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Sufficient conditions for regularity and uniqueness of a 3D nematic liquid crystal model
Author(s) -
Guillén–González F.,
Rodríguez–Bellido M. A.,
Rojas–Medar M. A.
Publication year - 2009
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200610776
Subject(s) - uniqueness , mathematics , liquid crystal , neumann boundary condition , dirichlet distribution , vector field , orientation (vector space) , boundary value problem , mathematical analysis , dirichlet boundary condition , boundary (topology) , velocity vector , dirichlet problem , pure mathematics , geometry , condensed matter physics , physics , classical mechanics
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 Δ d ) 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)