On the theory of divergence-measure fields and its applications

  • Gui-Qiang Chen
  • Hermano Frid
Article

Abstract

Divergence-measure fields are extended vector fields, including vector fields inLp 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 

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Copyright information

© Sociedade Brasileira de Matemática 2001

Authors and Affiliations

  • Gui-Qiang Chen
    • 1
  • Hermano Frid
    • 2
  1. 1.Department of MathematicsNorthwestern UniversityEvanstonUSA
  2. 2.Instituto de Matemática Pura e Aplicada-IMPARio de JaneiroBrazil

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