I^P/|| and ||/||, and therefore we have proved Lemma A.2.3. Let f e CP be defined for 0 < x < 1 , and lei 0 < 6 < ±. One can extend the definition of f to — oo < x < oc so that the extended function f has its support in -<5 < x < 1+6 and fulfills (A.2.3). The constant in (A.2.3) is independent of f. Interpolation 369 We consider now a discrete function. Let x,, = vh, v = 0, ±1 , ±2, . . . , h = I/TV, denote a grid and let /„ = /(£„), i/ = 0, 1,2, . . . , TV, be defined at the gridpoints 0 < xv < \ . We prove Lemma A.2.4. For any p we can extend the definition of /„ to the whole discrete line so that the support of ' jv is contained in the interval — \ < x < 1 4- f and The extension and the constant depend on p, but the constant does not depend on f or h. Proof. The proof proceeds similarly to the continous case. For 1 < x,, we define /„ = f(xv) preliminarily as the solution of the difference equation As in the continous case, the solution can be written down. It has the form where Pp_i(x) is a polynomial of degree < p- 1, whose coefficients are divided differences Di/i,, TV - p < v < TV, .;' = 0, 1, 2, . . . ,p - 1. For x < 0 we make the corresponding construction. Then ^/ is the desired extension. This extension is again denoted by /. By the discrete Sobolev inequality stated in Lemma A.3.18, we can estimate D^_fv in terms of ||^/||Q v_p and ||/||o.^, and therefore we obtain the estimate (A.2.4). We can consider the extended gridfunction / as a periodic function with period 2. If w denotes its Fourier interpolant, then The latter estimate follows from Theorem A.2.2 and formula (A.2.4). We have proved Theorem A.2.5. Let /„ = f(xv) denote a discrete function, where xv — vh, h = I/TV, j/ = 0, 1,2,..., TV. Let p denote a positive integer. We can interpolate f by a function w 6 C°°[0, 1] so that 370 Initial-Boundary Value Problems and the Navier-Stokes Equations The interpolant and the constant depend on p ; the constant is independent of f and h, however. Extensions of functions in a strip. We shall now generalize the results to a strip and start again with the continous case. Let / = f(x,y). f 6 C2p(I), and assume that / is 1 -periodic in y. For every fixed y we use the construction given above. For x > 1 we obtain Therefore, Thus, we need the LI -norms of the derivatives of order 2p to estimate \\B"f/W\\E- Lemma A.2.6. We can extend the definition of f so that The corresponding interpolation result is also clear: For every p we can find an interpolant whose derivatives up to order p can be estimated in terms of divided differences of order 2p. All these results can be extended in a straightforward way to functions in a strip of $ dimensions. APPENDIX 3 Sobolev Inequalities We show here some frequently used inequalities which express bounds of one norm of a function (or of a derivative of a function) in terms of other norms of the same function and its derivatives. Typical examples are so-called Sobolev inequalities which give bounds of the maximum norm of a function in terms of Z/2-norms of derivatives. It is important that the constants entering these estimates do not depend on any specific function under consideration, but are uniform for a certain class of functions. To derive such estimates, we may always assume the functions to be C~*- smooth, for convenience. (The functions take values in Cn.) Then the estimate extends to all (less smooth) functions which can be approximated by C30- functions w.r.t. the norms entering the estimate. L2-estimates of periodic functions. Let u e CX(R), u(x) = u(x + L); i.e., u is L-periodic. As usual, We first show Lemma A.3.1. For all e > 0 and all integers j, k > \ it holds that 371 372 Initial-Boundary Value Problems and the Navier-Stokes Equations Proof. By Fourier expansion, (A.3.1) and Parseval's relation yields that Therefore, the lemma is pro d if we can show that The function has its minimum at and Hence the assertion follows. A generalization to any number of space dimensions is straightforward. Let u € (/^(R*) be L-periodic in each variable x^ and denote We use the notations Sobolev Inequalities 