The Fractional Stefan Problem: Global Regularity of the Bounded Selfsimilar Solution"
Marcos Llorca, Juan Luis Vázquez
TL;DR
The paper analyzes the global regularity of bounded self-similar solutions to the one-phase fractional Stefan problem on the real line, with diffusion order s in (0,1) and nonlinearity Φ(h)=(h−L)_+. A key finding is the existence of a critical threshold at s=1/2: in the subcritical regime (0<s<1/2) the self-similar enthalpy H is globally in C^{1,α} with optimal α=1−2s at the free boundary, whereas in the critical and supercritical regimes (s≥1/2) the enthalpy is not Lipschitz at the free boundary and exhibits stronger singular behavior. The authors tackle the problem by introducing a regularized, smooth version of the Stefan problem, proving existence/uniqueness of very weak solutions, establishing self-similar reductions, and deriving uniform C^{1,α} estimates that survive the ε→0 limit via compactness (Fréchet–Kolmogorov). They also develop detailed lateral regularity and tail asymptotics, and employ Caffarelli–Silvestre extensions to handle the critical case, culminating in precise regularity and blow-up results at the free boundary across regimes. The work further provides asymptotic profiles at infinity, mass-transfer observations in subcritical regimes, and a suite of open problems guiding future study of fractional Stefan-type systems.
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 \[ \begin{cases} \partial_t h + (-Δ)^s Φ(h) = 0 & \text{in } \mathbb{R}^n \times (0,T),\\[2mm] h(\cdot,0) = h_0 & \text{in } \mathbb{R}^n , \end{cases} \] where $u = Φ(h) := (h-L)_+ = \max\{h-L,0\}$ 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 $C^{1,α}$ regularity, with Hölder exponent $α>0$. 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 $x \to \pm\infty$.