The Verigin problem with phase transition as a Wasserstein flow

TL;DR

This paper models a two-phase flow with phase transition in porous media using a novel Wasserstein gradient flow approach, establishing existence of weak solutions with optimal energy dissipation.

Contribution

It introduces a new variational framework and Wasserstein flow structure for the Verigin problem, enabling the construction of weak solutions with proven convergence.

Findings

Established a Wasserstein gradient flow formulation of the Verigin problem.

Proved convergence of the minimizing movement scheme to weak solutions.

Showed the limit solutions are characteristic functions of finite perimeter sets under certain conditions.

Abstract

We study the modeling of a compressible two-phase flow in a porous medium. The governing free boundary problem is known as the Verigin problem with phase transition. We introduce a novel variational framework to construct weak solutions. Our approach reveals the gradient-flow structure of the system by adopting a minimizing movement scheme using the Wasserstein distance. We prove the convergence of the scheme, obtaining ``relaxed" distributional solutions in the limit that satisfy an optimal energy-dissipation rate. Under the additional assumptions that $d \geq 3$ and that the discrete mass densities are uniformly Muckenhoupt weights, we show that the limit is the characteristic function of a set of finite perimeter in the region where there is no vacuum.