373 i.e., Jp measures the "dean" p-th derivatives whereas | • |//P measures all p-th derivatives. For any multi-index v, where the sum extends over all vectors uj = (u?i, . . . ,u;s) with integer compo- nents. If |^| = p then and therefore Thus one obtains the inclusion where K = K(p, s) is the number of multi-indices v of order p. A direct generalization of the previous lemma to s space dimensions is Lemma A.3.2. For all c > 0 and all integers j, k > 1 // /z0/d$ //jar The proof follows by Parseval's relation and the argument given in the proof of Lemma A.3.1. Because of the inclusion (A.3.2), we also have Lemma A.3.3. For all e > 0 and all integers j, k > 1 it holds that where C is independent of t, u, and L. Functions with compact support. Let u € Co^R*) and denote If L > 0 is so large that the support of u is contained in then we can identify u with an L-periodic function and apply the previous results. Since the estimates are independent of L, we obtain the inequalities (A.3.3), (A.3.4) for all u e Cjf. Initial-Boundary Value Problems and the Navier-Stokes Equations Maximum norm estimates of periodic functions. We start again with one space dimension and consider an L-periodic function For convenience we assume throughout that L > 1 ; we first show Lemma A.3.4. For each integer k > 1 there is a constant C* independent of Proof. From the representation of u(x) as a Fourier sum and the Cauchy- Schwarz inequality we obtain To estimate P\, we observe that and find, by Parseval's relation, To estimate PI* we write 374 Sobolev Inequalities 375 For the last estimate we used the restriction Hence it follows that and if we call* then the assertion follows. A key result in obtaining Sobolev inequalities in s space dimensions is for- mulated next; a proof can be given by transformation to polar coordinates. This result is responsible for the requirement k > s/2 in Theorem A.3.6 and it shows why maximum norm estimates depend on the number s of space dimensions. Lemma A.3.5. The integral is finite if and only ifk> s/2. Now let u € CX(R8) be L-periodic in each variable x/. The following result is (a version of) Sobolev 's inequality. Theorem A.3.6. For each integer k > [s/2] + 1 there is a constant independent of u, 0 < e < 1, and L > 1 with Proof. From the representation of u as a Fourier sum* we have that * We can assume 0 < e < 1 to be given and can solve for t > 0. Then 0 < e < I, if 4r^ > I. *The sums extend over all vectors u; = (u»i, ...,UJH) with integer components u>j. 36 Initial-Boundary Value Problems and the Navier-Stokes Equations Here To estimate PI, we first sum only over those u> for which all components u^ ^ 0. Then the sum can be bounded by an integral: It is not difficult to show that and using the previous lemma, we obtain that the above integral is finite. There- fore, If we sum the expressions in (A.3.5) over all u with u>a = 0, u>/ ^ 0 for / = 1, . . . , $ - 1, say, then we obtain in the same way a bound of the corresponding sum by Therefore, Sobolev Inequalities 377 Only in the latter estimate have we used that 0 < e < 1 < L. To summarize, and if we call then the assertion follows. One can also estimate derivatives of a function in maximum norm. The following result is a simple implication of the previous theorem and Lemma A.3.3. Again, we assume u € C°°(RS) to be L-periodic in each variable. Lemma A.3.7. Let \i/\ = j > 1, k > [s/2] + 1. There is a constant C independent of u, 0 < e < 1, and L > 1 with Proof. We apply the previous theorem to D"u instead of u and obtain If we choose then the result follows. Estimates by a product. It is sometimes convenient to have estimates by a product. For example, if we use Theorem A.3.6 for k = s, we obtain In the other case, \\u\\ > Js(u), we apply (A.3.6) with c = 1: 378 Initial-Boundary Value Problems and the Navier-Slokes Equations Thus we have proved We apply this result to Dvu instead of u and find Lemma A.3.8. There is a constant C = C(j, s) independent of u and L > 1 with Nonperiodic functions on an interval. Let u = u(x), 0 < x < L, denote a C00-function with values in C". There are points XQ,X\ with From we find that (A.3.8) for any € > 0. Similarly, and therefore for any e > 0, 77 > 0. We can use these formulae to show Lemma A.3.9. There is C > 0 independent <?