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 consists the Navier–Stokes–Cahn–Hilliard system Darcy–Cahn–Hilliard media coupled through set domain boundary conditions. These conditions include nonlinear Lions condition linear Beavers–Joseph–Saffman–Jones condition. establish global existence weak solutions three dimension. also show that strong solution if exists agrees with solutions.