Regularity criterion for 3D nematic liquid crystal flows in terms of finite frequency parts in $\dot{B}_{\infty,\infty }^{-1}$

In this paper, we establish the regularity criterion for the weak solution of nematic liquid crystal flows in three dimensions when the $L^{\infty }(0,T;\dot{B}_{\infty,\infty }^{-1})$ L ∞ ( 0 , T ; B ˙ ∞ , ∞ − 1 ) -norm of a suitable low frequency part of $(u,\nabla d)$ ( u , ∇ d ) is bounded by a scaling invariant constant and the initial data $(u_{0},\nabla d_{0})$ ( u 0 , ∇ d 0 ) . Our result refines the corresponding one in (Liu and Zhao in J. Math. Anal. Appl. 407:557-566, 2013) and that in (Ri in Nonlinear Anal. TMA 190:111619, 2020).