The bidimensional Stefan problem with convection: The time-dependent case

This paper considers the time dependent Stefan problem with convection in the fluid phase governed by the Stokes equation, and with adherence of the fluid on the lateral boundaries. The existence of a weak solution is obtained via the introduction of a temperature dependent penalty term in the fluid flow equation, together with the application of various compactness arguments. Consider a phenomenon (such as melting of ice) where there is a change of phase, say liquid-solid. In the liquid phase the thermal energy is transported both by diffusion and convection, and the effects of convection are reflected in the movement of the free-boundary separating the two phases. This paper shows that such a problem can be formulated mathematically and that it admits a solution in a week sense. Also investigated are some local regularity properties of the distribution of temperature and the field of velocities in the liquid phase.