The multiphase Muskat problem with equal viscosities in two dimensions

TL;DR

This paper analyzes the two-dimensional multiphase Muskat problem with equal viscosities, establishing its well-posedness, parabolic nature, and smoothing properties, while also preventing interface contact in certain regimes.

Contribution

It formulates the problem as a coupled evolution equation, proves its parabolicity and well-posedness, and shows smoothing effects and contact exclusion for non-global solutions.

Findings

The problem is of parabolic type.

Solutions exhibit smoothing properties.

Interface contact is prevented in specific regimes.

Abstract

We study the two-dimensional multiphase Muskat problem describing the motion of three immiscible fluids with equal viscosities in a vertical homogeneous porous medium identified with $\mathbb{R}^2$ under the effect of gravity. We first formulate the governing equations as a strongly coupled evolution problem for the functions that parameterize the sharp interfaces between the fluids. Afterwards we prove that the problem is of parabolic type and establish its well-posedness together with two parabolic smoothing properties. For solutions that are not global we exclude, in a certain regime, that the interfaces come into contact along a curve segment.