/0<e<l, L>1 and, with Sobolev Inequalities 379 Proof. From (A.3.8) follows In (A.3.9) we choose 77 = c/^/e, and estimate (w^ and (wi)2^ using the above formulae with Then the choice proves the result. In the following lemma we generalize the formulae (A.3.8), (A.3.10). The dependence of the constants Ci on e will be discussed below. Lemma A.3.10. Let j> 0, k > 1 denote integers, and letO<€< 1. There are constants Ci — C,0, A:, e), i = 1,2, independent ofL > 1 and u e C^tO, L] with Proo/. First let A: = 1. The estimates are shown for j — 0, and we assume that they are valid for some j. Then (by (A.3.10)) If we choose here e\ — e/(2C), then the desired estimate for HD-'+'ull follows. Similarly, (by (A.3.8)) 380 Initial-Boundary Value Problems and the Navier-Stokes Equations and the choice e\ = c2 finishes the induction w.r.t. j. The generalization from A; = 1 to arbitrary k is straightforward. In the previous lemma we did not keep track of the dependence of Ci on e. However, since the constants are independent of L > 1 , we can determine this dependence by scaling the x-variable. One obtains Theorem A.3.11. Let j > 0, k > 1 denote integers. There are constants Ci = Ci(j,k}, i = 1,2, independent ofQ < e < 1, L > 1, and u e C°*[0,L] with Proof. Suppose that w(x), 0 < x < L, is given; we define for 0 < h < 1 If one observes that and applies the previous lemma to u(x) with e = I , then the result follows. Li-estimates for functions in a strip. Suppose that u = u(x) is a C^-function defined in the strip which is 1 -periodic in X2,.-.,ara. For each fixed x\ one obtains, by Fourier where the sum extends over all vectors and Sobolev Inequalities 381 Parseval's relation yields with We first want to show that we can estimate the £2-norm of any derivative by clean" derivatives. As in the periodic case, let Lemm A.3.12. There is a constant C — C(j) independent of u with Proof. With Parseval's relation, Case 1. v\ = 0. From one obtains the estimate Case 2. v\ = j, thus v2 = . . . = vs = 0. Here trivially ||£>"u||2 = ||I>i'w||2. Case 3. I < v\ < j ' — 1. An application of Theorem A.3.1 1 yields We can assume that 382 Initial-Boundary Value Problems and the Navier-Stokes Equations and set Then we find from (A.3.11) If we set then and Therefore, and the result follows. Next we generalize Lemma A.3.2 to functions defined in a strip. Lemma A.3.13. Letj,k> 1 denote integers. There is a constant C = C(j, independent o/O < e < 1 and u with oof. We first estimate £^w: For each fixed o/_ we can apply Theorem A.3.11 to u(-,uO, and summation over w- gives the desired estimate of D\u. For J = 2, . . . , s, Sobolev Inequalities 383 and we can argue as in the proof on Lemma A.3. 1 . Maximum norm estimates for functions in a strip. As before, let u = u(x) denote a C30 -function defined in the strip which is 1 -periodic in #2, . . . , xa. The following est ate of |w|oo is sometimes useful since it requires only one differentiation in the nonperiodic x\ -direction. Specializations of the result to 2- and 3-dimensional strips will be given below. Theorem A.3.14. Let (3, 7 denote integers with There is a constant C = C(s) independent of u and with Here we have used the notation Proof. From the Fourier representation of As in the proof of Theorem A.3.6, 384 Initial-Boundary Value Problems and the Navier-Stokes Equations Also, by (A.3.8), This yields Furthermore, If we observe that 7 > a and sum over w_ , then the assenion follows. A 2D strip. In the previous theorem we set and obtain the estimate As an implication we note Lemma A .3.15. Consider a C™ -function u = w(3?, y) defined in the strip w/i/c/i /s \-periodic in y. Then Proof. If then (A.3.12) yields the estimate. Otherwise llw^H < ||u||, and we obtain the result from (A.3.12) upon choosing t~\. Sobolev Inequalities 385 A 3D strip. In Theorem A.3.14 we set and obtain, for 0 < e < 1, (A-3.13) Similarly as before, we can derive from this an estimate by products. Lemma A.3.16. Consider a Cx '-function u = u(x, y, z) defined in the strip which is I -periodic in y and in z. Then with Proof. If then (A.3.13) yields the estimate. Otherwise and we obtain the result from (A.3.13) upon choosing e = 1. Discrete Sobolev inequalities for periodic functions. Let m e { 1 , 2, . . .