Seminario Análisis y EDPs

The one-phase fractional Stefan problem

Ponente:  Jørgen Endal (UAM)
Fecha:  miércoles 05 de mayo de 2021 - 15:00
Lugar:  Online - Info:

We study the existence, properties of solutions, and free boundaries of the one-phase Stefan problem with fractional diffusion posed in $R^N$. 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 $xi_0>0$. This special solution will be an important tool to obtain that the temperature has finite speed of propagation while the enthalpy has infinite speed, and that the support of the temperature never recedes. Other interesting properties like e.g. $L o0^+$ and $L oinfty$ will also be discussed, and the theory itself is illustrated by convergent finite-difference schemes.