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 entropy kernel that generates the family of weak entropies is established by solving a new Euler-Poisson-Darboux equation, which is highly singular when the density of the fluid vanishes. New properties of cancellation of singularities in combinations of the entropy kernel and the associated entropy-flux kernel are found.
We prove the strong compactness of any sequence that is uniformly bounded in L ∞ and whose corresponding sequence of weak entropy dissipation measures is locally H -1 compact. The existence and large-time behavior of L ∞ 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 p-system of fluid dynamics in Lagrangian coordinates.
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Accepted: December 16, 1999
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Chen, GQ., LeFloch, P. Compressible Euler Equations¶with General Pressure Law. Arch. Rational Mech. Anal. 153, 221–259 (2000). https://doi.org/10.1007/s002050000091
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DOI: https://doi.org/10.1007/s002050000091