Existence of Weak Solutions for a General Porous Medium Equation with Nonlocal Pressure

We study the general nonlinear diffusion equation {u_t=\nabla\cdot (u^{m-1}\nabla (-∆)^{-s}u)} that describes a flow through a porous medium which is driven by a nonlocal pressure. We consider constant parameters {m > 1} and {0 < s < 1}, we assume that the solutions are non-negative and that the problem is posed in the whole space. In this paper we prove the existence of weak solutions for all integrable initial data {u_0 ≥ 0} and for all exponents {m > 1} by developing a new approximation method that allows one to treat the range {m≥q 3}, which could not be covered by previous works. We also extend the class of initial data to include any non-negative measure {μ} with finite mass. In passing from bounded initial data to measure data we make strong use of an L ¹-{L^∞} smoothing effect and other functional estimates. Finite speed of propagation is established for all {m ≥q 2}, and this property implies the existence of free boundaries. The authors had already proved that finite propagation does not hold for {m < 2}.