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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.



Geometrical and Analytical Questions In Nonlinear Elasticity

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 are written in a complicated indicial notation that boggles the eye and turns the stomach.
Stuart S. Antman

An Introduction to Euler’S Equations for an Incompressible Fluid

Euler’s equation for a fluid of constant density can be written in the form
$$\frac{D}{{Dt}}{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u}} + grad p = 0,$$
$$div {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u}} = 0,$$
Alexandre J. Chorin

Linearizing Flows and a Cohomology Interpretation of Lax Equations

A talk given by Phillip Griffiths in Chern’ Differential Equations Seminar on May 9, 1983
§ 1. By a Lax Equation with a parameter we shall mean an equation
$$\dot A\left( \xi \right) = \left[ {B\left( \xi \right),A\left( \xi \right)} \right]$$
S. S. Chern

The Ricci Curvature Equation

Let g = {gij be a Riemannian metric on a manifold M of dimension n. It is Ricci curvature Rc(g) = {Rij} is given by the formula
$$ {R_{{ij}}} = \frac{1}{{2(n - 1)}}{g^{{k2}}}\left[ {\frac{{{\partial ^{2}}}}{{\partial {x^{1}}\partial {x^{k}}}}{g_{{j2}}} + \frac{{{\partial ^{2}}}}{{\partial {x^{j}}\partial {x^{2}}}}{g_{{ik}}} - \frac{{{\partial ^{2}}}}{{\partial {x^{i}}\partial {x^{j}}}}{g_{{k2}}} - \frac{{{\partial ^{2}}}}{{\partial {x^{k}}\partial {x^{2}}}}gij} \right] + \frac{1}{{\left( {n - 1} \right)}}{g^{{k2}}}{g_{{pq}}}\left[ {r_{{ik}}^{p}r_{{j2}}^{q} - r_{{ij}}^{p}r_{{k2}}^{q}} \right] $$
$$ r_{{ij}}^{2} = \frac{1}{2}{g^{{k2}}}\left[ {\frac{\partial }{{\partial {x^{1}}}}{g_{{jk}}} + \frac{\partial }{{\partial {x^{J}}}}{g_{{ik}}} - \frac{\partial }{{\partial {x^{k}}}}{g_{{ij}}}} \right]. $$
Richard Hamilton

A Walk Through Partial Differential Equations

This is an informal talk for non-specialists dealing on an elementary level with some aspects of the theory of partial differential equations. In recent years progress in the theory has been tremendous, often in unexpected directions, while also solving classical problems in more general settings. New fields have been added, like the study of variational inequalities, of solitons, of wave front sets, of pseudo-differential operators, of differential forms on manifolds, etc. Much of the progress has been made possible by the use of functional analysis. However, in the process much of the original simplicity of the theory has been lost. This is perhaps connected with the emphasis on solving problems, which often requires the piling up mountains of a priori inequalities and the skillful juggling of function spaces to make ends meet. It is good to remember that mathematics is not only concerned with solving problems, but with studying the structure and behavior of the objects it creates. One of the best examples is the classical theory of functions of a complex variable. It, incidentally, does solve problems as in the Riemann mapping theorem. But much of its beauty lies in statements that can hardly be considered as “solving” anything, like the calculus of residues, or Picard’s theorem, or Cauchy’s formula
Fritz John

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

This paper is concerned with the question of convergence of the nonstationary, incompressible Navier-Stokes flow u = u v to the Euler flow u as the viscosity v tends to zero. If the underlying space domain is all of Rm, the convergence has been proved by several authors under appropriate assumptions on the convergence of the data (initial condition and external force); see Golovkin [1] and McGrath [2] for m = 2 and all time, and Swann [3] and the author [4,5] for m = 3 and short time. The case m ⩾ 4 can be handled in the same way; in fact, the simple method given in [5] applies to any dimension. All these results refer to strong solutions (or even classical solutions, depending on the data) of the Navier-Stokes equation.
Tosio Kato

Free Boundary Problems in Mechanics

Free boundary problems are defined and illustrated by several problems in mechanics. First the problem of finding the free surface of a liquid in hydrostatic equilibrium is considered. Then the effect of surface tension is taken into account. Finally, the contact of an inflated membrane, such as a balloon or tire, with a solid surface is formulated. This problem is solved by the method of matched asymptotic expansions when the contact area is small.
Joseph B. Keller

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

The solutions of a system of partial differential equations are frequently studied in two steps: first one proves the existence of a weak solution in a suitable Sobolev space; then one proves the regularity of this weak solution. For nonlinear systems the second step may be too difficult--indeed, it may be false, the solutions may have singularities. In such cases one seeks a partial regularity theorem, restricting the size of the set of possible singularities, and one attempts the local description, to leading order, of the behavior near a singularity.
Robert V. Kohn

Shock Waves, Increase of Entropy and Loss of Information

We present an informal review of the topics in the title as they pertain to solutions of hyperbolic systems of conservation laws. These are systems of the form
$$ u_{i}^{i} + f_{x}^{i} = 0,{\text{ i = 1,}} \ldots {\text{,n;}} $$
Peter D. Lax

