A first order phase transition with non-constant density

We introduce a new model for first order phase transitions accounting for non-constant densities of the phases during the process. The resulting initial and boundary value problem for a PDE system is recovered by thermodynamical principles. The resulting system presents some singularities and strong nonlinearities accounting for internal constraints, ensuring in particular the positivity of the pressure and the temperature. Physical consistency for the order parameter comes from a maximum principle argument. Existence of a weak solution is proved by a regularization-passage to the limit procedure