Logarithmically improved regularity criterion for the nematic liquid crystal flows in Ḃ∞,∞−1 space

In this work, we study the regularity criterion of the three-dimensional nematic liquid crystal flows. It is proved that if the vorticity satisfies ∫0T‖ω(t,⋅)‖B.∞,∞−121+log(e+‖ω(t,⋅)‖B.∞,∞−1)dt<∞, where Ḃ∞,∞−1 denotes the critical Besov space, then the solution (u,d)(u,d) becomes a regular solution on (0,T](0,T]. This result extends the recent regularity criterion obtained by Fan and Ozawa (2012) [11].