Calculus of Variations and Geometric Measure Theory
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M. Jacobs - I. Kim - A. R. Mészáros

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

created by mészáros on 24 May 2019
modified on 15 Sep 2020


Accepted Paper

Inserted: 24 may 2019
Last Updated: 15 sep 2020

Journal: Arch. Ration. Mech. Anal.
Year: 2020

improved version; numerical experiments included


Inspired by recent works on the threshold dynamics scheme for multi-phase mean curvature flow (by Esedoglu-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 Esedoglu-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.


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