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 $p_e$ 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 relaxation limit under such assumptions, and prove that the solutions previously constructed converge to weak solutions of the homogeneous system for the pure liquid phase.