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 Δd) must be imposed. On the other hand, when the Serrin's criterion u ∈ L 2p p-3 (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,+∞].