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

In [3], L. Berselli showed that the regularity criterion del u is an element of L2q/2q-3(0,T;L-q(Omega)), for some q is an element of (3/2, +infinity], 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 nernatic 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 Difichlet cases, the same result holds only for q is an element of (2, 3], whereas for q is an element of (3/2, 2) boolean OR (3, + infinity] additional regularity hypothesis for d (either on del d or Delta d) must be imposed. On the other hand, when the Serrin's criterion u is an element of L2p/p-3(0, T; L-p (Omega)) with some p is an element of (3, +infinity] ([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 is an element of (3, +infinity]. (C) 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim