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

TL;DR

This paper establishes decay estimates for derivatives of solutions to the nematic liquid crystal system in Besov spaces, extending previous work on space-time regularity and demonstrating unique spatial trajectories.

Contribution

It proves decay estimates for derivatives of Kock-Tataru solutions in Besov spaces, showing their temporal decay and spatial regularity in borderline function spaces.

Findings

Solutions satisfy decay estimates involving Besov space norms.

Solutions have unique, Hölder continuous spatial trajectories.

Results extend regularity properties of liquid crystal flows.

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.