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.

中文翻译：

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通过最优输运解决表面张力Muskat问题的弱解

受最近关于多相平均曲率流的阈值动力学方案（Esedoḡlu-Otto 和 Laux-Otto）的启发，我们引入了一种新颖的框架来近似解决具有表面张力的 Muskat 问题。我们的方法基于将 Muskat 问题解释为积 Wasserstein 空间中的梯度流。这种观点使我们能够通过最小化运动方案构建弱解决方案。我们不是直接使用奇异的表面张力，而是使用 Esedoḡlu-Otto 的热容能量近似来放松周长泛函。热容能量使我们能够显示相关的最小化运动方案在 Wasserstein 空间中的收敛性，并使该方案更易于数值模拟。在典型的能量收敛假设下，我们表明，我们的方案收敛到具有表面张力的 Muskat 问题的弱解。然后，我们通过对一些数值实验和平衡配置的讨论来结束本文。