Viscous displacement in porous media: the Muskat problem in 2D

TL;DR

This paper analyzes the Muskat problem for viscous displacement in 2D porous media, establishing well-posedness, smoothing effects, and criteria for global solutions with or without surface tension.

Contribution

It provides the first rigorous analysis of the Muskat problem in 2D, showing well-posedness and regularity results for large initial data with and without surface tension.

Findings

Muskat problem with surface tension is quasilinear parabolic.

Without surface tension, the Rayleigh-Taylor condition determines parabolicity.

Established local well-posedness for large initial data in appropriate Sobolev spaces.

Abstract

We consider the Muskat problem describing the viscous displacement in a two-phase fluid system located in an unbounded two-dimensional porous medium or Hele-Shaw cell. After formulating the mathematical model as an evolution problem for the sharp interface between the fluids, we show that Muskat problem with surface tension is a quasilinear parabolic problem, whereas, in the absence of surface tension effects, the Rayleigh-Taylor condition identifies a domain of parabolicity for the fully nonlinear problem. Based upon these aspects, we then establish the local well-posedness for arbitrary large initial data in $H^s$, $s>2$, if surface tension is taken into account, respectively for arbitrary large initial data in $H^2$ that additionally satisfy the Rayleigh-Taylor condition if surface tension effects are neglected. We also show that the problem exhibits the parabolic smoothing effect and we provide criteria for the global existence of solutions.