Sufficient conditions for regularity and uniqueness of a 3D nematic liquid crystal model

✍ Scribed by F. Guillén–González; M. A. Rodríguez–Bellido; M. A. Rojas–Medar

• Publisher: John Wiley and Sons
• Year: 2009
• Tongue: English
• Weight: 230 KB
• Volume: 282
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200610776

✦ Synopsis

Abstract

In [3], L. Berselli showed that the regularity criterion ∇u ∈ (0, T; L^q^ (Ω)), for some q ∈ (3/2, + ∞], implies regularity for the weak solutions of the Navier–Stokes equations, being u the velocity field. In this work, we prove that such hypothesis on the velocity gradient is also sufficient to obtain regularity for a nematic Liquid Crystal model (a coupled system of velocity u and orientation crystals vector d) when periodic boundary conditions for d are considered (without regularity hypothesis on d). For Neumann and Dirichlet cases, the same result holds only for q ∈ [2, 3], whereas for q ∈ (3/2, 2) ∪ (3, + ∞] additional regularity hypothesis for d (either on ∇ d or Δd) must be imposed.

On the other hand, when the Serrin's criterion u ∈ (0, T; L^p^ (Ω)) with some p ∈ (3, + ∞] ([16]) for u is imposed, we can obtain regularity of the system only in the problem of periodic boundary conditions for d. When Neumann and Dirichlet cases for d are considered, additional regularity for d must be imposed for each p ∈ (3, + ∞] (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)