  We study a porous medium equation with nonlocal diffusion effects given by an
inverse fractional Laplacian operator. More precisely, $$
u_t=\nabla\cdot(u\nabla (-\Delta)^{-s}u), \quad \ 0<s<1. $$ 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. Here we establish the boundedness and
$C^\alpha$ regularity of such weak solutions
