Nonlocal to Local Convergence of Stefan Problems Under Optimal Convergence 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 $$\hat{k}(ξ)=1-|ξ|^2+o(|ξ|^2)\ \ \mbox{ as }ξ\rightarrow 0,$$ 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.