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

Understand

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.