The multiphase Muskat problem with general viscosities in two dimensions

TL;DR

This paper investigates the two-dimensional multiphase Muskat problem with three immiscible fluids of arbitrary viscosities, establishing well-posedness and smoothing properties under certain physical conditions.

Contribution

It introduces new analytical techniques to handle general viscosities and recasts the problem as a strongly coupled nonlinear nonlocal evolution equation.

Findings

Well-posedness in all L2-subcritical Sobolev spaces.

Parabolic smoothing properties of the solution.

Applicability of Rellich identities and Neumann series in analysis.

Abstract

In this paper we study the two-dimensional multiphase Muskat problem describing the motion of three immiscible fluids with general viscosities in a vertical homogeneous porous medium under the influence of gravity. Employing Rellich type identities in the regime where the fluids are ordered according to their viscosities, respectively a Neumann series argument when the fluids are not ordered by viscosity, we may recast the governing equations as a strongly coupled nonlinear and nonlocal evolution problem for the functions that parameterize the sharp interfaces that separate the fluids. This problem is of parabolic type if the Rayleigh-Taylor condition is satisfied at each interface. Based on this property, we then show that the multiphase Muskat problem is well-posed in all $L_2$-subcritical Sobolev spaces and that it features some parabolic smoothing properties.