A gradient flow approach to a thin film approximation of the Muskat problem

TL;DR

This paper introduces a gradient flow framework for a thin film approximation of the Muskat problem, utilizing a variational scheme and Liapunov functionals to construct weak solutions.

Contribution

It presents a novel interpretation of the coupled system as a gradient flow in the Wasserstein space and develops a variational scheme for solution construction.

Findings

Established a gradient flow formulation for the thin film Muskat problem.

Developed a variational scheme to construct weak solutions.

Identified key Liapunov functionals for regularity and analysis.

Abstract

A fully coupled system of two second-order parabolic degenerate equations arising as a thin film approximation to the Muskat problem is interpreted as a gradient flow for the 2-Wasserstein distance in the space of probability measures with finite second moment. A variational scheme is then set up and is the starting point of the construction of weak solutions. The availability of two Liapunov functionals turns out to be a central tool to obtain the needed regularity to identify the Euler-Lagrange equation in the variational scheme.