The Verigin problem with phase transition as a Wasserstein flow

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.