On a regularized family of models for the full Ericksen–Leslie system

We consider a general family of regularized systems for the full Ericksen–Leslie model for the hydrodynamics of liquid crystals in nn-dimensional compact Riemannian manifolds. The system we consider consists of a regularized family of Navier–Stokes equations (including the Navier–Stokes-αα-like equation, the Leray-αα equation, the Modified Leray-αα equation, the Simplified Bardina model, the Navier–Stokes–Voigt model and the Navier–Stokes equation) for the fluid velocity uu suitably coupled with a parabolic equation for the director field dd. We establish existence, stability and regularity results for this family. We also show the existence of a finite dimensional global attractor for our general model, and then establish sufficiently general conditions under which each trajectory converges to a single equilibrium by means of a Lojasiewicz–Simon inequality.