A regularity criterion for the solution of nematic liquid crystal flows in terms of the Ḃ∞,∞−1-norm

In this paper, we investigate the regularity criterion for the solution of nematic liquid crystal flows in three dimensions (n=3n=3) and in two dimensions (n=2n=2). We prove that the solution (u,d)(u,d) is smooth up to time TT provided that there exists a positive constant ε0>0ε0>0 such that (i) for n=3n=3, ‖(u,∇d)‖L∞(0,T;Ḃ∞,∞−1)≤ε0, and (ii) for n=2n=2, ‖∇d‖L∞(0,T;Ḃ∞,∞−1)≤ε0.