The Stefan problem with convection and Joule's heating

Abstract

We establish the existence and partial regularity of a capacity solution to a coupled, degenerate, strongly nonlinear system of PDE's which models the melting of a solid due to volume electric heating. The system generalizes the usual Stefan problem, the evolutionary thermistor problem, and the spot welding problem. We allow temperature dependence for the electrical conductivity---which may lead to degeneracy---and take fully into account the flow of the fluid, which we model with the Navier-Stokes system. Existence is proved by considering a sequence of approximate problems, for which a priori estimates are obtained. Then the limit provides a capacity solution for the original problem. The approximate problems are obtained by smoothing, time-retardation and penalization. Of special interest is the fact that the set where the material is above its melting temperature is open, since only there the Navier-Stokes equations hold. The question of the behavior of solutions in mushy regions, regions where the temperature is identically the melting temperature, is left open.