Weak Solutions to the Muskat Problem with Surface Tension Via Optimal Transport

Abstract

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.

A variational finite volume scheme for Wasserstein gradient flows

“…Beside this simple example studied for instance in [10,34], many problems have been proven to exhibit the same variational structure. Porous media flows [15,38,53], magnetic fluids [52], superconductivity [3,4], crowd motions [47], aggregation processes in biology [9,22], semiconductor devices modelling [36], or multiphase mixtures [18,33] are just few examples of problems that can be represented as gradient flows in the Wasserstein space. Designing efficient numerical schemes for approximating their solutions is therefore a major issue and our leading motivation.…”

“…Next proposition provides a finer energy / energy dissipation estimate than (33), which can be thought as discrete counterpart to the energy / energy dissipation inequality (EDI) which is a characterization of generalized gradient flows [2,48]. Proposition 2.7 Given ρ n−1 ∈ P T , let ρ n be the unique solution to (32) and letρ n be a solution to (49), then…”

De Giorgi’s inequality for the thresholding scheme with arbitrary mobilities and surface tensions

“…It allowed Otto and one of the authors to prove convergence results in the multiphase setting [19,20], which lies beyond the reach of the more classical viscosity approach based on the comparison principle implemented in [3,9,13]. Also in different frameworks, this variational viewpoint turned out to be useful, such as MCF in higher codimension [23] or the Muskat problem [14]. The only downside of the generalization [8] are the somewhat unnatural effective mobilities μ i j = 1 σ i j .…”

A fast approach to optimal transport: the back-and-forth method

“…For instance, the back-and-forth method can be readily adapted to compute Wasserstein gradient flows. This opens the door to large-scale simulations of a wide class of important and interesting PDEs, [6,16,17,21] to name just a few. In addition, the method may prove useful to solve computational problems arising from the burgeoning area of mean field games [15,18].…”

“…Here we introduced a Bregman divergence J (•|•) (see Definition 4). Let F be defined by (15), then as previously noted in (16)…”