BKM s criterion for the 3D nematic liquid crystal flows in Besov spaces of negative regular index

In this paper, we investigate the blow-up criterion of a smooth solution of the nematic liquid crystal flow in three- 2 dimensional space. More precisely, We prove that if ʃT0 ( ‖ ω ‖ 2/2−α + ∇d )dt ∞, 0 α 2, then the solution (u, d) 0 B˙ −α ∞ ∞ B˙ 0 , ∞ , ∞ ∞ can be extended smoothly beyond t = T.