The Stefan problem with small surface tension

Abstract

The Stefan problem with small surface tension ε \varepsilon is considered. Assuming that the classical Stefan problem (with ε = 0 \varepsilon = 0 ) has a smooth free boundary Γ \Gamma , we denote the temperature of the solution by θ 0 {\theta _0} and consider an approximate solution θ 0 + ε u {\theta _0} + \varepsilon u for the case where ε ≠ 0 \varepsilon \ne 0 , ε \varepsilon small. We first establish the existence and uniqueness of u u , and then investigate the effect of u u on the free boundary Γ \Gamma . It is shown that small surface tension affects the free boundary Γ \Gamma radically differently in the two-phase problem than in the one-phase problem.