Compressible Euler equations with general pressure law

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-infinity and whose corresponding sequence of weak entropy dissipation measures is locally H-1 compact. The existence and large-time behavior of L-infinity 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.