Boletim da Sociedade Brasileira de Matemática - Bulletin/Brazilian Mathematical Society

, Volume 32, Issue 3, pp 401–433

On the theory of divergence-measure fields and its applications

Authors

  • Gui-Qiang Chen
    • Department of MathematicsNorthwestern University
  • Hermano Frid
    • Instituto de Matemática Pura e Aplicada-IMPA
Article

DOI: 10.1007/BF01233674

Cite this article as:
Chen, G. & Frid, H. Bol. Soc. Bras. Mat (2001) 32: 401. doi:10.1007/BF01233674

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 fieldsnormal tracesGauss-Green theoremproduct rulesRadon measuresconservation lawsEuler equationsgas dynamicsentropy solutionsentropy inequalitystabilityuniquenessvacuumCauchy probleminitial layersboundary layersinitial-boundary value problems

Mathematical subject classification

Primary: 00-0226B2028C0535L6535B1035B35Secondary: 26B3526B1235L67

Copyright information

© Sociedade Brasileira de Matemática 2001