Regularity of solutions of the fractional porous medium flow

Abstract

We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is

The problem is posed in $\{x\in {\mathbb{R}}^n,t\in \mathbb{R}\}$ with nonnegative initial data $u(x,0)$ that are integrable and decay at infinity. A previous paper has established the existence of mass-preserving, nonnegative weak solutions satisfying energy estimates and finite propagation. As main results we establish the boundedness and $C^{\alpha }$ regularity of such weak solutions. Finally, we extend the existence theory to all nonnegative and integrable initial data.