On the Theory of Divergence-Measure Fields and Its Applications

Authors: Gui-Qiang Chen and Hermano Frid

Title: On the Theory of Divergence-Measure Fields and Its Applications

Abstract
Divergence-measure fields are extended vector fields, including vector fields in $L^p$ and vector-valued Radon measures, whose divergences are Radon measures. Such fields arise naturally in the study of entropy solutions of nonlinear conservation laws and other areas. In this paper, a theory of divergence-measure fields is presented and analyzed, in which normal traces, a generalized Gauss-Green theorem, and product rules, among others, are established. Some applications of this theory to several nonlinear problems in conservation laws and related areas are discussed. In particular, with the aid of this theory, we prove the stability of Riemann solutions, which may contain rarefaction waves, contact discontinuities, and/or vacuum states, in the class of entropy solutions of the Euler equations for gas dynamics.

This article has appeared in:
Boletim da Sociedade Brasileira de Matem\'{a}tica (Bol. Soc. Bras. Math.) vol. 32 pages 1-33 (2001)

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Author Address
    Gui-Qiang Chen
    Department of Mathematics
    Northwestern University
    Evanston, IL 60208-2730
    gqchen@math.nwu.edu