Alinhac S. Hyperbolic Partial Differential Equations

Подождите немного. Документ загружается.

Universitext

For other titles in this series, go to

http://www.springer.com/series/223

Serge Alinhac

Hyperbolic Partial

Differential Equations

123

Serge Alinhac

Université Paris-Sud XI

Orsay Cedex 91405

France

serge.alinhac@math.u-psud.fr

Editorial Board:

Sheldon Axler, San Francisco State University

Vincenzo Capasso, Università degli Studi di Milano

Carles Casacuberta, Universitat de Barcelona

Angus MacIntyre, Queen Mary, University of London

Kenneth Ribet, University of California, Berkeley

Claude Sabbah, CNRS, École Polytechnique

Endre Süli, University of Oxford

Wojbor Woyczyński, Case Western Reserve University

ISBN 978-0-387- 87822-5 e-ISBN 978-0-387-87823-2

DOI 10.1007/978-0-387-87823-2

Springer Dordrecht Heidelberg London New York

Library of Congress Control Number:

Mathematics Subject Classification (2000): 35Lxx

permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York,

NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in

connection with any form of information storage and retrieval, electronic adaptation, computer

software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.

The use in this publication of trade names, trademarks, service marks, and similar terms, even if they

are not identified as such, is not to be taken as an expression of opinion as to whether or not they are

subject to proprietary rights.

