## Compressible Euler Equations with General Pressure Law

#### Title: Compressible Euler Equations with General Pressure Law

Abstract
We study the hyperbolic system of Euler equations for an isentropic, compressible fluid governed by a general pressure law. The existence and regularity of the {\it entropy kernel\/} that generates the family of weak entropies is established by solving a new Euler-Poisson-Darboux equation, which is {\it highly singular\/} when the density of the fluid vanishes. New properties of {\it cancellation of singularities\/} in combinations of the entropy kernel and the associated entropy-flux kernel are found. We prove the {\it strong compactness\/} of any sequence that is uniformly bounded in $L^\infty$ and whose weak entropy dissipation measures are locally $H^{-1}$ compact. The {\it existence\/} and {\it large-time behavior\/} of $L^\infty$ entropy solutions of the Cauchy problem are established. This is based on a reduction theorem for Young measures, whose proof is new even for the polytropic perfect gas. The existence result also extends to the {\it p-system\/} of fluid dynamics in Lagrangian coordinates.

Archive for Rational Mechanics and Analysis, vol 153, pages 221-259 (2000)
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Author Address
Gui-Qiang Chen
Department of Mathematics
Northwestern University
Evanston, IL 60208-2730
gqchen@math.nwu.edu