Well-posedness of the nonhomogeneous incompressible Navier-Stokes/Allen-Cahn system

In this paper, we investigate a system coupled by nonhomogeneous incompressible Navier-Stokes equations and Allen-Cahn equations describing a diffuse interface for two-phase flow of viscous fluids with different densities in a bounded domain $\Omega\subset\mathbb R^d (d=2, 3)$. The mobility is allowed to depend on phase variable but non-degenerate. We first prove the existence of global weak solutions to the initial boundary value problem in 2D and 3D cases. Then we obtain the existence of local in time strong solutions in 3D case as well as the global strong solutions in 2D case. Moreover, by imposing smallness conditions on the initial data, the 3D local in time strong solution is extended globally, with an exponential decay rate for perturbations. At last, we show the weak-strong uniqueness.