ENTROPIES AND WEAK SOLUTIONS OF THE COMPRESSIBLE ISENTROPIC EULER EQUATIONS

In this Note, we study the system of isentropic Euler equations for co mpressible fluids, with a general equation of state. We establish the existence of the fundamental kernel that generates the family of weak entropies, and study its singularities. The kernel is the solution of an equation of Euler-Poisson-Darboux type, and its partial derivative with respect to the density variable tends to a Dirac measure as the d ensity approaches zero. We prove a new reduction theorem for the Young measures associated with the compressible Euler system. From these re sults, we deduce the existence, compactness, and asymptotic decay of m easurable and bounded entropy solutions.