Well-posedness for the classical Stefan problem and the zero surface tension limit

We develop a framework for a unified treatment of well-posedness for the Stefan problem with or without surface tension. In the absence of surface tension, we establish well-posedness in Sobolev spaces for the classical Stefan problem. We introduce a new velocity variable which extends the velocity of the moving free-boundary into the interior domain. The equation satisfied by this velocity is used for the analysis in place of the heat equation satisfied by the temperature. Solutions to the classical Stefan problem are then constructed as the limit of solutions to a carefully chosen sequence of approximations to the velocity equation, in which the moving free-boundary is regularized and the boundary condition is modified in a such a way as to preserve the basic nonlinear structure of the original problem. With our methodology, we simultaneously find the required stability condition for well-posedness and obtain new estimates for the regularity of the moving free-boundary. Finally, we prove that solutions of the Stefan problem with positive surface tension $\sigma$ converge to solutions of the classical Stefan problem as $\sigma \to 0$.Comment: Various typos corrected and references adde

A new equivalence of Stefan's problems for the Time-Fractional-Diffusion Equation

A fractional Stefan problem with a boundary convective condition is solved, where the fractional derivative of order $\alpha \in (0,1)$ is taken in the Caputo sense. Then an equivalence with other two fractional Stefan problems (the first one with a constant condition on $x = 0$ and the second with a flux condition)is proved and the convergence to the classical solutions is analyzed when $\alpha \nearrow 1$ recovering the heat equation with its respective Stefan condition.Comment: This paper was already accepted to be published in the in the journal "Fractional Calculus and Applied Analysis". arXiv admin note: substantial text overlap with arXiv:1306.175