Shock Waves and Reaction--Diffusion Equations

ISBN-13: 9781461269298 / Angielski / Miękka / 2012 / 634 str.

Joel Smoller

Opis

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.

Spis treści

1 Ill-Posed Problems.- 2 Characteristics and Initial-Value Problems.- 3 The One-Dimensional Wave Equation.- 4 Uniqueness and Energy Integrals.- 5 Holmgren’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.- 10 Comparison Theorems and Monotonicity Methods.- 11 Linearization.- 12 Topological Methods.- 13 Bifurcation Theory.- 14 Systems of Reaction-Diffusion Equations.- 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.- 22 The Conley Index.- 23 Index Pairs and the Continuation Theorem.- 24 Travelling Waves.- 25 Recent Results.- Author Index.