LOCAL EXISTENCE AND UNIQUENESS OF SOLUTIONS OF THE STEFAN PROBLEM WITH SURFACE-TENSION AND KINETIC UNDERCOOLING

We consider the Stefan Problem with surface tension and kinetic undercooling effects, that is, with the temperature u satisfying the condition u = -σκ - βV on the interface Γ(σ, β = const. >0), (*) where κ and V are the mean curvature and the normal velocity of Γ, respectively. We establish short time existence and uniqueness of classical solutions for the resulting free-boundary problem. The key observation is that, when translated to local coordinates, equation (*) is a quasi-linear parabolic equation on a manifold (without boundary). To prove existence and uniqueness we look for the solution u as a fixed point of a contracting map R. We start by solving the equation of motion (*) for a given left-hand side u. This provides us with an interface Γ which we use to solve the parabolic problem in the bulk (with the conservation of energy condition across Γ), thereby obtaining a new temperature function ū=R(u). Finally, it is the regularizing character of the operator R that allows us to show that R is a contraction on a small time interval. © 1992.