Nonlocal to Local Convergence of Stefan Problems Under Optimal Convergence Condition

TL;DR

This paper establishes the optimal condition under which solutions to nonlocal Stefan problems converge to classical Stefan solutions, even without symmetry or compactness assumptions on the kernel.

Contribution

It introduces an optimal convergence condition for nonlocal Stefan problems and proves convergence to classical solutions without symmetry or compactness assumptions.

Findings

Identified an optimal convergence condition for nonlocal kernels.

Proved convergence of nonlocal solutions to classical Stefan solutions.

Provided an equivalent characterization of the optimal condition.

Abstract

In this paper, we consider a free boundary problem with a nonlocal diffusion kernel function $k(x)$. Due to the long distance exchange effect of nonlocal diffusion, the free boundary can expand discontinuously, which makes the problem rather complicated. Among other things, we propose the optimal convergence condition without assuming the symmetry or compactness of $k$, i.e., the Fourier transform of $k$ satisfies $$\hat{k}(\xi)=1-|\xi|^2+o(|\xi|^2)\ \ \mbox{ as }\xi\rightarrow 0,$$ and discover an equivalent characterization of this optimal condition. More importantly, by the employment of the variational inequality, the apriori estimates and the Fourier transform, we demonstrate that, along a series of properly rescaled kernel functions, the corresponding solutions to the nonlocal free boundary problems converge to the solution of the classical Stefan problem under the proposed optimal condition.