Classical solutions for a logarithmic fractional diffusion equation

Abstract

We prove global existence and uniqueness of strong solutions to the logarithmic porous medium type equation with fractional diffusion $${\partial }_{t}u+(-\Delta {)}^{1/2}\log (1+u)=0,$$ posed for $x\in \mathbb{R}$, with nonnegative initial data in some function space of $L \logL$ type. The solutions are shown to become bounded and $C^{\infty }$ smooth in $(x,t)$ for all positive times. We also reformulate this equation as a transport equation with nonlocal velocity and critical viscosity, a topic of current relevance. Interesting functional inequalities are involved.

30 pages, 2 figures