The one-phase fractional Stefan problem. (English) Zbl 1473.80010

del Teso, Félix; Endal, Jørgen; Vázquez, Juan Luis

Math. Models Methods Appl. Sci. 31, No. 1, 83-131 (2021).

Summary: We study the existence and properties of solutions and free boundaries of the one-phase Stefan problem with fractional diffusion posed in \(\mathbb{R}^N\). In terms of the enthalpy \(h(x,t)\), the evolution equation reads \(\partial_th+ (- \Delta)^s\Phi(h)=0\), while the temperature is defined as \(u:=\Phi(h):=\max\{h-L,0\}\) for some constant \(L>0\) called the latent heat, and \((- \Delta)^s\) stands for the fractional Laplacian with exponent \(s\in(0,1)\).
We prove the existence of a continuous and bounded selfsimilar solution of the form \(h(x,t)=H(x t^{- 1 / ( 2 s )})\) which exhibits a free boundary at the change-of-phase level \(h(x,t)=L\). This level is located at the line (called the free boundary) \(x(t)= \xi_0 t^{1 / ( 2 s )}\) for some \(\xi_0>0\). The construction is done in 1D, and its extension to \(N\)-dimensional space is shown.
We also provide well-posedness and basic properties of very weak solutions for general bounded data \(h_0\) in several dimensions. 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. Besides, we 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 \(L\to 0^+\), \(L\to+\infty\), \(s\to 0^+\) and \(s\to 1^-\) 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.

MSC:

80A22	Stefan problems, phase changes, etc.
35D30	Weak solutions to PDEs
35K15	Initial value problems for second-order parabolic equations
35K65	Degenerate parabolic equations
35R09	Integro-partial differential equations
35R11	Fractional partial differential equations
65M06	Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12	Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs

Keywords:

Stefan problem; phase transition; long-range interactions; free boundaries; nonlinear and nonlocal equation; fractional diffusion

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