Stefan problem with a kinetic condition at the free boundary

Abstract

In the one-dimensional two-phase Stefan problem, the standard equilibrium condition θ=0 at the free boundary x=s(t) is replaced by the kinetic law {Mathematical expression} here β is a continuous and increasing function R→R and β(0)=0. This introduces supercooled and superheated states. Existence of at least one solution is proved. Then (1) is replaced by {Mathematical expression} and it is shown that as e{open} → 0+ a subsequence of the corresponding solutions (θε, sε) converges to a solution (θ, s) of the reduced problem, which is characterised by the free boundary condition {Mathematical expression} Then the case of a radially symmetric multidimensional system is dealt with, taking also account of the surface tension effect. Denoting by s(t) the radial co-ordinate of the free boundary, the following linearized kinetics is considered for a water ball surrounded by ice {Mathematical expression} An existence result is proved for the problem obtained by coupling (4) with the heat equation. © 1985 Fondazione Annali di Matematica Pura ed Applicata.