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

This paper concerns an adaptive finite element method for the Stefan one-phase problem. We derive a parabolic variational inequality using the Duvaut transformation [1]. In each time-step we consider an adaptive algorithm based on a combination of the Uzawa method associated with the corresponding multivalued operator and a convergent adaptive method for the linear problem. We justify the conve...