Properties of Rankine-Hugoniot curves for van der waals fluids

We consider the Euler system made of three conservation laws modeling one-dimensional, inviscid, compressible fluid flows. Considering first a general equation of state, we reformulate the standard condition that the specific entropy be increasing at a shock. The new formulation turns out to be easier to check in concrete examples when searching for admissible shock waves. Then, restricting attention to van der Waals fluids, we first determine regions in the phase space in which the system is hyperbolic or elliptic, or fails to be genuinely nonlinear. Second, based on our reformulation of the entropy condition, we provide a complete description of all admissible shock waves, classified in two distinct categories: the compressive shocks satisfying standard (Liu, Lax) entropy criteria, and undercompressive shocks violating these criteria and requiring a kinetic relation.