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

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)