Besov空间中液晶系统Kock-Tataru解的时空导数估计
Space-time derivative estimates of the Kock-Tataru solutions to the nematic liquid crystal system in Besov spaces
摘要：
在最近的论文\cite{DW1}（Y. Du和K.王，液晶方程的Kock $\&$ Tataru解的时空正则性，SIAM数学分析杂志，\textbf{45}（6），3838- 3853.），作者证明了在$BMO^{-1}\times BMO$中具有小初始数据$(u_{0},d_{0})$的$n$维不可压缩液晶流的整体时间Koch-Tataru型解$(u,d)$在所谓的Koch-Tataru空间范数中具有任意的时空导数估计。本文的目的是证明Koch-Tataru型解满足衰减估计的任何时空导数涉及一些边界Besov空间范数。更精确地说，对于具有初始数据$(u_{0},d_{0})\in BMO^{-1}\times BMO$和$\|u_{0}\|_{BMO^{-1}}+[d_{0}]_{ BMO}\leq \varepsilon$的液晶流的整体时间Koch-Tataru型解$(u,d)$，对于一些足够小的$\varepsilon>0$，以及对于任何正整数$k$和$m$，我们有\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*}。
英文摘要：
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 Hölder continuous with respect to space variables.
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