Global existence of weak solutions for the incompressible Navier-Stokes-Cahn-Hilliard system with unmatched densities and mixed boundary conditions

This paper is concerned with the Abels-Garcke-Gr"{u}n (AGG) model for two-phase flows of two viscous incompressible system with different densities. The AGG model consists of a Navier-Stokes-Cahn-Hilliard system characterized by a mixture density depending on a volume fraction and an additional flux term due to interface diffusion. We establish the global existence of weak solutions to the initial boundary value problem with the mixed boundary conditions, including the velocity, pressure and Navier-slip conditions for fluid, and the Dirichlet and Neumann conditions for volume fraction and chemical potential, together, in a cubic domain of $\mathbb{R}^3$.