Seminar on Nonlinear Partial Differential Equations

375 Pages·1984·14.292 MB·English

by  Stuart S. Antman (auth.), S. S. Chern (eds.)

Mathematical Sciences Research Institute 2
Publications
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 Four
Manifolds
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 of graduate students of the University and mature
mathematicians who are not experts in the field. This volume contains
18 of these lectures. 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
Geometrical and Analytical Questions Stuart S. Antman 1
in Nonlinear Elasticity
An Introduction to Euler's Equations Alexandre J. Chorin 31
for an Incompressible Fluid
Linearizing Flows and a Cohomology Phillip Griffiths 37
Interpretation of Lax Equations
The Ricci Curvature Equation Richard Hamilton 47
A Walk Through Partial Differential Fritz John 73
Equations
Remarks on Zero Viscosity Limit for Tosio Kato 85
Nonstationary Navier-Stokes Flows
with Boundary
Free Boundary Problems in Mechanics Joseph B. Keller 99
The Method of Partial Regularity as Robert V. Kohn 117
Applied to the Navier-Stokes Equations
Shock Waves. Increase of Entropy Peter D. Lax 129
and Loss of Information
Stress and Riemannian Metrics in Jerrold E. Mal'sden 173
Nonlinear Elasticity
The Cauchy Problem and Propagation Richard Melrose 185
of Singularities
Analytical Theories of Vortex Motion John Neu 203
The Minimal Surface Equation R. Osserman 237
A Survey of Removable Singularities John C. Polking 261
Applications of the Maximum Principle M. H. Protter 293
Minimax Methods and Their Application Paul H. Rabinowitz 307
to Partial Differential Equations
Analytic Aspects of the Harmonic Richard M. Schoen 321
Map Problem
Equations of Plasma Physics Alan Weinstein 359
GEOMETRICAL AND ANALYTICAL QUESTIONS IN NONLINEAR
ELASTICITY
by
Stuart S. Antman
Department of Mathematics
and Institute for Physical Science and Technology
University of Maryland
College Park, MD 20742
1. Introduction
There are many reasons why nonlinear elasticity is not widely
known in the scientific community: (i) It is basically a new science
whose mathematical structure is only now becoming clear. (ii) Reliable
expositions of the theory often take a couple of hundred pages to get
to the heart of the matter. (iii) Many expositions alre written in a
complicated indicial notation that boggles the eye and turns the
stomach.
One may think of nonlinear elasticity as characterized by a
quasilinear system of partial differential equations. But there is more
to the subject t.han the mere analysis of these equations, however
difficult it may be: These equations are augmented by a variety of
subsidiary conditions reflecting that nonlinear elasticity is meant to
describe the deformation of three-dimensional bodies in accord with
certain laws of physics. Thus the subject includes all the richness
and complexity of three-dimensional differential geometry.
In this article, I give a concise presentation of the basic
theory of nonlinear elasticity. The mathematical structure of the
equations is examined with emphasis placed on the inextricable linking
of geometry, analysis, and mechanics. The article ends with a
comparison of nonlinear elasticity with the much better known theory
of incompressible Newtonian fluids.
To remove some of the mystery from nonlinear elasticity, we
give a very brief sketch of how the conceptual foundations lead to
the governing equations. The rest of this article is devoted to an
elaboration of these ideas, which will show why the subject is rich,
fascinating, and challenging.
We identify material points of a body by their positions x in
some reference configuration. Let p(x,t) be the position occupied by x
at time t. p may be thought of as the basic unknown of the problem.
Its derivative ap(x,t) at (x,t) describes the local way the material is
ax
deformed and rotated. Force intensity per unit reference area at (x,t)
is measured by the Piola-Kirchhoff stress-tensor T(x,t). The
requirement that the force on each part of the body equal the time
rate of change of the linear momentum leads the equations of motion
= 2
(Ll) Div T + f p a p,
at2
where f is a given force intensity (such as gravity) applied to the
body and p is a mass density. The form of the divergence Div T is
discussed in Section 3. The mechanical behavior of a material is
specified by a relation between ap/ax and T. A material is elastic if
T
there is a function such that
_ .... [ ap ]
(1.2) T(x,t) - T ax (x,t), x .
The substitution of (1.2) into (1.1) yields a quasilinear system of
partial differential equations for p. This system is the object of our
study. We now examine with care each of the three central concepts
used to systematize the theory of continuum mechanics, namely, the
geometry of deformation, stress and the equations of motion, and the
T.
specification of material properties by means of Note that the
T
form of completely determines the analytical properties of the
system (1.1), (1.2) since it is the only thing that can be appreciably
varied.
We usually denote scalars by lower case Greek letters, vectors
2
(i.e.. elements of Buclidean 3-space E 3) by lower case Latin letters,
and tensors (Le., linear transformations from E3 to itself) by upper
case Latin letters.
3