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

  • Author(s): Shkoller, Steve
  • et al.

Published Web Location

https://arxiv.org/pdf/math/0101203.pdf
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Abstract

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.

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