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∞ and whose weak entropy dissipation measures are locally H−1 compact. The existence and large-time behavior of L∞ entropy solutions of the Cauchy problem are established. The existence result also extends to the p-system of fluid dynamics in Lagrangian coordinates. Our proof of the reduction theorem for Young measures also further simplifies the proof known for the polytropic perfect gas.