**Courant Lecture Notes**

Volume: 14;
2006;
217 pp;
Softcover

MSC: Primary 35;

Print ISBN: 978-0-8218-3576-0

Product Code: CLN/14

List Price: $36.00

Individual Member Price: $28.80

**Electronic ISBN: 978-1-4704-3114-3
Product Code: CLN/14.E**

List Price: $36.00

Individual Member Price: $28.80

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#### Supplemental Materials

# Hyperbolic Partial Differential Equations

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*Peter D. Lax*

with an appendix by Cathleen S. Morawetz

A co-publication of the AMS and the Courant Institute of Mathematical Sciences at New York University

Peter D. Lax is the winner of the 2005 Abel Prize

The theory of hyperbolic equations is a large subject,
and its applications are many: fluid dynamics and aerodynamics, the theory of
elasticity, optics, electromagnetic waves, direct and inverse scattering, and
the general theory of relativity. This book is an introduction to most facets
of the theory and is an ideal text for a second-year graduate course on the
subject.

The first part deals with the basic theory: the relation of hyperbolicity to
the finite propagation of signals, the concept and role of characteristic
surfaces and rays, energy, and energy inequalities. The structure of solutions
of equations with constant coefficients is explored with the help of the
Fourier and Radon transforms. The existence of solutions of equations with
variable coefficients with prescribed initial values is proved using energy
inequalities. The propagation of singularities is studied with the help of
progressing waves.

The second part describes finite difference approximations of hyperbolic
equations, presents a streamlined version of the Lax-Phillips scattering
theory, and covers basic concepts and results for hyperbolic systems of
conservation laws, an active research area today.

Four brief appendices sketch topics that are important or amusing, such as
Huygens' principle and a theory of mixed initial and boundary value problems. A
fifth appendix by Cathleen Morawetz describes a nonstandard energy identity and
its uses.

Peter D. Lax is the winner of the 2005 Abel Prize.
Read more here.

Titles in this series are co-published with the Courant Institute of Mathematical Sciences at New York University.

#### Table of Contents

# Table of Contents

## Hyperbolic Partial Differential Equations

- Cover Cover11
- List of editors 33
- Title page 44
- Contents 66
- Foreword 88
- Basic notions 1010
- Finite speed of propagation of signals 1414
- Hyperbolic equations with constant coefficients 2424
- Hyperbolic equations with variable coefficients 4646
- Pseudodifferential operators and energy inequalities 6464
- Existence of solutions 7070
- Waves and rays 7878
- Finite difference approximation to hyperbolic equations 9292
- Scattering theory 130130
- Hyperbolic systems of conservation laws 174174
- Appendix A. Huygens’ principle for the wave equation on odd-dimensional spheres 210210
- Appendix B. Hyperbolic polynomials 214214
- Appendix C. The multiplicity of eigenvalues 216216
- Appendix D. Mixed initial and boundary value problems 220220
- Appendix E. Energy decay for star-shaped obstacles 224224
- Back Cover Back Cover1234

#### Readership

Graduate students and research mathematicians interested in hyperbolic equations.