The Fractional Stefan Problem: Global Regularity of the Bounded Selfsimilar Solution"

TL;DR

This paper investigates the regularity properties of self-similar solutions to the fractional one-phase Stefan problem, revealing a critical threshold at s=1/2 that determines different regularity regimes.

Contribution

It establishes the regularity classification of solutions based on the fractional parameter s, identifying a critical threshold at s=1/2 and analyzing boundary and asymptotic behaviors.

Findings

For 0 < s < 1/2, solutions are at least C^{1,α} regular.

At s=1/2, the enthalpy is not Lipschitz continuous at the free boundary.

For 1/2 < s < 1, the enthalpy lacks Lipschitz regularity at the free boundary.

Abstract

We study the regularity of the bounded self-similar solution to the one-phase Stefan problem with fractional diffusion posed on the whole line. In terms of the enthalpy $h(x,t)$, the evolution problem reads \[ \begin{cases} \partial_t h + (-\Delta)^s \Phi(h) = 0 & \text{in } \mathbb{R}^n \times (0,T),\\[2mm] h(\cdot,0) = h_0 & \text{in } \mathbb{R}^n , \end{cases} \] where $u = \Phi(h) := (h-L)_+ = \max\{h-L,0\}$ denotes the temperature, $L>0$ is the latent heat, and $s \in (0,1)$. We prove that the regularity of the self-similar solution depends on $s$, with a critical threshold at $s = 1/2$. More precisely, in the subcritical case $0 < s < 1/2$, the self-similar solution exhibits at least $C^{1,\alpha}$ regularity, with H\"older exponent $\alpha >0$. In contrast, we show that the enthalpy of the self-similar solution is not Lipschitz continuous at the free boundary in the critical case $s=1/2$, as well as in the supercritical case $1/2 < s < 1$. Additional results are also established concerning the lateral regularity at the free boundary and the asymptotic behavior of the solution profile as $x \to \pm\infty$.