Nonlinear porous medium flow with fractional potential pressure

TL;DR

This paper investigates a nonlinear porous medium equation incorporating nonlocal fractional diffusion, demonstrating finite speed propagation and establishing the existence of weak, bounded solutions for all time.

Contribution

It introduces a novel porous medium model with inverse fractional Laplacian diffusion and proves key properties like finite speed propagation and solution existence.

Findings

Solutions exist for all time with finite speed propagation.

The model exhibits bounded, weak solutions under initial conditions.

Nonlocal fractional effects are incorporated into porous medium flow.

Abstract

We study a porous medium equation, with nonlocal diffusion effects given by an inverse fractional Laplacian operator. We pose the problem in n-dimensional space for all t>0 with bounded and compactly supported initial data, and prove existence of a weak and bounded solution that propagates with finite speed, a property that is nor shared by other fractional diffusion models.