Existence and Weak-Strong Uniqueness of Solutions to the Cahn-Hilliard-Navier-Stokes-Darcy System in Superposed Free Flow and Porous Media

We study a diffuse interface model for two-phase flows of similar densities in superposed free flow and porous media. The model consists of the Navier-Stokes-Cahn-Hilliard system in free flow and the Darcy-Cahn-Hilliard system in porous media coupled through a set of domain interface boundary conditions. These domain interface boundary conditions include the nonlinear Lions interface condition and the linear Beavers-Joseph-Saffman-Jones interface condition. We establish global existence of weak solutions in three dimension. We also show that the strong solution if exists agrees with the weak solutions