The Stefan Problem with Small Surface Tension

The Stefan problem with small surface tension e is considered. Assuming that the classical Stefan problem (with s = 0) has a smooth free boundary T, we denote the temperature of the solution by 60 and consider an approximate solution 60 + su for the case where e ^ 0, e small. We first establish the existence and uniqueness of u , and then investigate the effect of u on the free boundary T. It is shown that small surface tension affects the free boundary T radically differently in the two-phase problem than in the one-phase problem. 0. Introduction In the classical formulation of the two-phase Stefan problem the temperatures 8W and 8{ of water and ice satisfy the following conditions on the interface {OO,0 = 0}; (0.1) Vx8w-V^-Vx8lV^ = ^t, (0.2) *„ = *, = (); the first condition is the cbnservation of energy. The functions 6W , 8i further satisfy the heat equation in the water and ice sets, respectively, as well as initial and boundary conditions on the fixed portions of the boundary. For definiteness we shall take in this paper the initial geometry to be as in Figure 1, namely, the fixed boundary (dD) is surrounded by water (region G = GQ\D) and the water is surrounded by ice. The fixed boundary does not change in time, but the free boundary (the water-ice interface) will of course change with time. The one-phase Stefan problem arises when 8i = 0 in the ice region, or 8w = 0 in the water region. Taking for definiteness the case 6i = 0, the interface conditions are