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. More precisely, $u_t=\nabla\cdot(u\nabla (-Δ)^{-s}u), \quad \ 0<s<1.$ 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 $C^α$ regularity of such weak solutions

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