Global existence of BV solutions and relaxation limit for a model of multiphase reactive flow

We consider a strictly hyperbolic system of balance laws in one space variable, that represents a simple model for a fluid flow in presence of phase transitions. The state variables are specific volume, velocity and mass-density fraction $\lambda$ of the vapor in the fluid. A reactive source term drives the dynamics of the phase mixtures; such term depends on a relaxation parameter and involves an equilibrium pressure, allowing for metastable states. First we prove the global existence of weak solutions to the Cauchy problem, where the initial datum for $\lambda$ is close either to $0$ or $1$ (the pure phases) and has small total variation, while the initial variations of pressure and velocity are not necessarily small. Then we consider the relaxation limit and prove that the weak solutions of the full system converge to those of the reduced system