In this paper, we investigate blow up criteria for the local smooth solutions to the 3D incompressible nematic liquid crystal flows via the components of the gradient velocity field ∇u\documentclass[12pt]{minimal}
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\begin{document}$\nabla u$\end{document} and the gradient orientation field ∇d\documentclass[12pt]{minimal}
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\begin{document}$\nabla d$\end{document}. More precisely, we show that 0<T∗<+∞\documentclass[12pt]{minimal}
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\begin{document}$0< T_{ \ast}<+\infty$\end{document} is the maximal time interval if and only if ∫0T∗∥∥∂iu∥Lxiγ∥Lxjxkαβ+∥∇d∥L∞83dt=∞, with 2α+2β≤3α+24α, and 1≤γ≤α,2<α≤+∞,\documentclass[12pt]{minimal}
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\begin{document} $$\begin{aligned} & \int_{0}^{T_{\ast}} \bigl\Vert \Vert \partial_{i}u\Vert _{L_{x_{i}} ^{\gamma}} \bigr\Vert _{L_{x_{j}x_{k}}^{\alpha}}^{\beta}+ \|\nabla d\| _{L^{\infty}}^{\frac{8}{3}}\mathrm{d}t=\infty, \\ &\quad\text{ with } \frac{2}{\alpha}+\frac{2}{\beta}\leq\frac{3\alpha +2}{4\alpha}, \text{ and } 1\leq\gamma\leq\alpha,2< \alpha\leq+\infty, \end{aligned}$$ \end{document} or ∫0T∗∥∂3u3∥Lαβ+∥∇d∥L∞83dt=∞,with 3α+2β≤3(α+2)4α, and 2<α≤∞,\documentclass[12pt]{minimal}
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\begin{document} $$\begin{aligned} \int_{0}^{T_{\ast}}\|\partial_{3}u_{3} \|^{\beta}_{L^{\alpha}}+\| \nabla d\|^{\frac{8}{3}}_{L^{\infty}} \mathrm{d}t=\infty,\quad\text{with } \frac{3}{\alpha}+\frac{2}{\beta}\leq \frac{3(\alpha+2)}{4 \alpha}, \text{ and } 2< \alpha\leq\infty, \end{aligned}$$ \end{document} where i,j,k∈{1,2,3}\documentclass[12pt]{minimal}
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\begin{document}$i,j,k\in\{1,2,3\}$\end{document}, i≠j\documentclass[12pt]{minimal}
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\begin{document}$i\neq j$\end{document}, i≠k\documentclass[12pt]{minimal}
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\begin{document}$i\neq k$\end{document}, and j≠k\documentclass[12pt]{minimal}
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\begin{document}$j\neq k$\end{document}.
