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 u_t=\nabla\cdot(u\nabla (-\Delta)^{-s}u), \quad \ 0<s<1. 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.