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

Abstract

We study the general nonlinear diffusion equation $${u_t=\nabla\cdot (u^{m-1}\nabla (-\Delta)^{-s}u)}$$ 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 $${u_0 \ge 0}$$ and for all exponents $${m > 1}$$ by developing a new approximation method that allows one to treat the range $${m\geqq 3}$$ , which could not be covered by previous works. We also extend the class of initial data to include any non-negative measure $${\mu}$$ with finite mass. In passing from bounded initial data to measure data we make strong use of an L1- $${L^\infty}$$ smoothing effect and other functional estimates. Finite speed of propagation is established for all $${m \geqq 2}$$ , and this property implies the existence of free boundaries. The authors had already proved that finite propagation does not hold for $${m < 2}$$ .