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

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 $x \to \pm\infty$.

💡 Research Summary

This paper investigates the regularity of the unique bounded self‑similar solution to the one‑phase Stefan problem with fractional diffusion on the whole real line. The evolution equation is
∂ₜh + (−Δ)^{s} Φ(h) = 0, h(·,0)=h₀, Φ(h) = (h−L)_{+},
with latent heat L>0 and fractional exponent s∈(0,1). The authors consider step‑function initial data h₀(x)=L+P₁ for x≤0 and 0 for x>0. Building on the existence and uniqueness results of