Existence of weak solutions for a general porous medium equation with nonlocal pressure

TL;DR

This paper proves the existence of weak solutions for a nonlinear nonlocal porous medium equation with broad initial data, introduces a new approximation method, and establishes finite speed of propagation for certain parameters.

Contribution

It develops a novel approximation approach to prove weak solutions for all $m>1$, including $m extgreater 3$, and extends initial data to measures, also establishing finite propagation for $m extgreater 2$.

Findings

Existence of weak solutions for all integrable initial data and $m>1$.

Extension to initial data as non-negative measures with finite mass.

Finite speed of propagation established for all $m extgreater 2$.

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 $m<2$.