Ecuaciones diferenciales

COURANT
PETER D. LAX
LECTURE NOTES

Hyperbolic Partial Differential Equations

American Mathematical Society

Courant Institute of Mathcmati:al Science.

AMS

Courant Lecture Notes in Mathematics
Executive Editor Jalal Shatah

Managing Editor Paul D. Monsour
Assistant Editors Reeva Goldsmith Suzan Toma
Copy Editors Will Klump Marc Nirenberg Joshua Singer

Hyperbolic PartialDifferential Equations

Peter D. Lax
Courant Institute of Mathematical Sciences
With an Appendix by Cathleen S. Morawetz

14

Hyperbolic Partial Differential Equations

Courant Institute of Mathematical Sciences
New York University New York, New York

American Mathematical Society
Providence, Rhode Island

2000 Mathematics Subject Classification. Primary 35L05, 35L10, 35L15,35L20, 35L25, 35L30, 35L35, 35L40, 35L45, 35L50, 35L55, 35L60, 35L65, 35L67, 35P25.

For additional information and updates on this book, visit

www.ams.org/bookpages/ein-14

Library of Congress Cataloging-in-Publication Data
Lax, Peter D. Hyperbolic partial differential equations / Peter D. Lax. p. cm. - (Courant lecture notes, ISSN 1529-9031 ; 14) Includes bibliographical references.
ISBN-13:978-0-8218-3576-0 (alk. paper)

1. Differential equations, Hyperbolic.
QA377.L387 2006 515'.3535 - dc22

I. Title.

2006050151

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111009080706Contents
Foreword
vii

Chapter 1. Chapter 2.

Basic Notions

1

Finite Speed of Propagation of Signals

5
14

References

Chapter 3. Hyperbolic Equations with Constant Coefficients 3.1. The Domain of Influence 3.2. Spacelike Hypersurfaces 3.3. The Initial Value Problem on Spacelike Hypersurfaces 3.4. Characteristic Surfaces 3.5. Solution of the Initial Value Problem by the RadonTransform 3.6. Conservation of Energy References Chapter 4. Hyperbolic Equations with Variable Coefficients 4.1. Equations with a Single Space Variable 4.2. Characteristic Surfaces 4.3. Energy Inequalities for Symmetric Hyperbolic Systems 4.4. Energy Inequalities for Solutions of Second-Order Hyperbolic Equations 4.5. Energy Inequalities for Higher-Order Hyperbolic Equations References Chapter 5.Pseudodifferential Operators and Energy Inequalities References Chapter 6. Existence of Solutions 6.1. Equivalence of the Initial Value Problem and the Periodic Problem 6.2. Negative Norms 6.3. Solution of the Periodic Problem 6.4. A Local Uniqueness Theorem References Chapter 7. Waves and Rays Introduction 7.1. The Initial Value Problem for Distributions 7.2. Progressing Waves 7.3. Integrals of CompoundDistributions
V

15 15 19

23 25

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69 69
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vi

CONTENTS

7.4.

An Approximate Riemann Function and the Generalized Huygens Principle References
Chapter 8. Finite Difference Approximation to Hyperbolic Equations 8.1. Consistency 8.2. Domain of Dependence 8.3. Stability and Convergence...