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.