Extended Divergence-Measure Fields and the Euler Equations for Gas Dynamics |
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Authors: | Gui-Qiang Chen Hermano Frid |
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Institution: | (1) Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730, USA. E-mail: gqchen@math.northwestern.edu, US;(2) Instituto de Matemática Pura e Aplicada – IMPA, Estrada Dona Castorina, 110, Rio de Janeiro, RJ 22460-320, Brazil. E-mail: hermano@impa.br, BR |
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Abstract: | A class of extended vector fields, called extended divergence-measure fields, is analyzed. These fields include vector fields
in L
p
and vector-valued Radon measures, whose divergences are Radon measures. Such extended vector fields naturally arise in the
study of the behavior of entropy solutions of the Euler equations for gas dynamics and other nonlinear systems of conservation
laws. A new notion of normal traces over Lipschitz deformable surfaces is developed under which a generalized Gauss-Green
theorem is established even for these extended fields. An explicit formula is obtained to calculate the normal traces over
any Lipschitz deformable surface, suitable for applications, by using the neighborhood information of the fields near the
surface and the level set function of the Lipschitz deformation surfaces. As an application, we prove the uniqueness and stability
of Riemann solutions that may contain vacuum in the class of entropy solutions of the Euler equations for gas dynamics.
Received: 7 May 2002 / Accepted: 2 December 2002
Published online: 2 April 2003
Communicated by P. Constantin |
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Keywords: | |
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