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

2009928133

Département de Mathématiques

Contents

Introduction ix

1 Vector Fields and Integral Curves 1

1.1 FirstDeﬁnitions ........................ 1

1.2 Flows .............................. 2

1.3 DirectionalDerivatives..................... 3

1.4 LevelSurfaces.......................... 4

1.5 BracketofTwoFields ..................... 5

1.6 Cauchy Problem and Method of Characteristics . . . . . . . 5

1.7 StoppingTime ......................... 8

1.8 StraighteningOutofaField.................. 8

1.9 Propagation of Regularity . . . . . . . . . . . . . . . . . . . 9

1.10Exercises ............................ 9

2 Operators and Systems in the Plane 13

2.1 Operators in the Plane: First Deﬁnitions . . . . . . . . . . . 13

2.2 Systems in the Plane: First Deﬁnitions . . . . . . . . . . . . 15

2.3 ReducinganOperatortoaSystem.............. 16

2.4 GronwallLemma........................ 18

2.5 Domains of Determination I (A priori Estimate) . . . . . . 19

vi Contents

2.6 Domains of Determination II (Existence) . . . . . . . . . . 21

2.7 Exercises ............................ 22

3 Nonlinear First Order Equations 27

3.1 Quasilinear Scalar Equations . . . . . . . . . . . . . . . . . 27

3.2 EikonalEquations ....................... 30

3.3 Exercises ............................ 35

3.4 Notes .............................. 40

4 Conservation Laws in One-Space Dimension 41

4.1 FirstDeﬁnitionsandExamples ................ 41

4.2 Examples of Singular Solutions . . . . . . . . . . . . . . . . 42

4.3 SimpleWaves.......................... 44

4.4 RarefactionWaves ....................... 46

4.5 RiemannInvariants....................... 47

4.6 ShockCurves .......................... 48

4.7 Lax Conditions and Admissible Shocks . . . . . . . . . . . . 50

4.8 ContactDiscontinuities .................... 52

4.9 RiemannProblem ....................... 53

4.10ViscosityandEntropy ..................... 54

4.10.1 Entropy......................... 55

4.10.2 LimitsofViscositySolutions ............. 55

4.10.3 Application to the Case of a Shock . . . . . . . . . . 56

4.11Exercises ............................ 58

4.12Notes .............................. 62

5 The Wave Equation 65

5.1 ExplicitSolutions........................ 65

5.1.1 FourierAnalysis .................... 66

Contents vii

5.1.2 Solutions as Spherical Means . . . . . . . . . . . . . 67

5.1.3 Finite Speed of Propagation, Domains of

Determination ..................... 68

5.1.4 StrongHuygensPrinciple ............... 69

5.1.5 AsymptoticBehaviorofSolutions........... 69

5.2 GeometryofTheWaveEquation............... 70

5.2.1 Rotations and Scaling Vector Field . . . . . . . . . . 71

5.2.2 Hyperbolic Rotations and Lorentz Fields . . . . . . 74

5.2.3 Light Cones, Null Frames, and Good Derivatives . . 75

5.2.4 Klainerman’s Inequality . . . . . . . . . . . . . . . . 77

5.3 Exercises ............................ 79

5.4 Notes .............................. 82

6 Energy Inequalities for the Wave Equation 83

6.1 StandardInequalityinaStrip ................ 83

6.2 ImprovedStandardInequality................. 86

6.3 InequalitiesinaDomain.................... 88

6.4 GeneralMultipliers....................... 90

6.5 MorawetzInequality ...................... 92

6.6 KSSInequality ......................... 94

6.7 ConformalInequality...................... 98

6.8 Exercises ............................ 102

6.9 Notes .............................. 110

7 Variable Coeﬃcient Wave Equations and Systems 111

7.1 WhatisaWaveEquation? .................. 111

7.2 Energy Inequality for the Wave Equation . . . . . . . . . . 112

7.3 SymmetricSystems ...................... 115

7.3.1 DeﬁnitionsandExamples ............... 115

viii Contents

7.3.2 EnergyInequality ................... 117

7.4 Finite Speed of Propagation . . . . . . . . . . . . . . . . . . 118

7.5 Klainerman’sMethod ..................... 120

7.6 Existence of Smooth Solutions . . . . . . . . . . . . . . . . 123

7.7 GeometricalOptics....................... 126

7.7.1 An Algebraic Computation . . . . . . . . . . . . . . 126

7.7.2 Formal and Actual Geometrical Optics . . . . . . . . 127

7.7.3 Parametrics for the Cauchy Problem . . . . . . . . . 128

7.8 Exercises ............................ 130

7.9 Notes .............................. 135

Appendix 137

A.1 Ordinary Diﬀerential Equations . . . . . . . . . . . . . . . . 137

A.1.1 CauchyProblem .................... 137

A.1.2 Flows .......................... 140

A.1.3 Lower and Upper Fences . . . . . . . . . . . . . . . . 141

A.2 Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 141

A.2.1 FirstDeﬁnitions .................... 141

A.2.2 Submanifolds Deﬁned by Equations . . . . . . . . . 142

A.2.3 ParametrizedSurfaces................. 143

A.2.4 Graphs ......................... 144

A.2.5 Weaving......................... 145

A.2.6 StokesFormula..................... 145

References 147

Index 149

Introduction

The aim of this book is to present hyperbolic partial diﬀerential equations

at an elementary level. In fact, the required mathematical background is

only a third year university course on diﬀerential calculus for functions

of several variables. No functional analysis knowledge is needed, nor any

distribution theory (with the exception of shock waves mentioned below).

All solutions appearing in the text are piecewise classical C

solutions.

Beyond the simpliﬁcations it allows, there are several reasons for this

choice: First, we believe that all main features of hyperbolic partial dif-

ferential equations (PDE) (well-posedness of the Cauchy problem, ﬁnite

speed of propagation, domains of determination, energy inequalities, etc.)

can be displayed in this context. We hope that this book itself will prove our

belief. Second, all properties, solution formulas, and inequalities established

here in the context of smooth functions can be readily extended to more

general situations (solutions in Sobolev spaces or temperate distributions,

etc.) by simple standard procedures of functional analysis or distribution

theory, which are “external” to the theory of hyperbolic equations: The

deep mathematical content of the theorems is already to be found in

the statements and proofs of this book. The last reason is this: We do

hope that many readers of this book will eventually do research in the ﬁeld

that seems to us the natural continuation of the subject: nonlinear hyper-

bolic systems (compressible ﬂuids, general relativity theory, etc.). In this

area, a large part of the work is devoted to prove global existence in time

of classical solutions, in which case the whole work is about understanding

the behavior and decay of smooth solutions.

There are of course many excellent books and textbooks partially or com-

pletely devoted to the subject of hyperbolic equations, some of which are

quoted in the References at the end. But having discarded the books clearly

too diﬃcult to read for a ﬁrst approach or that use abundantly distribution

theory and Sobolev spaces, we found it somewhat hard to indicate refer-

ences providing an easy introductory exposition of such subjects as, for