Shock Waves and Reaction—Diffusion Equations / by Joel Smoller.

Format:

Book

Author/Creator:

Smoller, Joel, Author.

Series:

Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics, 2196-9701 ; 258

Language:

English

Subjects (All):

Mathematical analysis.

Analysis.

Local Subjects:

Analysis.

Physical Description:

1 online resource (XXIII, 634 p.)

Edition:

2nd ed. 1994.

Place of Publication:

New York, NY : Springer New York : Imprint: Springer, 1994.

Language Note:

English

Summary:

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.

Contents:

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.

Notes:

Bibliographic Level Mode of Issuance: Monograph

Includes bibliographical references and index.

ISBN:

1-4612-0873-4