The Two-Phase Stefan Problem with Anomalous Diffusion

TL;DR

This paper studies a non-local two-phase Stefan problem involving anomalous diffusion, proving the continuity of weak solutions and connecting it to the broader class of filtration problems like the Porous Medium Equation.

Contribution

It introduces a formulation of the non-local Stefan problem as a nonlinear integro-differential equation and proves the continuity of its weak solutions, linking it to general filtration problems.

Findings

Weak solutions are unique and continuous.

The problem is formulated as a singular nonlinear parabolic integro-differential equation.

Connections are established between Stefan and Porous Media problems.

Abstract

The non-local in space two-phase Stefan problem (a prototype in phase change problems) can be formulated via a singular nonlinear parabolic integro-differential equation which admits a unique weak solution. This formulation makes Stefan problem to be part of the General Filtration Problems; a class which includes the Porous Medium Equation. In this work, we prove that the weak solutions to both Stefan and Porous Media problems are continuous.