# Weak solutions to the Muskat problem with surface tension via optimal transport

## TL;DR

This paper introduces a novel optimal transport-based framework to approximate weak solutions of the Muskat problem with surface tension, enabling convergence analysis and numerical simulations.

## Contribution

It presents a new gradient flow approach in Wasserstein space for the Muskat problem, using heat content energy to relax surface tension effects and prove convergence.

## Key findings

- Convergence of the scheme to weak solutions under energy assumptions
- Numerical simulations demonstrating the scheme's effectiveness
- Analysis of equilibrium configurations

## Abstract

Inspired by recent works on the threshold dynamics scheme for multi-phase mean curvature flow (by Esedo\={g}lu-Otto and Laux-Otto), we introduce a novel framework to approximate solutions of the Muskat problem with surface tension. Our approach is based on interpreting the Muskat problem as a gradient flow in a product Wasserstein space. This perspective allows us to construct weak solutions via a minimizing movements scheme. Rather than working directly with the singular surface tension force, we instead relax the perimeter functional with the heat content energy approximation of Esedo\={g}lu-Otto. The heat content energy allows us to show the convergence of the associated minimizing movement scheme in the Wasserstein space, and makes the scheme far more tractable for numerical simulations. Under a typical energy convergence assumption, we show that our scheme converges to weak solutions of the Muskat problem with surface tension. We then conclude the paper with a discussion on some numerical experiments and on equilibrium configurations.