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

Abstract

In recent paper \cite{DW1} (Y. Du and K. Wang, Space-time regularity of the Kock $\&$ Tataru solutions to the liquid crystal equations, SIAM J. Math. Anal., \textbf{45}(6), 3838--3853.), the authors proved that the global-in-time Koch-Tataru type solution $(u,d)$ to the $n$-dimensional incompressible nematic liquid crystal flow with small initial data $(u_{0},d_{0})$ in $BMO^{-1}\times BMO$ has arbitrary space-time derivative estimates in the so called Koch-Tataru space norms. The purpose of this paper is to show that the Koch-Tataru type solution satisfies the decay estimates for any space-time derivative involving some borderline Besov space norms. More precisely, for the global-in-time Koch-Tataru type solution $(u,d)$ to the nematic liquid crystal flow with initial 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} \nabla d)\|_{\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 unique trajectory which is H\"{o}lder continuous with respect to space variables.