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

TL;DR

This paper establishes well-posedness for the classical Stefan problem in Sobolev spaces, introduces a new velocity variable for analysis, and proves convergence of solutions with surface tension to the classical case as surface tension vanishes.

Contribution

It develops a unified framework for well-posedness with or without surface tension, introducing a new velocity variable and a limiting process for surface tension.

Findings

Well-posedness in Sobolev spaces for the classical Stefan problem.

A new velocity variable extending into the interior domain.

Convergence of solutions with positive surface tension to the classical problem as surface tension tends to zero.

Abstract

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 $\sigma \to 0$.