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

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 $u=\Phi (h):=(h-L{)}_{+}=\max \{h-L,0\}$ 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 Hölder 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 $x\to \pm \infty$.

60 pages, 3 figures