Salvador Moll: Moll Flux saturated diffusion equations


Flux-saturated diffusion equations are a class of second order parabolic equations of the form \begin{equation}\label{general} u_t=\mathrm{div\,} \mathbb{a}(u,\nabla u), \end{equation} which are characterized by a hyperbolic scaling for large values of the modulus of the gradient, in the sense that \begin{equation}\label{def-varphi} \frac{1}{\psi_{0}(\mathbb{v})}\lim_{t\to +\infty} \mathbb{a}(z,t\mathbb{v})\cdot\mathbb{v} =: \varphi(z) \quad \mbox{for all }\ z\ge 0, \end{equation} where $\psi_{0}:\mathrm{R^n}\mapsto [0,+\infty)$ is a positively $1$-homogeneous convex function, with $\psi_0(0)=0$ and $\psi_0>0$ otherwise, accounting for possible anisotropy effects. Well known examples are the so called relativistic porous medium equation: \begin{equation}\label{m} u_t= \nu \mathrm{div} \left(\frac{u^m \nabla u}{\sqrt{u^2+\nu^2 c^{-2}|\nabla u|^2}}\right), \quad m\in (1,+\infty), \end{equation} and the \emph{speed limited porous medium equation}: \begin{equation}\label{M} u_t=\nu \mathrm{div} \left(\frac{u \nabla u^{M-1}}{\sqrt{1+\nu^2 c^{-2}|\nabla u^{M-1}|^2}}\right),\quad M\in (1,+\infty)\,, \end{equation} where $\nu>0$ is a kinematic viscosity constant and $c>0$ represents a characteristic limiting speed. In this talk, I will introduce this type of equations and I will make an overview of the state of art, including well posedness and some qualitative properties such as regularity, finite speed of propagation of the support of the solutions and waiting time phenomena.