Porous medium equation with nonlocal pressure

TL;DR

This paper studies a nonlinear diffusion equation with nonlocal pressure in porous media, establishing existence, uniqueness, and asymptotic behavior of solutions, including new results on self-similar solutions in specific parameter ranges.

Contribution

It provides new existence results for self-similar solutions and analyzes the long-term behavior of solutions in one-dimensional cases.

Findings

Existence of self-similar solutions for N=1 and m>2.

Asymptotic behavior of solutions in one dimension.

Relation of the model to other porous medium equations.

Abstract

We provide a rather complete description of the results obtained so far on the nonlinear diffusion equation $u_t=\nabla\cdot (u^{m-1}\nabla (-\Delta)^{-s}u)$, 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.