Hyperbolic Conservation Laws and the Compensated Compactness Method
The method of compensated compactness as a technique for studying hyperbolic conservation laws is of fundamental importance in many branches of applied mathematics. Until now, however, most accounts of this method have been confined to research papers. Offering the first comprehensive treatment, Hyperbolic Conservation Laws and the Compensated Compactness Method gathers together into a single volume the essential ideas and developments.
The authors begin with the fundamental theorems, then consider the Cauchy problem of the scalar equation, build a framework for L8 estimates of viscosity solutions, and introduce the Invariant Region Theory. The study then turns to methods for symmetric systems of two equations and two equations with quadratic flux, and the extension of these methods to the Le Roux system. After examining the system of polytropic gas dynamics (g-law), the authors first study two special systems of one-dimensional Euler equations, then consider the general Euler equations for one-dimensional compressible fluid flow, and extend that method to systems of elasticity in L8 space. Weak solutions for the elasticity system are introduced and an application to adiabatic gas flow through porous media is considered. The final four chapters explore applications of the compensated compactness method to the relaxation problem.
With its careful account of the underlying ideas, development of applications in key areas, an inclusion of the author's own contributions to the field, this monograph will prove a welcome addition to the literature and to your library.
Table of Contents
Introduction to Compensated Compactness. The Cauchy Problem with Initial Data. Strictly Hyperbolic Systems with Linear Degeneracy. Polytropic Gas Dynamics: The Gamma Law. A One-Dimensional System of Euler Equations. The Case of Quadratic Flux. The Entropy Condition for Reduced Systems. Other Applications.
Yunguang Lu is a Professor of Mathematics at the University of Columbia, Bogota and at the University of Science and Technology of China, Hefei.