Solutions for a hyperbolic model of multi-phase flow

We discuss a model for the ow of an inviscid uid admitting liquid and vapor phases, as well as a mixture of them. The ow is modeled in one spatial dimension; the state variables are the specic volume, the velocity and the mass density fraction of vapor in the uid. The equation governing the time evolution of contains a source term, which enables metastable states and drives the uid towards stable pure phases. We rst discuss, for the homogeneous system, the BV stability of Riemann solutions generated by large initial data and check the validity of several sucient conditions that are known in the literature. Then, we review some recent results about the existence of solutions, which are globally dened in time, for close either to 0 or to 1 (corresponding to almost pure phases). These solutions possibly contain large shocks. Finally, in the relaxation limit, solutions are proved to satisfy a reduced system and the related entropy condition.