Convex entropies and hyperbolicity for general Euler equations

The compressible Euler equations possess a family of generalized entropy densities of the form rho f(sigma), where rho is the mass density, sigma is the specific entropy, and f is an arbitrary function. Entropy inequalities associated with convex entropy densities characterize physically admissible shocks. For polytropic gases, Harten has determined which rho f(sigma) are strictly convex. In this paper we extend this determination to gases with an arbitrary equation of state. Moreover, we show that at every state where the sound speed is positive (i.e., where the Euler equations are hyperbolic) there exist rho f(sigma) that are strictly convex, thereby establishing the converse of the general fact that the existence of a strictly convex entropy density implies hyperbolicity.