Stationary solutions of a singular Navier-Stokes enthalpy-heat conduction system

Abstract

We present a result on existence of stationary solutions of a system coupling singular Navier-Stokes equations to a enthalpy-heat equation. This system may model the solidification process of certain classes of materials by taking into consideration the possibility of flow in the melt; thus, the singular Navier-Stokes equations only holds in the a priori unknown molten region and one has a free-boundary value problem to be solved. To obtain solutions for such system, we initially consider a sequence of approximate problems associated to an appropriate regularizations of the original one, which are obtained by a suitable modification such that the approximate equations for the flow hold in the entire domain. After analyzing these approximate problems, by using compactness arguments, we take limits to obtain generalized solutions of the original problem.