On the theory of divergence-measure fields and its applications Gui-Qiang Chen Hermano Frid Article

Received: 09 December 2001 DOI :
10.1007/BF01233674

Cite this article as: Chen, GQ. & Frid, H. Bol. Soc. Bras. Mat (2001) 32: 401. doi:10.1007/BF01233674
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Abstract Divergence-measure fields are extended vector fields, including vector fields inL ^{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.

Keywords divergence-measure fields normal traces Gauss-Green theorem product rules Radon measures conservation laws Euler equations gas dynamics entropy solutions entropy inequality stability uniqueness vacuum Cauchy problem initial layers boundary layers initial-boundary value problems

Mathematical subject classification Primary: 00-02 26B20 28C05 35L65 35B10 35B35 Secondary: 26B35 26B12 35L67 Dedicated to Constantine Dafermos on his 60^{th} birthday

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Authors and Affiliations Gui-Qiang Chen Hermano Frid 1. Department of Mathematics Northwestern University Evanston USA 2. Instituto de Matemática Pura e Aplicada-IMPA Rio de Janeiro Brazil