}, h = (2m + I)"1, and define a grid xv = vh, v = 0, ±1, ±2,... . Suppose that u — u(x) is a 1 -periodic gridfunction, u(x) = u(x + 1), which is defined at all gridpoints x = xv. We want to show that discrete analogs of the Sobolev in- equalities hold with constants independent of h. For notations, see Section 3.2.2. Lemma A.3.17. For e > 0, j, k > 1 it holds that with constants Cj.2(e) independent of h. 386 Initial-Boundary Value Problems and the Navier-Stokes Equations Proof. Denote by w = w(x) the Fourier interpolant of u. Since for some point £„, we have iD^ujgc < \Djw\oc. Now the second inequality follows from Lemma A.3.7 (with s=l) and Theorem A.2.2. Clearly, the first inequality follows from the second. In the same way we can obtain discrete Sobolev inequalities for periodic functions in more space dimensions. Nonperiodic discrete functions on an interval Now we assume that u^ — u(xv), xv = i/h, v = 0, 1, 2, . . . , N, Nh = 1, is defined for gridpoints xv with 0 < xv < 1. As before, D+u,, = («,,+! — u^)jh for v = 0, . . . , N — 1, and we use the notation We have (A.3.14) where Let and assume for definiteness that I > k. By (A.3.14), Sobolev Inequalities 387 This yields, for any p > 0, If we apply (A.3.14) with v = £>_w, we obtain Therefore, Now we can proceed as in the proof of Lemma A.3.9 and obtain As in the continuous case, we can use (A.3.15), (A.3.16) to prove Lemma A.3.18. Let h = \/N and let u ...,N, denote a gridfunction. Then the following discrete Sobolev inequalities hold: with constants independent ofh. Discrete analogs of our estimat for functions defined in a strip are also valid; this can be shown by Fourier interpolation in the periodic directions. This page intentionally left blank APPENDIX 4 Application of the Arzela-Ascoli Theorem We will use the following result proved in analysis. It is (a special case of) the Arzela-Ascoli Theorem. Theorem A.4.1. Let ft C Rs be a closed bounded set, and let um : Q — > C" denote a sequence of functions with the following properties: (i) For each e > 0, there is a 6 > 0 independent ofm with (ii) There is a K independent of m with Then there is a continuous function u : fl! — »• Cn, and a sequence of indices nij — > CXD with To apply the result, assume that we have a sequence of functions 389 390 Initial-Boundary Value Problems and the Navier-Stokes Equations which is uniformly smooth; i.e., um € C00 for all m, and for all nonnegative integers p, q there is a constant C(p, q) independent of m with We want to show: Theorem A.4.2. There is a function u € C°°, and a sequence m,j —» oo with In short, a subsequence ofum converges along with all its derivatives to a C°°- limit. Proof. By the Arzela-Ascoli Theorem there is a continuous u and a sequence Now apply the Arzela-Ascoli result to the sequence and obtain the existence of a continuous v and a sequence jk —» oo with From we conclude that for m = m Hence u is differentiable w.r.t. x, and L«t us show that (A.4.2) Application of the Arzela-Ascoli Theorem 391 i.e., we have convergence as nij -* oo, not just for the subsequence mjfc. This follows from a general argument which we formulate for a sequence of numbers only. The generalization to a sequence of functions is straightforward. Lemma A.4.3. Let bj denote a sequence of numbers, and suppose that (i) Each subsequence bjk has a convergent subsequence. (ii) All convergent subsequences of bj converge to the same limit b. Then bj —» 6 as j —>• oo. Proof. If bj does not converge to 6, then there is an e > 0 and a sequence jk —»• oo with However, according to (i), (ii) there is a subsequence of bjk converging to 6, a contradiction. 