A Mathematical Analysis of a Phase Field Type Model for Solidification with Convection: pure materials in the two dimensional case

We investigate the existence and regularity of weak solutions of a phase field type model for pure material solidification in presence of natural convection. We assume that the nostationary solidification process occurs in a bounded domain, which for technical reasons are restricted to be two dimensional. The governing equations of the model are the following: the phase field equation coupled with a nonlinear heat equation and modified Navier-Stokes equations which include buoyancy forces modeled by Boussinesq approximation and a Carman-Koseny term to model the flow in mushy regions. Since this modified Navier-Stokes equations only hold in a priori unknown non-solid regions, we actually have a free boundary value problem.