Porous medium equation with nonlocal pressure

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We provide a rather complete description of the results obtained so far on the nonlinear diffusion equation uₜ=∇· (uᵐ⁻¹∇ (-Δ)⁻ˢu), which describes a flow through a porous medium driven by a nonlocal pressure. We consider constant parameters m>1 and 0<s<1, we assume that the solutions are non-negative, and the problem is posed in the whole space. We present a theory of existence of solutions, results on uniqueness, and relation to other models. As new results of this paper, we prove the existence of self-similar solutions in the range when N=1 and m>2, and the asymptotic behavior of solutions when N=1. The cases m = 1 and m = 2 were rather well known.