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In recent paper \cite{DW1} (Y. Du and K. Wang, Space-time regularity of theKock $\&$ Tataru solutions to the liquid crystal equations, SIAM J. Math.Anal., \textbf{45}(6), 3838--3853.), the authors proved that the global-in-timeKoch-Tataru type solution $(u,d)$ to the $n$-dimensional incompressible nematicliquid crystal flow with small initial data $(u_{0},d_{0})$ in $BMO^{-1}\timesBMO$ has arbitrary space-time derivative estimates in the so called Koch-Tataruspace norms. The purpose of this paper is to show that the Koch-Tataru typesolution satisfies the decay estimates for any space-time derivative involvingsome borderline Besov space norms. More precisely, for the global-in-timeKoch-Tataru type solution $(u,d)$ to the nematic liquid crystal flow withinitial data $(u_{0},d_{0})\in BMO^{-1}\times BMO$ and$\|u_{0}\|_{BMO^{-1}}+[d_{0}]_{ BMO}\leq \varepsilon$ for some small enough$\varepsilon>0$, and for any positive integers $k$ and $m$, one has\begin{align*} \|t^{\frac{k}{2}+m}(\partial^{k}_{t}\nabla^{m} u,\partial^{k}_{t}\nabla^{m} \nablad)\|_{\widetilde{L}^{\infty}(\mathbb{R}_{+},\dot{B}^{-1}_{\infty,\infty})\cap\widetilde{L}^{1}(\mathbb{R}_{+};\dot{B}^{1}_{\infty,\infty})}\leq \varepsilon.\end{align*} Furthermore, we shall give that the solution admits an uniquetrajectory which is H\"{o}lder continuous with respect to space variables.

Space-time derivative estimates of the Kock-Tataru solutions to the nematic liquid crystal system in Besov spaces