Long-time dynamics of a regularized family of models for homogeneous incompressible two-phase flows

In this article we consider a general family of regularized models for incompressible two-phase flows based on the Allen-Cahn formulation in n-dimensional compact Riemannian manifolds, for d = 2, 3. The system we consider consists of a regularized family of Navier-Stokes equations for the fluid velocity u coupled with a convective Allen-Cahn equation for the order (phase) parameter phi. We discretize these equations in time using the implicit Euler scheme and we prove that the discrete attractors generated by the numerical scheme converge to the global attractor of the continuous system as the time-step approaches zero.