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2 Editors

S.S. Chern 1. Kaplansky c.c. Moore 1.M. Singer

Mathematical Sciences Research Institute Publications

Volume 1

D. Freed and K. Uhlenbeck: Instantons and FourManifolds

Volume 2

S.S. Chern (ed.): Seminar on Nonlinear Partial Differential Equations

Forthcoming

J.

Lepowsky, G. Mandelstam, and I.M. Singer (eds.): Vertex Theory in Mathematics and Physics

S.S. Chern and P. Griffiths: Essays on Exterior Differential Systems

Seminar on N onlinear Partial Differential Equations Edited by s.s. Chern

With 35 Illustrations

Springer Science+Business Media, LLC

S.S. Chem Department of Mathematics University of California Berkeley, CA 94720 U.S.A.

Mathematical Sciences Research Institute 2223 Fulton Street, Room 603 Berkeley, CA 94720 U.S.A.

AMS Subject Classification: 35-02, 35AXX, 35BXX, 35QXX, 73C50, 76-XX

Library of Congress Cataloging in Publication Data Main entry under title: Seminar on nonlinear partial differential equations. (Mathematical Sciences Research Institute publications; 2) Includes bibliographies. 1. Differential equations, Partial-Addresses, essays, lectures. 2. Differential equations, Nonlinear-Addresses, essays, lectures. 1. Chem, Chiing-S hen. II. Series. QA377.S45 1984 515.3'53 84-20210 The Mathematical Sciences Researeh Institute wishes to acknowledge support from the National Science Foundation. © 1984 by Springer Science+Business Media New York Originally published by Springer-Verlag New York Tne. in 1984 Softcover reprint ofthe hardcover Ist edition 1984 AII rights reserved. No part of this book may be translated or reproduecd in any form without written permission from Springer Scienee+Business Media, LLC,

9 8 765 4 3 2 1 ISBN 978-1-4612-7013-3 ISBN 978-1-4612-1110-5 (eBook) DOI 10.1007/978-1-4612-1110-5

Foreword

When the Mathematical Sciences Research Institute was started in the Fall of 1982, one of the programs was "non-linear partial differential equations".

A seminar was organized whose audience

consisted

students

of

graduate

of

the

University

mathematicians who are not experts in the field. 18 of these lectures.

and

mature

This volume contains

An effort is made to have an adequate

Bibliography for further information.

The Editor wishes to take this

opportunity to thank all the speakers and the authors of the articles presented in this volume for their cooperation. S. S. Chern, Editor

Table of Contents

1

Geometrical and Analytical Questions in Nonlinear Elasticity

Stuart S. Antman

An Introduction to Euler's Equations for an Incompressible Fluid

Alexandre J. Chorin

31

Linearizing Flows and a Cohomology Interpretation of Lax Equations

Phillip Griffiths

37

The Ricci Curvature Equation

Richard Hamilton

47

A Walk Through Partial Differential Equations

Fritz John

73

Remarks on Zero Viscosity Limit for Nonstationary Navier-Stokes Flows with Boundary

Tosio Kato

85

Free Boundary Problems in Mechanics

Joseph B. Keller

99

The Method of Partial Regularity as Applied to the Navier-Stokes Equations

Robert V. Kohn

117

Shock Waves. Incre