The one-phase fractional Stefan problem

TL;DR

This paper investigates the fractional Stefan problem, establishing existence of self-similar solutions with free boundaries, analyzing properties of weak solutions, and exploring limits and numerical schemes for fractional diffusion with phase change.

Contribution

It introduces the first construction of self-similar solutions with free boundaries for the fractional Stefan problem and extends analysis to multiple dimensions.

Findings

Existence of self-similar solutions with free boundaries.

Finite speed of propagation for temperature u.

Infinite speed of propagation for enthalpy h.

Abstract

We study the existence and properties of solutions and free boundaries of the one-phase Stefan problem with fractional diffusion posed in $\mathbb{R}^N$. In terms of the enthalpy $h(x,t)$, the evolution equation reads $\partial_t h+(-\Delta)^s\Phi(h) =0$, while the temperature is defined as $u:=\Phi(h):=\max\{h-L,0\}$ for some constant $L>0$ called the latent heat, and $(-\Delta)^s$ stands for the fractional Laplacian with exponent $s\in(0,1)$. We prove the existence of a continuous and bounded selfsimilar solution of the form $h(x,t)=H(x\,t^{-1/(2s)})$ which exhibits a free boundary at the change-of-phase level $h(x,t)=L$. This level is located at the line (called the free boundary) $x(t)=\xi_0 t^{1/(2s)}$ for some $\xi_0>0$. The construction is done in 1D, and its extension to $N$-dimensional space is shown. We also provide well-posedness and basic properties of very weak solutions for general bounded data $h_0$ in several dimensions. The temperatures $u$ of these solutions are continuous functions that have finite speed of propagation, with possible free boundaries. We obtain estimates on the growth in time of the support of $u$ for solutions with compactly supported initial temperatures. Besides, we show the property of conservation of positivity for $u$ so that the support never recedes. On the contrary, the enthalpy $h$ has infinite speed of propagation and we obtain precise estimates on the tail. The limits $L\to0^+$, $L\to +\infty$, $s\to0^+$ and $s\to 1^-$ are also explored, and we find interesting connections with well-studied diffusion problems. Finally, we propose convergent monotone finite-difference schemes and include numerical experiments aimed at illustrating some of the obtained theoretical results, as well as other interesting phenomena.