Stress and Riemannian Metrics in Nonlinear Elasticity

In Doyle and Ericksen [1956, p. 77] it is observed that the Cauchy stress tensor σ can be derived by varying the internal energy e with respect to the Riemannian metric on space: σab = 2pбe/σgab Their formula has gone virtually unnoticed in the elasticity literature. In this lecture we shall explain some of the reasons why this formula is, in fact, of fundamental significance. Some additional reasons for its importance follow. First of all, it allows for a rational derivation of the Duhamel-Neumann hypothesis on a decomposition of the rate of deformation tensor see Sokolnikoff [1956, p. 359]), which is useful in the identification problem for constitutive functions. This derivation, due to Hughes, Marsden and Pister, is described in Marsden and Hughes [1983, p. 204–207]. Second, it is used in extending the Noll-Green-Naghdi-Rivlin balance of energy principle (using invariance under rigid body motions) to a covariant theory which allows arbitrary mappings. This is described in Section 2.4 of Marsden and Hughes [1983] and is closely related to the discussion herein. Finally, in classical relativistic field theory, it has been standard since the pioneering work of Belinfante [1939] and Rosenfeld [1940] to regard the stress-energy-momentum tensor as the derivative of the Lagrangian density with respect to the spacetime (Lorentz) metric; see for example, Hawking and Ellis [1973, Sect. 3.3] and Misner, Thorne and Wheeler [1973, Sect. 21.3]. This modern point of view has largely replaced the construction of “canonical stress-energy-momentum tensors”. Thus, for the Lagrangian formulation of elasticity (relativistic or not) the Doyle-Ericksen formulation plays the same role as the Belinfante-Rosenfeld formula and brings it into line with developments in other areas of classical field theory
Jerrold E. Marsden

The Cauchy Problem and Propagation of Singularities

This lecture was intended as an introduction to some of the recent progress in characterizing those linear partial differential operators which are hyperbolic in the sense that the Cauchy problem is locally well-posed in distributions or in C. Only single differential operators are considered, the same problems for determined systems are less well understood.
Richard Melrose

Analytical Theories of Vortex Motion

In an incompressible Newtonian fluid, the physical principles of mass and momentum conservation lead directly to the Navier Stokes equations for the velocity u and pressure p:
John Neu

The Minimal Surface Equation

The minimal surface equation is a system of non-linear elliptic partial differential equations of the form
$$ \sum\limits_{{i,j + 1}}^{n} {{{a}_{{ij}}}\frac{{{{\partial }^{2}}{{y}_{k}}}}{{\partial {{x}_{i}}\partial {{x}_{j}}}} = 0,{\text{ k = 1,}} \ldots {\text{,m,}}} $$
R. Osserman

A Survey of Removable Singularities

Suppose \( P\left( {x,D} \right) = \sum\limits_{{1\alpha 1m}} {{{a}_{\alpha }}\left( x \right){{D}^{\alpha }}} \) is a linear partial differential operator defined on an open set ⋂ ⊂ ℝ n, and that A ⊂ ⋂ is closed.
John C. Polking

Applications of the Maximum Principle

The connection between analytic functions of a complex variable and partial differential equations in the real domain is usually expressed in the observation that u and v, the real and imaginary parts of an analytic function f = u + iv, satisfy the Cauchy-Riemann equations
$$ {{u}_{x}} - {{v}_{y}} = 0,{\text{ }}{{{\text{v}}}_{x}} + {{u}_{y}} = 0. $$
M. H. Protter

Minimax Methods and Their Application to Partial Differential Equations

This research was sponsored in part by the National Science Foundation under Grant No. MCS-8110556. Reproduction in whole or in part is permitted for any purpose of the United States Government.
Paul H. Rabinowitz

Analytic Aspects of the Harmonic Map Problem

A fundamental nonlinear object in differential geometry is a map between manifolds. If the manifolds have Riemannian metrics, then it is natural to choose representaives for maps which respect the metric structures of the manifolds. Experience suggests that one should choose maps which are minima or critical points of variational integrals. Of the integrals which have been proposed, the energy has attracted most interest among analysts, geometers, and mathematical physicists. Its critical points, the harmonic maps, are of some geometric interest. They have also proved to be useful in applications to differential geometry. Particularly one should mention the important role they play in the classical minimal surface theory. Secondly, the applications to Kahler geometry given in [S], [SiY] illustrate the usefulness of harmonic maps as analytic tools in geometry. It seems to the author that there is good reason to be optimistic about the role which the techniques and results related to this problem can play in future developments in geometry.
Richard M. Schoen

Equations of Plasma Physics

A plasma is a gas of charged particles under conditions where collective electromagnetic interactions dominate over interactions between individual particles. Plasmas have been called the fourth state of matter [1]. As one adds heat to a solid, it undergoes a phase transition (melting) to become a liquid. More heat causes the liquid to boil into a gas. Adding still more energy causes the gas to ionize (i.e. some of the negative electrons become dissociated from their gas atoms, leaving positively charged ions). Above 100,000 °K, most matter ionizes into a plasma. While the earth is a relatively plasma-free bubble (aside from fluorescent lights, lightning discharges, and magnetic fusion energy experiments) 99.9% of the universe is in the plasma state (e.g. stars and most of interstellar space).
Alan Weinstein
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