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Author Index Agmon, S., 384 Agranovich, M.S., 321, 324Friedman, 119 Fnedrichs, K.O., 119, 176, 323 Brenner, P., 119 Carrier, G.F., 119, 272 Chorin, A., 9 Courant, R., 119, 176Girding, L., 79, 97 Gilbarg, D., 176 Gindinkin, S., 324 Glimm, J., 176 Gurtin, M., 16 Gustafsson, B., 119, 322, 324 D Di Pema, R., 176 Eidelman, S.D., 119, 324 Eskin, G.I., 187, 320, 324Hadamard, J., 79 Henshaw, W.D., 336 Hemquist, C.D., 202 Hersch, R., 323 Hilbert, D., 119 Hopf, E., 156 Hughes, T.J.R., 12 399A B CF G H E 400 Initial-Boundary Value Problems and the Navier-Stokes Equations John, R, 176, 202 Pearson, C.E., 119, 272 Pctrovskii. I.O.. 79. 119.202 Protter, M.H., 156 Kasahara, K., 79 Krciss, H.-O.. 79. 119. 202. 273. 319. 324. 336, 359. 360Ralston. J.V.. 324 Rauch, J., 319, 321,324 Reyna, L.G., 336 Richtmeyer, R.D., 118 Ladyzhenskaya, O.A., 176, 323, 325 Landau, L.D., 176 Lax, P.O., 176 Leray, J., 202 Lifshitz, E.M., 176 Lions, J.L., 323 Liu, T.P., 176 Lorenz, J., 360 M Majda, A., 176, 321 Marsden, J.E., 9, 12 Martins, L., 16 Matano, J., 156 Meyer, R., 9 M.chelson, D., 118, 322, 323, 324 Mizohata, S., 202 Morton, K.W., 119 Naughton, M., 360 Nirenberg, L., 187, 320Sakamoto, R.. 324 Serrin, J., 9 Sjoberg, A., 157 Smoller, J., 176 Solonnikov, V.A., 176, 324 Strong, W.G., 79 Strikwerda, J., 322, 323 Sundstrom, A., 119, 321, 322, 324 Temam, R., 325 Urakeva, N.N., 176 Vishik, M.I.. 324 Yolevich, L., 324 von Wahl. W., 325, 343 W Weinberger, H.F., 156 Whitham, G.B., 156, 157, 176 Oleinik, O.A., 157 Oliger, J., 119,321 Osher. S., 118,321,324 Yamaguti. M.. 79J K LP R S T U V YN O Subject Index Arzela-Ascoli theorem, 389 Asymptotic expansion, 168Deformation tensor, IS Difference approximation (scheme), 89,127, 179, 183, 207, 219, 288 Dirichlet boundary condition, 214, 226 Duhamel's principle, 70 Bounded derivative principle, 359 Burgers' equation, 4, 122, 141 Cauchy-Riemann system, 327, 338 Characteristic (line), 103, 107, 142, 254 characteristic boundary, 266, 283 noncharacteristic boundary, 255, 259 characteristic variables, 254 Continuity equation, 2, 13 Couette experiment, 16Energy norm, 70 Entropy condition, 157 Eulerian description, 11 Euler equations, 15, 59 compressible, 59, 197 incompressible, 3, 15 linearized, 59, 302 D D/Dt operator, 2, 12Finite speed of propagation, 110, 182, 288 Fourier interpolation, 365 Fourier transform, 35 401A B CE F 402 Initial-Boundary Value Problems and the Navier-Stokes Equations Generalized solution, 42, 98, 135, 207 Gronwall's lemma, 84 H Heat equation, 25, 43, 204 backward, 25 Hyperbolic system, 29, 55, 75, 161 strictly hyperbolic, 57, 102 strongly hyperbolic, 31, 57, 75, 102, 186 symmetric hyperbolic, 57, 60, 100, 181, 191, 283 weakly hyperbolic, 29, 57 1 Initialization, 358 Ill-posed problem, 29, 36. 39, 248Parabolic system, 32, 61, 77, 96, 180 strongly parabolic. 82. 178. 276 Parseval's relation. 35 Picard's lemma. 108 Poisson's equation. 349 Pseudodifferential operator, 187. 320 Rankine-Hugoniot jump condition, 150 Rarefaction, 152 Resolvent condition, 45 Reynolds number, 18 Korteweg-de Vries equation, 157 linearized, 116,271 Lagrangian description, 11 Laplace's equation, 349 Laplace transform, 235, 306 solution via, 239 Linearization, 21 Localization principle, 21, 77 M Mach number, 7 Material derivative, 12 Matrix theorem, 45, 54 Maximum principle, 132, 335 Mixed (hyperbolic-parabolic) system, 62, 111, 166, 172, 188, 262 Momentum equation, 2, 14 Mo-solution, 36 space, 34 N Navier-Stokes equations, 2, 16, 62 compressible, 2. 16, 62. 167 incompressible, 342 linearized, 62, 113, 188, 267, 355 Neumann boundary condition, 214, 226, 227, 277Schrodinger equation. 78, 117 Schur's theorem, 362 Semibounded operator, 70, 269 maximal semibounded, 270 Sobolev inequality, 94, 367 discrete, 385 Symbol (of a differential operator) 27, 75, 76 Symmetrizer, 75, 102, 161, 202, 317, 320 Transport theorem, 12 Traveling wave, 144, 176 Vorticity. 327, 330, 335, 337 Wave equation, 29, 105 Weak solution, 5, 148, 176 Well-posedness, 19, 39, 54, 74, 223, 225 operational definition of. 28 strongly well-posed, 224 strongly well-posed in the generalized sense, 227, 313 weakly well-posed. 30. 39. 228G P R KS L T V W