Well-posedness and Global Attractors for Liquid Crystals on Riemannian Manifolds

We study the coupled Navier-Stokes Ginzburg-Landau model of nematic liquid crystals introduced by F.H. Lin, which is a simplified version ofthe Ericksen-Leslie system. We generalize the model to compact n-dimensional Riemannian manifolds, deriving the system from a variational principle, and provide a very simple proof of local well-posedness for this coupled system using a contraction mapping argument. We then prove that this system is globally well-posed and has compact global attractors when the dimension of the manifold M is two. A small data result in n dimensions follows easily. 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 threedimensional.