Well-posedness and global attractors for liquid crystals on Riemannian manifolds

Journal article

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.

Authors: Shkoller, S

• Journal: Communications in Partial Differential Equations
• Volume: 27
• Issue: 5-6
• Pages: 1103-1137
• Publication date: 2001-01-24
• DOI: 10.1081/PDE-120004895
• ISSN: 0360-5302
• Language: English
• Keywords: math.AP; 35Q30, 35Q72