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Nonlocal to Local Convergence of Stefan Problems Under Optimal Convergence Condition

Xinfu Chen, Fang Li, Maolin Zhou

TL;DR

This work analyzes a nonlocal version of the one‑phase Stefan problem with a diffusion kernel $k$, addressing when and how nonlocal dynamics converge to the classical local Stefan model. It establishes wellposedness, comparison and maximum principles, and robust apriori estimates for the nonlocal problem, and provides an exact equivalence between the optimal convergence condition $\hat{k}(\xi)=1-|\xi|^2+o(|\xi|^2)$ and moment constraints on the kernel. By recasting the nonlocal problem via a variational inequality and employing Fourier analysis, the authors prove that, under the stated optimal condition and suitable rescaling, the nonlocal Stefan problem converges to the classical Stefan problem; this includes a careful treatment of energy-like terms and weak convergence of source terms. The paper also reveals that lack of symmetry or convexity can cause jumping phenomena in the free boundary, highlighting the delicate balance between kernel structure and geometry in nonlocal free boundary problems. Overall, it provides a rigorous framework for nonlocal-to-local convergence and clarifies when nonlocal effects dissipate in the limit to the classical Stefan dynamics.

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.

Nonlocal to Local Convergence of Stefan Problems Under Optimal Convergence Condition

TL;DR

This work analyzes a nonlocal version of the one‑phase Stefan problem with a diffusion kernel , addressing when and how nonlocal dynamics converge to the classical local Stefan model. It establishes wellposedness, comparison and maximum principles, and robust apriori estimates for the nonlocal problem, and provides an exact equivalence between the optimal convergence condition and moment constraints on the kernel. By recasting the nonlocal problem via a variational inequality and employing Fourier analysis, the authors prove that, under the stated optimal condition and suitable rescaling, the nonlocal Stefan problem converges to the classical Stefan problem; this includes a careful treatment of energy-like terms and weak convergence of source terms. The paper also reveals that lack of symmetry or convexity can cause jumping phenomena in the free boundary, highlighting the delicate balance between kernel structure and geometry in nonlocal free boundary problems. Overall, it provides a rigorous framework for nonlocal-to-local convergence and clarifies when nonlocal effects dissipate in the limit to the classical Stefan dynamics.

Abstract

In this paper, we consider a free boundary problem with a nonlocal diffusion kernel function . 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 , i.e., the Fourier transform of satisfies 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.
Paper Structure (13 sections, 11 theorems, 182 equations)

This paper contains 13 sections, 11 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Wellposedness and preliminaries
  3. Local and global existence
  4. The maximum principle
  5. Apriori estimates
  6. Fundamental properties of nonlocal Stefan problem
  7. Expansion, boundedness and continuity
  8. Jumping phenomena
  9. Proofs of main results
  10. Equivalent characterization of optimal convergence condition
  11. The nonlocal to local convergence of Stefan problem
  12. Proof of Theorem \ref{['thm-expansion-boundedness']}
  13. Proof of Theorem \ref{['thm-continuity']}

Key Result

Theorem 1.1

Assume that in the problem (one-nonlocalstefan), the kernel function $k$ satisfies the assumption (K), the condition (initialdata) is valid and the initial data satisfies (initial-onephase). Then the problem (one-nonlocalstefan) admits a unique classical solution $\gamma(t,\cdot) \in L^{\infty}(\mat Moreover, if $\gamma_0|_{\bar{\Omega}_0}\in C(\bar{\Omega}_0)$, then $\gamma(t, \cdot)$ is continuo

Theorems & Definitions (22)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • proof : Proof of Theorem \ref{['thm-wellposedness']}
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • ...and 12 more