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

Jacobs, Matt and Kim, Inwon and Mészáros, Alpár R. (2020) 'Weak solutions to the Muskat problem with surface tension via optimal transport.', Archive for rational mechanics and analysis., 239 (1). pp. 389-430.


Inspired by recent works on the threshold dynamics scheme for multi-phase mean curvature flow (by Esedoḡlu–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 Esedoḡlu–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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Date accepted:15 September 2020
Date deposited:23 October 2020
Date of first online publication:14 October 2020
Date first made open access:23 October 2020

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