The Muskat problem in 2D: equivalence of formulations, well-posedness, and regularity results

TL;DR

This paper analyzes the 2D Muskat problem, proving its formulation as a parabolic evolution equation, establishing local well-posedness for large initial data, and demonstrating instant analyticity of solutions.

Contribution

It shows the equivalence of different formulations of the 2D Muskat problem and proves well-posedness and regularity results in specific functional spaces.

Findings

The problem can be formulated as a quasilinear parabolic evolution equation.

Local well-posedness holds for large initial data in specified Sobolev spaces.

Solutions become instantly real-analytic in time and space.

Abstract

In this paper we consider the Muskat problem describing the motion of two unbounded immiscible fluid layers with equal viscosities in vertical or horizontal two-dimensional geometries. We first prove that the mathematical model can be formulated as an evolution problem for the sharp interface separating the two fluids, which turns out to be, in a suitable functional analytic setting, quasilinear and of parabolic type. Based upon these properties, we then establish the local well-posedness of the problem for arbitrary large initial data and show that the solutions become instantly real-analytic in time and space. Our method allows us to choose the initial data in the class $H^s,$ $s\in(3/2,2)$, when neglecting surface tension, respectively in $H^s,$ $s\in(2,3),$ when surface tension effects are included. Besides, we provide new criteria for the global existence of solutions.