Porous medium equation with nonlocal pressure

We provide a rather complete description of the results obtained so far on the nonlinear diffusion equation u_t=∇· (u^m-1∇ (-Δ)^-su), which describes a flow through a porous medium driven by a nonlocal pressure. We consider constant parameters m>1 and 02, and the asymptotic behavior of solutions when N=1. The cases m = 1 and m = 2 were rather well known.