Weak Solutions to the Muskat Problem with Surface Tension Via Optimal Transport
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Weak Solutions to the Muskat Problem with Surface Tension Via Optimal Transport Matt Jacobs, Inwon Kim & Alpár R. Mészáros Communicated by F. Otto
Abstract Inspired by recent works on the threshold dynamics scheme for multi-phase mean curvature flow (by Esedo¯glu–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¯glu–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.
1. Introduction The Muskat problem was first introduced by Morris Muskat [30] as a model for the flow of two immiscible fluids through a porous medium. Since its introduction, this problem has received sustained attention in a variety of fields. It is used to model flows in oil reservoirs (water is injected into the oil well to drive oil extraction), and in hydrology to model flows of groundwater through aquifers. In this paper we are interested in obtaining the global existence of weak solutions for the Muskat problem with surface tension, based on its gradient flow structure. We begin by introducing a variational formulation of the problem, which will motivate our subsequent analysis. The fluid evolution can be written as Darcy’s law vi + bi−1 ∇δρi E(ρ) = 0,
(1)
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coupled with the continuity equation ∂t ρi + ∇ · (ρi vi ) = 0,
(2)
where vi is the velocity of phase i, ρ = (ρ1 , ρ2 ) is the collection of relative concentrations for each phase, ∇(δρi E) denotes the spacial gradient of the classical first variation of the free energy with respect to ρi , and bi > 0 (i = 1, 2) denotes constant mobilities. For convenience, throughout the rest of the paper, we will refer to ρ as a collection of density functions; however, one should note that ρ only encodes information about the volume occupied by the fluids and nothing about their mass. The physical setting for our problem is a bounded, convex open domain ⊂ Rd with smooth boundary. We shall suppose that the two fluids fill the entire domain, and that they are confined to for all time. We then take the internal energy to be a sum of three distinct terms: E(ρ) = E p (ρ) + Es (ρ) + (ρ).
(3)
The first term in the energy, describing incompressibility and containment of the fluids, is given by 0, if ρ1 (x) + ρ2 (x) = 1 for a.e. x ∈ E p (ρ) = (4) +∞, otherwise.
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