Shock Waves and Reaction—Diffusion Equations

Couverture
Springer Science & Business Media, 6 déc. 2012 - 581 pages
. . . the progress of physics will to a large extent depend on the progress of nonlinear mathe matics, of methods to solve nonlinear equations . . . and therefore we can learn by comparing different nonlinear problems. WERNER HEISENBERG I undertook to write this book for two reasons. First, I wanted to make easily available the basics of both the theory of hyperbolic conservation laws and the theory of systems of reaction-diffusion equations, including the generalized Morse theory as developed by C. Conley. These important subjects seem difficult to learn since the results are scattered throughout the research journals. 1 Second, I feel that there is a need to present the modern methods and ideas in these fields to a wider audience than just mathe maticians. Thus, the book has some rather sophisticated aspects to it, as well as certain textbook aspects. The latter serve to explain, somewhat, the reason that a book with the title Shock Waves and Reaction-Diffusion Equations has the first nine chapters devoted to linear partial differential equations. More precisely, I have found from my classroom experience that it is far easier to grasp the subtleties of nonlinear partial differential equations after one has an understanding of the basic notions in the linear theory. This book is divided into four main parts: linear theory, reaction diffusion equations, shock wave theory, and the Conley index, in that order. Thus, the text begins with a discussion of ill-posed problems.
 

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Table des matières

CHAPTER
13
CHAPTER 4
26
An InitialValue Problem for a Hyperbolic Equation
39
B Fundamental Solutions
52
CHAPTER 8
64
CHAPTER 9
78
CHAPTER 10
92
Linearization
106
D The Riemann Problem for a Scalar Conservation Law
301
B Shocks and Simple Waves
320
C Solution of the General Riemann Problem
335
C Interaction of Shock Waves
358
B The Difference Approximation
377
CHAPTER 20
391
D Instability of Rarefaction Shocks
414
B Artificial Viscosity
431

The KreinRutman Theorem
122
B The LeraySchauder Degree
139
D A Rapid Course in Topology
156
SB Stability of Bifurcating Solutions
176
CHAPTER 14
192
C A Comparison Theorem
213
E A Lyapunov Function for Contracting Rectangles
227
CHAPTER 15
239
C Evolutionary Systems
254
B Uniqueness of the Entropy Solution
281
CHAPTER 22
448
B Isolated Invariant Sets and Isolating Blocks
461
CHAPTER 23
478
C Continuation
494
CHAPTER 24
507
C Periodic Travelling Waves
521
E Instability of Equilibrium Solutions of the Neumann Problem
542
Bibliography
557
Author Index 571
570
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