Global solutions for a hyperbolic model of multiphase flow

We study a strictly hyperbolic system of three balance laws arising in the modelling of fluid flows, in one space dimension. The fluid is a mixture of liquid and vapor, and pure phases may exist as well. The flow is driven by a reaction term depending either on the deviation of the pressure p from an equilibrium value pe and on the mass density fraction of the vapor in the fluid; this makes possible for metastable regions to exist. A relaxation parameter is also involved in the model. First, for the homogeneous system, we review a result about the global existence of weak solutions to the initial-value problem, for initial data with large variation. Then we focus on the inhomogeneous case. For initial data sufficiently close to the stable liquid phase we prove, through a fractional step algorithm, that weak global solutions still exist. At last, we study the relax- ation limit under such assumptions, and prove that the solutions previously constructed converge to weak solutions of the homogeneous system for the pure liquid phase.