Abstract

In [3], L. Berselli showed that the regularity criterion ? u ? (0, T;L q(O)), for some q ? (3/2,+8], 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,+8] additional regularity hypothesis for d (either on ?d or ?d) must be imposed. On the other hand, when the Serrin's criterion u ? L 2p p-3 (0, T;L p(O)) with some p ? (3,+8] ([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,+8]. © 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.