Global well-posedness of the nonhomogeneous incompressible Navier-Stokes-Cahn-Hilliard system with Landau Potential

We study the nonhomogeneous incompressible Navier-Stokes-Cahn-Hilliard system in a bounded smooth domain in $R^3$, which describes the dynamics of nonhomogeneous incompressible two-phase viscous flows. We first give a blow-up criterion of local strong solution to the initial-boundary-value problem for the case of initial density away from zero. After establishing some key a priori with the help of the Landau Potential, we obtain the global existence and the decay-in-time of strong solution, provided that the initial datum ${\Vert \nabla u_0\Vert }_{L^2\left(\Omega \right)}+{\Vert \nabla {\mu }_0\Vert }_{L^2\left(\Omega \right)}+{\Vert {\rho }_0\Vert }_{L^{\infty }\left(\Omega \right)}$ is suitably small. Precisely, we provide a systematic analysis of the Navier-Stokes-Cahn-Hilliard system through detailed a priori estimates, covering blow-up criterion, global existence and decay behavior of strong solutions.