The confined Muskat problem: differences with the deep water regime

TL;DR

This paper compares the behavior of the Muskat problem in confined porous strips versus unbounded domains, revealing that boundaries reduce diffusion and make the system more singular, highlighting key differences in fluid interface evolution.

Contribution

It provides a detailed analysis of how boundary confinement alters the qualitative properties of the Muskat problem compared to the unbounded case.

Findings

Boundaries decrease diffusion rate.

Confined system is more singular.

Qualitative differences in interface evolution.

Abstract

We study the evolution of the interface given by two incompressible fluids with different densities in the porous strip $\RR\times[-l,l]$. This problem is known as the Muskat problem and is analogous to the two phase Hele-Shaw cell. The main goal of this paper is to compare the qualitative properties between the model when the fluids move without boundaries and the model when the fluids are confined. We find that, in a precise sense, the boundaries decrease the diffusion rate and the system becomes more singular.