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.
Highlights
The Muskat problem was first introduced by Morris Muskat [30] as a model for the flow of two immiscible fluids through a porous medium
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
Recall that the setting for our problem is a smooth convex domain ⊂ Rd and without loss of generality by scaling we assume that L d ( ) = 2
Summary
The Muskat problem was first introduced by Morris Muskat [30] as a model for the flow of two immiscible fluids through a porous medium. Remarks on our results (1) Let us underline the fact that our discrete-time scheme (9) produces minimizers that are characteristic functions of a partition of To prove this fact, we exploit the strict concavity of the heat content along the admissible set. While this discussion remains at the heuristic level, our conjectures are supported by the equilibrium states attained in our numerical experiments We end the paper with an appendix section, where we recall the results from [22] that are used when passing to the limit the weak curvature equation
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
