The Fractional Stefan Problem: Global Regularity of the Bounded Selfsimilar Solution"
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2025
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| author | Llorca, Marcos Vázquez, Juan Luis |
| author_facet | Llorca, Marcos Vázquez, Juan Luis |
| contents | 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 $x \to \pm\infty$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_17725 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Fractional Stefan Problem: Global Regularity of the Bounded Selfsimilar Solution" Llorca, Marcos Vázquez, Juan Luis Analysis of PDEs 35K55, 35K65, 35R11, 35C06. 35B40. 35K55, 35K65, 35R11, 35C06. 35B40. 35K55, 35K65, 35R11, 35C06. 35B40 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 $x \to \pm\infty$. |
| title | The Fractional Stefan Problem: Global Regularity of the Bounded Selfsimilar Solution" |
| topic | Analysis of PDEs 35K55, 35K65, 35R11, 35C06. 35B40. 35K55, 35K65, 35R11, 35C06. 35B40. 35K55, 35K65, 35R11, 35C06. 35B40 |
| url | https://arxiv.org/abs/2512.17725 |