Exponential stability of a diffuse interface model of incompressible two-phase flow with phase variable dependent viscosity and vacuum

This paper is concerned with a simplified model for two-phase fluids with diffuse interface. The model couples the nonhomogeneous incompressible Navier-Stokes equations with the Allen-Cahn equation. The viscosity coefficient is allowed to depend both on the phase variable and on the density. Under some smallness assumptions on initial data, the global existence of unique strong solutions to the 3D Cauchy problem and the initial boundary value problem is established. Meanwhile, we obtain the exponential decay-in-time properties of the solutions. Here, the initial vacuum is allowed and no compatibility conditions are required for the initial data via time weighted techniques.