Crandall / Rabinowitz / Turner Directions in Partial Differential Equations

Proceedings of a Symposium Conducted by the Mathematics Research Center, the University of Wisconsin-Madison, October 28-30, 1985
1. Auflage 2014
ISBN: 978-1-4832-6924-5
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 258 Seiten, Web PDF

Directions in Partial Differential Equations covers the proceedings of the 1985 Symposium by the same title, conducted by the Mathematics Research Center, held at the University of Wisconsin, Madison. This book is composed of 13 chapters and begins with reviews of the calculus of variations and differential geometry. The subsequent chapters deal with the study of development of singularities, regularity theory, hydrodynamics, mathematical physics, asymptotic behavior, and critical point theory. Other chapters discuss the use of probabilistic methods, the modern theory of Hamilton-Jacobi equations, the interaction between theory and numerical methods for partial differential equations. The remaining chapters explore attempts to understand oscillatory phenomena in solutions of nonlinear equations. This book will be of great value to mathematicians and engineers.

Autoren/Hrsg.

Crandall, Michael G.

Rabinowitz, Paul H.

Turner, E. L.

Inhaltsverzeichnis

1;Front Cover;1
2;Directions in Partial Differential Equations;4
3;Copyright Page;5
4;Table of Contents;8
5;Dedication;6
6;Preface;10
7;Symposium Speakers;12
8;Contributors;14
9;Chapter 1. Singular Minimizers and their Significance in Elasticity;16
9.1;1.
INTRODUCTION;16
9.2;2.
MINIMIZATION OF THE ENERGY IN ELASTICITY;17
9.3;3.
SMOOTH MINIMZERS;19
9.4;4.
MINIMIZERS SINGULAR ONLY ON THE BOUNDARY;20
9.5;5.
LIPSCHITZ MINIMIZERS;22
9.6;6. CONTINUOUS
MINIMIZERS WITH UNBOUNDED DERIVATIVES;24
9.7;7.
MINIMIZERS DISCONTINUOUS AT A POINT OR LINE;25
9.8;8.
CODIMENSION 1 DISCONTINUITIES;27
9.9;REFERENCES;27
10;Chapter 2. Nonlinear Elliptic Equations Involving the Critical Sobolev Exponent—Survey and Perspectives;32
10.1;1. INTRODUCTION;32
10.2;2. PROBLEM
( I ) WHEN N > 4;34
10.3;3. PROBLEM (
I ) WHEN N = 3;37
10.4;4. PROBLEM (
II );38
10.5;5. THE EFFECT OF TOPOLOGY:
THE WORK OF BAHRI-CORON;40
10.6;6. VARIOUS RELATED PROBLEMS;43
10.7;REFERENCES;48
11;Chapter 3. The Differentiability of the Free Boundary for the n-Dimensional Porous Media Equation;52
11.1;REFERENCES;57
12;Chapter 4. Oscillations and Concentrations in Solutions to the Equations of Mechanics;58
12.1;Definition. A mapping;59
12.2;REFERENCES;67
13;Chapter 5. The Connection Between the Navier-Stokes Equations, Dynamical Systems, and Turbulence Theory;70
13.1;INTRODUCTION;70
13.2;2. THE
NAVIER–STOKES EQUATONS AND THEIR ABSTRACT FRAMEWORK;72
13.3;3. THE UNIVERSAL ATTRACTOR;76
13.4;4. ESTIMATES OF THE FRACTAL
DIMENSION;80
13.5;5. INERTIAL MANIFOLDS AND INERTIAL FORMS;84
13.6;REFERENCES;86
14;Chapter 6. Blow-Up of Solutions of Nonlinear Evolution Equations;90
14.1;INTRODUCTION;90
14.2;§1.
Nonlinear parabolic equations;91
14.3;§2.
Degenerate nonlinear parabolic equations;94
14.4;§3.
Nonlinear wave equations;96
14.5;§4. The Hamilton-Jacobi equation;98
14.6;§5. Other equations;99
14.7;REFERENCES;101
15;Chapter 7. Coherence and Chaos in the Kuramoto-Velarde Equation;104
15.1;I. INTRODUCTION;104
15.2;II. OVERVIEW OF COMPUTATIONAL SIMULATIONS AND THEORETICAL RESULTS;108
15.3;III. THE K-V EQUATION BIFURCATION INTERVALS;114
15.4;IV. SUMMARY;123
15.5;ACKNOWLEDGMENTS;123
15.6;REFERENCES;123
16;Chapter 8. Einstein Geometry and Hyperbolic Equations;128
16.1;REFERENCES;155
17;Chapter 9. Recent Progress on First Order Hamilton-Jacobi Equations;160
17.1;1.
INTRODUCTION;160
17.2;2.
DEFINITIONS OF VISCOSITY SOLUTIONS;163
17.3;3.
HOW TO PROVE UNIQUENESS OF VISCOSITY SOLUTIONS;165
17.4;4.
EXAMPLES OF EXISTENCE AND UNIQUENESS RESULTS;168
17.5;REFERENCES;171
18;Chapter 10. The Focusing Singularity of the Nonlinear
Schrödinger Equation;174
18.1;1. Introduction;174
18.2;2. Invariants;175
18.3;3. Symmetries;175
18.4;4. Global existence in the subcritical case;176
18.5;5. The variance argument for blowup;177
18.6;6. The R profile;177
18.7;7. Singular solutions by perturbation;178
18.8;8. Dynamic reseating;179
18.9;9. The Q profile;182
18.10;10. The Numerical Scheme for the Transformed Equations;183
18.11;11. Evaluation of the discretized
e differential operators, .(N) and .e(N);185
18.12;12. Evaluation of
e integrals;186
18.13;13. Evaluation of L and
t;186
18.14;14. Numerical Results;187
18.15;References;213
19;Chapter 11. A Probabilistic Approach to Finding Estimates for the Heat Kernel Associated with a
Hörmander Form Operator;218
19.1;0.
Introduction;218
19.2;1.
Outline of the Proof;221
19.3;REFERENCES;224
20;Chapter 12. Discontinuities and Oscillations;226
20.1;1.
DISCONTINUITIES; PART ONE;228
20.2;2.
DISCONTINUITIES; PART TWO;231
20.3;3.
DISCONTINUITIES; PART THREE;232
20.4;4.
OSCILLATIONS; PART ONE;235
20.5;5.
OSCILLATIONS; PART TWO;237
20.6;6.
OSCILLATIONS; PART THREE;238
20.7;7. ARE OSCILXATIONS REAL?;240
20.8;8.
CONCLUSION;242
20.9;REFERENCES;243
21;Chapter 13. The Structure of Manifolds with Positive Scalar Curvature;250
21.1;Theorem
1;251
21.2;Proposition
1;251
21.3;Theorem
2;252
21.4;Theorem
3;252
21.5;Theorem
4;253
21.6;Theorem
5;253
21.7;Theorem
6;253
21.8;Theorem
7;255
21.9;Theorem
8;256
21.10;Theorem
9;256
21.11;Theorem
10;256
21.12;References;256
22;Index;258