Shock Waves and Reaction―Diffusion Equations

✍ Scribed by Joel Smoller

• Publisher: Kluwer Academic Publishers
• Year: 1994
• Tongue: English
• Leaves: 659
• Series: Grundlehren der mathematischen Wissenschaften, 258; 258
• Category: Library

✦ Synopsis

For this edition, a number of typographical errors and minor slip-ups have been corrected. In addition, following the persistent encouragement of Olga Oleinik, I have added a new chapter, Chapter 25, which I titled "Recent Results." This chapter is divided into four sections, and in these I have discussed what I consider to be some of the important developments which have come about since the writing of the first edition. Section I deals with reaction-diffusion equations, and in it are described both the work of C. Jones, on the stability of the travelling wave for the Fitz-Hugh-Nagumo equations, and symmetry-breaking bifurcations. Section II deals with some recent results in shock-wave theory. The main topics considered are L. Tartar's notion of compensated compactness, together with its application to pairs of conservation laws, and T.-P. Liu's work on the stability of viscous profiles for shock waves. In the next section, Conley's connection index and connection matrix are described; these general notions are useful in con­ structing travelling waves for systems of nonlinear equations. The final sec­ tion, Section IV, is devoted to the very recent results of C. Jones and R. Gardner, whereby they construct a general theory enabling them to locate the point spectrum of a wide class of linear operators which arise in stability problems for travelling waves. Their theory is general enough to be applica­ ble to many interesting reaction-diffusion systems.

✦ Table of Contents

Cover
Title
Copyright
Dedication
Acknowledgment
Preface to the Second Edition
Preface to the First Edition
Contents
List of Frequently Used Symbols
PART I Basic Linear Theory
CHAPTER I Ill-Posed Problems
A. Some Examples
B. Lewy's Example
CHAPTER 2 Characteristics and Initial-Value Problems
CHAPTER 3 The One-Dimensional Wave Equation
CHAPTFR 4 Uniqueness and Energy Integrals
CHAPTER 5 Holmgrcn's Uniqueness Theorem
CHAPTER 6 An Initial-Value Problem for a Hyperbolic Equation
CHAPTER 7 Distribution Theory
A. A Cursory View
B. Fundamental Solutions
C. Appendix
CHAPTER 8 Second-Order Linear Elliptic Equations
A. The Strong Maximum Principle
B. A-Priori Estimates
C. Existence of Solutions
D. Elliptic Regularity
CHAPTER 9 Second-Order Linear Parabolic Equations
A. The Heat Equation
B. Strong Maximum Principles
PART II Reaction- Diffusion Equations
CHAPTER 10 Comparison Theorems and Monotonicity Methods
A. Comparison Theorems for Nonlinear Equations
B. Upper and Lower Solutions
C. Applications
CHAPTER 11 Linearization
A. Spectral Theory for Self-Adjoint Operators
B. Linearized Stability
C. Appendix: The Krein-Rutman Theorem
CHAPTER 12 Topological Methods
A. Degree Theory in Rn
B. The Leray Schauder Degree
C. An Introduction to Morse Theory
D. A Rapid Course in Topology
CHAPTER 13 Bifurcation Theory
A. The Implicit Function Theorem
B. Stability of Bifurcating Solutions
C. Some General Bifurcation Theorems
D. Spontaneous Bifurcation; An Example
CHAPTER 14 Systems of Reaction-Diffusion Equations
A. Local Existence of Solutions
B. Invariant Regions
C. A Comparison Theorem
D. Decay to Spatially Homogeneous Solutions
E. A Lyapunov Function for Contracting Rectangles
F. Applications to the Equations of Mathematical Ecology
PART III The Theory of Shock Waves
CHAPTER 15 Discontinuous Solutions of Conservation Laws
A. Discontinuous Solutions
B. Weak Solutions of Conscrvation Laws
C. Evolutionary Systems
D. The Shock Inequalities
E. Irreversibility
CHAPTER 16 The Single Conservation Law
A. Existence of an Entropy Solution
B. Uniqueness of the Entropy Solution
C. Asymptotic Behavior of the Entropy Solution
D. The Riemann Problem for a Scalar Conservation Law
CHAPTER 17 The Riemann Problem for Systems of Conservation Laws
A. The p-System
B. Shocks and Simple Waves
C. Solution of the General Ricmann Problem
CHAPTER 18 Applications to Gas Dynamics
A. The Shock Inequalities
B. The Ricmann Problem in Gas Dynamics
C. Interaction of Shock Waves
CHAPTER 19 The Glimm Difference Scheme
A. The Interaction Estimate
B. The Ditrcrcnce Approximation
C. Convergence
Chapter 20 Riemann Invariants, Entropy, and Uniqueness
A. Riemann Invariants
B. A Concept of Entropy
C. Solutions with Big Data
D. Instability of Rarefaction Shocks
E. Oleinik's Uniqueness Theorem
Chapter 21 Quasi-Linear Parabolic Systems
A. Gradient Systems
B. Artificial Viscosity
C. Iscntropic Gas Dynamics
PART IV The Conley Index
CHAPTER 22 The Conley Index
A. An Impressionistic Overview
B. Isolated Invariant Sets and Isolating Blocks
C. The Homotopy Index
CHAPTER 23 Index Pairs and the Continuation Theorem
A. Morse Decompositions and Index Pairs
B. The Conlcy Index of an Isolated Invariant Set
C. Continuation
D. Some Further Remarks
CHAPTER 24 Travelling Waves
A. The Structure of Weak Shock Waves
B. The Structure of Magnetohydrodynamic Shock Waves
C. Periodic Travelling Waves
D. Stability of Steady-State Solutions
E. Instability of Equilibrium Solutions of the Neumann Problem
F. Appendix: A Criterion for Nondegeneracy
CHAPTER 25 Recent Results
Section I. Reaction-Diffusion Equations
A. Stability of the Fitz-Hugh-Nagumo Travelling Pulse
B. Symmetry-Breaking
C. A Bifurcation Theorem
D. Equivariant Conley Index
E. Application to Scmilincar Elliptic Equations
Concluding Remarks
Section II. Theory of Shock Waves
A. Compensated Compactness
B. Stability of Shock Waves
C. Miscellaneous Results
Section III. Conley Index Theory
A. The Connection Index
B. Conley's Connection Matrix
C. Miscellaneous Results
Section IV. Stability of Travelling Waves-A Topological Approach
A. Introduction
B. The Search for 6,(L)
C. Applications to Fast-Slow Systems
References
Bibliography
Author Index
Subject Index