Global well-posedness of the one-phase Muskat problem with surface tension

TL;DR

This paper proves the global existence and uniqueness of solutions for the one-phase Muskat problem with surface tension under small initial conditions, showing solutions decay over time.

Contribution

It establishes the first global well-posedness result for the one-phase Muskat problem with surface tension for small initial data.

Findings

Unique global strong solution exists for small initial data.

Solution converges to zero in Lipschitz norm as time approaches infinity.

First such global well-posedness result with surface tension.

Abstract

In this paper, we establish the global well-posedness of the one-phase Muskat problem with surface tension for small initial data. This problem describes the motion of the interface separating a wet region from a dry region within a porous medium, a process governed by Darcy's law. Although physically essential, the inclusion of surface tension introduces an additional challenge. We prove that if the initial free boundary is sufficiently small in $H^s$, $s>d/2+1$, then the problem admits a unique global strong solution. Moreover, the solution converges to zero in Lipschitz norm as $t\rightarrow\infty$. To the best of our knowledge, this work constitutes the first global well-posedness result for the one-phase Muskat problem with surface tension.