Traveling wave solutions to the one-phase Muskat problem: existence and stability

TL;DR

This paper establishes the existence and stability of traveling wave solutions for the one-phase Muskat problem under small external forces and pressures, revealing new stable configurations in fluid interface dynamics.

Contribution

It proves the existence and stability of nontrivial traveling wave solutions for the Muskat problem with external forces and pressures, a novel result in the field.

Findings

Existence of locally unique traveling wave solutions under small data.

Periodic traveling waves induced by external pressure are asymptotically stable.

First class of nontrivial stable solutions for the Muskat problem.

Abstract

We study the Muskat problem for one fluid in arbitrary dimension, bounded below by a flat bed and above by a free boundary given as a graph. In addition to a fixed uniform gravitational field, the fluid is acted upon by a generic force field in the bulk and an external pressure on the free boundary, both of which are posited to be in traveling wave form. We prove that for sufficiently small force and pressure data in Sobolev spaces, there exists a locally unique traveling wave solution in Sobolev-type spaces. The free boundary of the traveling wave solutions is either periodic or asymptotically flat at spatial infinity. Moreover, we prove that small periodic traveling wave solutions induced by external pressure only are asymptotically stable. These results provide the first class of nontrivial stable solutions for the problem.