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 the problem is posed in the whole space. In this paper we prove 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 to treat the range $m\ge 3$ that 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 $L^1$-$L^{\infty }$ smoothing effect and other functional estimates. Finite speed of propagation is established for all $m\ge 2$, and this property implies the existence of free boundaries. The authors had already proved that finite propagation does not hold for $m<2$.

42 pages, 8 figures. To appear in Archive for Rational Mechanics and Analysis. Minor typos corrected. This version additionally includes more details in the proofs of some results of the paper