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AuthorSmoller, Joel. author
TitleShock Waves and Reactionโ{128}{148}Diffusion Equations [electronic resource] / by Joel Smoller
ImprintNew York, NY : Springer US, 1983
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. . . the progress of physics will to a large extent depend on the progress of nonlinear matheยญ 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 matheยญ 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, reactionยญ diffusion equations, shock wave theory, and the Conley index, in that order. Thus, the text begins with a discussion of ill-posed problems


I Basic Linear Theory -- 1 Ill-Posed Problems -- 2 Characteristics and Initial-Value Problems -- 3 The One-Dimensional Wave Equation -- 4 Uniqueness and Energy Integrals -- 5 Holmgrenโ{128}{153}s Uniqueness Theorem -- 6 An Initial-Value Problem for a Hyperbolic Equation -- 7 Distribution Theory -- 8 Second-Order Linear Elliptic Equations -- 9 Second-Order Linear Parabolic Equations -- II Reactionโ{128}{148}Diffusion Equations -- 10 Comparison Theorems and Monotonicity Methods -- 11 Linearization -- 12 Topological Methods -- 13 Bifurcation Theory -- 14 Systems of Reactionโ{128}{148}Diffusion Equations -- III The Theory of Shock Waves -- 15 Discontinuous Solutions of Conservation Laws -- 16 The Single Conservation Law -- 17 The Riemann Problem for Systems of Conservation Laws -- 18 Applications to Gas Dynamics -- 19 The Glimm Difference Scheme -- 20 Riemann Invariants, Entropy, and Uniqueness -- 21 Quasi-Linear Parabolic Systems -- IV The Conley Index -- 22 The Conley Index -- 23 Index Pairs and the Continuation Theorem -- 24 Travelling Waves -- Author Index

Physics Acoustics Physics Theoretical Mathematical and Computational Physics Acoustics


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