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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