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

Abstract

In [3] L. C. Berselli, On a Regularity Criterion for the Solutions to the 3D Navier-Stokes Equations, Diff. and Integral Eq., Vol. 15, Number 9, 1129-1137 (2002). , L. Berselli showed that the additional regularity hypothesis for the velocity gradient ∇u ∈ L 2q 2q−3 (0, T;L q (Ω)), for some q ∈ (3/2, +∞], implies the strong regularity for the weak solutions of the Navier-Stokes equations. In this work, we prove that such hypothesis is also sufficient in order to obtain the strong solution 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. For Neumann and Dirichlet boundary conditions, we obtain the same result only for the cases of q ∈ [2, 3], whereas in the cases q ∈ (3/2, 2) ∪ (3, +∞], we also need to impose an additional regularity hypothesis for d (either on ∇d or ∆d). On the other hand, when the following hypothesis for u of Serrin’s type is imposed ([18] J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rat. Mech. Anal. 9 (3), 187-195 (1962): u ∈ L 2p p−3 (0, T;L p (Ω)) for some p ∈ (3, +∞], we can obtain strong regularity only in the case of periodic boundary conditions for d. For Neumann or Dirichlet boundary conditions, additional regularity for d must be imposed in all cases.