Regularized family of models for incompressible Cahn–Hilliard two-phase flows

We consider a general family of regularized models for incompressible two-phase flows based on the Cahn–Hilliard formulation in nn-dimensional compact Riemannian manifolds (with or without boundary) for n=2,3n=2,3. The system we consider consists of a regularized family of Navier–Stokes equations (including the Navier–Stokes-αα-like model, the Leray-αα model, the Modified Leray-αα model, the Simplified Bardina model, the Navier–Stokes–Voight model, the Navier–Stokes model, and many others) for the fluid velocity uu suitably coupled with a convective Cahn–Hilliard equation for the order (phase) parameter ϕϕ. We give a unified analysis of the entire three-parameter family of two-phase models. We first establish existence, stability and regularity results. Then, we show the existence of a global attractor and exponential attractor for our general model, and then establish precise conditions under which each trajectory (u,ϕ)(u,ϕ) converges to a single equilibrium by means of a Lojasiewicz–Simon inequality.