BKM's criterion for the 3D nematic liquid crystal flows via two velocity components and molecular orientations

In this paper we provide a sufficient condition, in terms of the horizontal gradient of two horizontal velocity components and the gradient of liquid crystal molecular orientation field, for the breakdown of local in time strong solutions to the three-dimensional incompressible nematic liquid crystal flows. More precisely, let $T_{*}$ be the maximal existence time of the local strong solution $(u, d)$, then $T_{*}<+\infty$ if and only if \begin{align*} \int_{0}^{T_*} \big( \|\nabla_h u^h\|_{\dot{B}^0_{p,\frac{2p}{3}}}^q + \|\nabla d\|_{\dot{B}^0_{\infty,\infty}}^2 \big)dt = \infty\ \ \text{with}\ \ \ \frac{3}{p} +\frac{2}{q} = 2,\ \ \frac{3}{2} < p\leq\infty, \end{align*} where $u^{h}=(u^{1},u^{2})$, $\nabla_{h}=(\partial_{1}, \partial_{2})$. This result can be regarded as the generalization of the BKM's criterion in \cite{HW12}, and is even new for the three-dimensional incompressible Navier-Stokes equations.