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

Abstract

We consider a general family of regularized systems for the full Ericksen–Leslie model for the hydrodynamics of liquid crystals in n-dimensional compact Riemannian manifolds. The system we consider consists of a regularized family of Navier–Stokes equations (including the Navier–Stokes-α-like equation, the Leray-α equation, the Modified Leray-α equation, the Simplified Bardina model, the Navier–Stokes–Voigt model and the Navier–Stokes equation) for the fluid velocity u suitably coupled with a parabolic equation for the director field d. 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.