  We study the coupled Navier-Stokes Ginzburg-Landau model of nematic liquid
crystals introduced by F.H. Lin, which is a simplified version of the
Ericksen-Leslie system. We generalize the model to compact n-dimensional
Riemannian manifolds, and show that the system comes from a variational
principle. We present a new simple proof for the local well-posedness of this
coupled system without using the higher-order energy law. We then prove that
this system is globally well-posed and has compact global attractors when the
dimension of the manifold M is two.Finally, we introduce the Lagrangian
averaged liquid crystal equations, which arise from averaging the Navier-Stokes
fluid motion over small spatial scales in the variational principle. We show
that this averaged system is globally well-posed and has compact global
attractors even when M is three-dimensional.
