An existence result for a two-phase Stefan problem arising in metal casting

We present a model arising from the thermal modelling of two metal casting processes. We consider an enthalpy formulation for this two-phase Stefan problem in a time varying three-dimensional domain and consider convective heat transfer in the liquid phase. Then, we introduce a weak formulation in a fixed domain, by means of a suitable transformation. Existence of solution is obtained by applying an abstract theorem. The proof of this theorem is done by taking an implicit discretization in time together with a regularization. By passing to the limit in the regularization parameter and in the time step, we obtain the existence of solution of the continuous problem.