Existence and Uniqueness of Solution to the Two-Phase Stefan Problem with Convection

The well posedness of the two-phase Stefan problem with convection is established in $$L^{1}$$ . First we consider the case with a singular enthalpy and we fix the convection velocity. In the second part of the paper we study the case of a smoothed enthalpy, but the convection velocity is the solution to a Navier-Stokes equation. In the last section we give some numerical illustrations of a physical case simulated using the models studied in the paper.