On the theory of divergence-measure fields and its applications |
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Authors: | Gui-Qiang Chen Hermano Frid |
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Affiliation: | (1) Department of Mathematics, Northwestern University, 2033 Sheridan Road, 60208-2730 Evanston, Illinois, USA;(2) Instituto de Matemática Pura e Aplicada-IMPA, Estrada Dona Castorina, 110, 22460-320 Rio de Janeiro, RJ, Brazil |
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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.Dedicated to Constantine Dafermos on his 60th birthday |
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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 |
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