Well-posedness of the Muskat problem with $H^2$ initial data

TL;DR

This paper proves well-posedness and regularity results for the Muskat problem with initial data in the Sobolev space H^2, including global solutions for small perturbations and local solutions for larger data.

Contribution

It establishes the well-posedness and regularity of the Muskat problem with H^2 initial data, extending previous results to this Sobolev space setting.

Findings

Global well-posedness and decay for small H^2 perturbations in the two-phase case

Local well-posedness for arbitrary size H^2 initial data in the one-phase case

Solutions become infinitely smooth instantaneously

Abstract

We study the dynamics of the interface between two incompressible fluids in a two-dimensional porous medium whose flow is modeled by the Muskat equations. For the two-phase Muskat problem, we establish global well-posedness and decay to equilibrium for small $H^2$ perturbations of the rest state. For the one-phase Muskat problem, we prove local well-posedness for $H^2$ initial data of arbitrary size. Finally, we show that solutions to the Muskat equations instantaneously become infinitely smooth.