A nonlocal one-phase Stefan problem that develops mushy regions

TL;DR

This paper introduces a nonlocal one-phase Stefan problem model that naturally develops mushy regions, analyzing its properties, convergence to classical models, and long-term behavior through an obstacle problem.

Contribution

It presents a novel nonlocal Stefan problem model with mushy region development and analyzes its well-posedness, properties, and convergence to local models.

Findings

Mushy regions form dynamically in the nonlocal model.

Rescaling the kernel causes the solution to converge to the classical Stefan problem.

Long-time behavior characterized by a nonlocal obstacle problem.

Abstract

We study a nonlocal version of the one-phase Stefan problem which develops mushy regions, even if they were not present initially, a model which can be of interest at the mesoscopic scale. The equation involves a convolution with a compactly supported kernel. The created mushy regions have the size of the support of this kernel. If the kernel is suitably rescaled, such regions disappear and the solution converges to the solution of the usual local version of the one-phase Stefan problem. We prove that the model is well posed, and give several qualitative properties. In particular, the long-time behavior is identified by means of a nonlocal mesa solving an obstacle problem.