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• W2989761313 abstract "We study the existence and properties of solutions and free boundaries of the one-phase Stefan problem with fractional diffusion posed in $mathbb{R}^N$. The equation for the enthalpy $h$ reads $partial_t h+ (-Delta)^s Phi(h) =0$ where the temperature $u:=Phi(h):=max{h-L,0}$ is defined for some constant $L>0$ called the latent heat, and $(-Delta)^s$ is the fractional Laplacian with exponent $sin(0,1)$. We prove the existence of a continuous and bounded selfsimilar solution of the form $h(x,t)=H(x,t^{-1/(2s)})$ which exhibits a free boundary at the change-of-phase level $h(x,t)=L$ located at $x(t)=xi_0 t^{1/(2s)}$ for some $xi_0>0$. We also provide well-posedness and basic properties of very weak solutions for general bounded data $h_0$. The temperatures $u$ of these solutions are continuous functions that have finite speed of propagation, with possible free boundaries. We obtain estimates on the growth in time of the support of $u$ for solutions with compactly supported initial temperatures. We also show the property of conservation of positivity for $u$ so that the support never recedes. On the contrary, the enthalpy $h$ has infinite speed of propagation and we obtain precise estimates on the tail. The limits $Lto0^+$, $Lto +infty$, $sto0^+$ and $sto 1^-$ are also explored, and we find interesting connections with well-studied diffusion problems. Finally, we propose convergent monotone finite-difference schemes and include numerical experiments aimed at illustrating some of the obtained theoretical results, as well as other interesting phenomena." @default.
• W2989761313 created "2019-12-05" @default.
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• W2989761313 date "2019-11-30" @default.
• W2989761313 modified "2023-10-01" @default.
• W2989761313 title "The one-phase fractional Stefan problem" @default.
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