On clogged and fast diffusions in porous media with fractional pressure

TL;DR

This paper investigates the behavior of solutions in porous media models with fractional pressure, focusing on existence, infinite-speed propagation, and solutions in clogged media with degenerate mobility.

Contribution

It introduces a maximum principle approach for fractional Laplacian to analyze solutions, including instantaneously bounded solutions in clogged porous media.

Findings

Established existence and propagation properties of solutions.

Derived lower bounds on solutions using maximum principles.

Constructed solutions that become instantaneously bounded in clogged media.

Abstract

We study the existence and infinite-speed propagation of solutions to models arising in porous media, when the mobility is highly degenerate (inverse power law). The approach is based on maximum principles for the fractional Laplacian, and allows to derive lower bounds on solutions in a straightforward manner. Finally, in the case of clogged porous media, where the mobility vanishes at points of unbounded density, solutions that become instantaneously bounded are constructed.