Regularity of solutions of the fractional porous medium flow

We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is u_t=∇·(u∇ (-Δ)⁻ˢu), \ 0 The problem is posed in \{x∈ R^n, t∈ 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^α regularity of such weak solutions. Finally, we extend the existence theory to all nonnegative and integrable initial data.