Sufficient conditions for regularity and uniqueness of a 3D nematic liquid crystal model |
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| Authors: | F. Guillén–González M. A. Rodríguez–Bellido M. A. Rojas–Medar |
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| Affiliation: | 1. Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain;2. Phone: +34 954 557 010, Fax: +34 954 552 898;3. Dpto. de Ciencias Básicas, Facultad de Ciencias, Universidad del Bío Bío, Campus Fernando May, Casilla 447, Chillán, Chile;4. Phone: +56 42 253 261, Fax: +56 42 253 046 |
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| 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 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) |
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| Keywords: | Liquid Crystal system sufficient hypothesis of regularity strong solution uniqueness regularity criterion |
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