# Regularity of solutions of the fractional porous medium flow

### Luis A. Caffarelli

University of Texas at Austin, USA### Fernando Soria

Universidad Autónoma de Madrid, Spain### Juan Luis Vázquez

Universidad Autónoma de Madrid, Spain

## 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 $\{x\in\ren, t\in \re\}$ 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.

## Cite this article

Luis A. Caffarelli, Fernando Soria, Juan Luis Vázquez, Regularity of solutions of the fractional porous medium flow. J. Eur. Math. Soc. 15 (2013), no. 5, pp. 1701–1746

DOI 10.4171/JEMS/401