A thin film approximation of the Muskat problem with gravity and capillary forces

TL;DR

This paper proves the existence of solutions for a coupled thin film model of the Muskat problem incorporating gravity and capillary effects, using a gradient flow approach in the Wasserstein space.

Contribution

It introduces a novel weak solution existence proof for a complex, coupled fourth-order degenerate parabolic system modeling two-fluid interface evolution.

Findings

Existence of nonnegative weak solutions established.

Model captures effects of gravity and capillarity in fluid layers.

System viewed as a gradient flow in Wasserstein space.

Abstract

Existence of nonnegative weak solutions is shown for a thin film approximation of the Muskat problem with gravity and capillary forces taken into account. The model describes the space-time evolution of the heights of the two fluid layers and is a fully coupled system of two fourth order degenerate parabolic equations. The existence proof relies on the fact that this system can be viewed as a gradient flow for the 2-Wasserstein distance in the space of probability measures with finite second moment.