Table of contents for Hyperbolic partial differential equations / Peter D. Lax ; with an appendix by Cathleen S. Morawetz.


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Chapter 1. Basic Notions                                            1
Chapter 2. Finite Speed of Propagation of Signals                   5
References                                                      14
Chapter 3  Hyperbolic Equations with Constant Coefficients         15
3.1  The Domain of Influence                                     15
3.2  Spacelike Hypersurfaces                                     19
3.3- The Initial Value Problem on Spacelike Hypersurfaces        23
3.4. Characteristic Surfaces                                    25
315. Solution of the Initial Value Problem by the Radon Transform  29
3,6. Conservation of Energy                                      33
References                                                       34
Chapter 4. Hyperbolic Equations with Variable Coefficients         37
4.1. Equations with a Single Space Variable                      37
4.2. Characteristic Surfaces                                     39
4.3. Energy Inequalities for Symmetric Hyperbolic Systems        41
4.4. Energy inequalities for Solutions
of Second-Order Hyperbolic Equations                       45
4.5. Energy Inequalities for Higher-Order Hyperbolic Equations   46
References                                                       53
Chapter 5. Pseudodifferential Operators and Energy Inequalities    55
References                                                      60
Chapter 6. Existence of Solutions                                  61
6.1  Equivalence of the Initial Value Problem and the Periodic Problem  61
6.2. Negative Norms                                              63
6.3. Solution of the Periodic Problem                            65
6.4. A Local Uniqueness Theorem                                  66
References                                                       67
Chapter 7. Waves and Rays                                          69
Introduction                                                     69
7.1. The Initial Value Problem for Distributions                 71
7.2. Progressing Waves                                           74
7.3. Integrals of Compound Distributions                         77
74   An A pproximate Riemann Function
and the Generalized Huygens Principle                     79
References                                                      82
Chapter 8  Finite Difference Approximation to Hyperbolic Equations  83
8, . Consistency                                                83
8.2. Domain of Dependence                                       84
8.3  Stability and Convergence                                  85
8.4. Higher-Order Schemes and Their Stability                   88
8.5. The Gibbs Phenomenon                                       96
8.6. The Computation of Discontinuous Solutions
of Linear Hyperbolic Equations                            98
87. Schemes in More Than One Space Variable                    103
.8   The Stability of Difference Schemes                       108
References                                                     119
Chapter 9  Scattering Theory                                     121
9.. Asymptotic Behavior of Solutions of the Wave Equation      121
9.2  The Lax-Phillips Scatering Theory                         125
9.3  The Associated Semigroup                                  129
9.4. Back to the Wave Equation in the Exteior of an Obstacle   132
95. The Semigroup Associated with Scattering by an Obstacle    139
9.6. Analytic Form of the Scattering Matrix                    144
9.7. Scattering of Automorphic Waves                           154
References                                                     163
Chaper 10. Hyperbolic Systems of Conservation Laws               165
10.   Scalar Equations; Basics                                 165
10.2. The Initial Value Problem for Admissible Solutions       169
10.3  Hyperbolic Systems of Conversation Laws                  178
10.4. The Viscosity Method and Entropy                         184
105. Finite Difference Methods                                 189
10 6. The Flow of Compressible luids                           193
Reterences                                                     197



Library of Congress subject headings for this publication: Differential equations, Hyperbolic