Cauchy problem for quasilinear hyperbolic systems. (English) Zbl 0959.35003

Kong, De Xing

MSJ Memoirs. 6. Tokyo: Mathematical Society of Japan. vii, 213 p. (2000).

This Memoir provides a careful and comprehensive account of the Cauchy problem for quasilinear hyperbolic systems. Included among the many applications of such systems are gas dynamics, shallow water theory, nonlinear elasticity and acoustics. The questions addressed in this Memoir, that contains four major chapters, are:
1. Under what conditions does the Cauchy problem for such systems admit a unique global classical solution
2. Under what conditions does the Cauchy problem for such systems blow up, and when and where does this occur
3. How do singularities develop from smooth initial data, and what can be said about the stability of such singularities, included amongst which are shocks.
The account that is clearly written, and contains a comprehensive bibliography, provides an excellent introduction to the subject.

Reviewer: A.Jeffrey (Newcastle upon Tyne)

MSC:

35-02	Research exposition (monographs, survey articles) pertaining to partial differential equations
35L45	Initial value problems for first-order hyperbolic systems
35L67	Shocks and singularities for hyperbolic equations
35L60	First-order nonlinear hyperbolic equations

Keywords:

gas dynamics; shallow water theory; nonlinear elasticity; acoustics; unique global classical solution; blow up