Gevrey analyticity of solutions to the 3D nematic liquid crystal flows in critical Besov space

Abstract

We show that the solution to the Cauchy problem of the 3D nematic liquid crystal flows, with initial data belongs to a critical Besov space, belongs to a Gevrey class. More precisely, it is proved that for any $(u_{0},d_{0} - \overline{d}_{0}) \in \dot{B}^{\frac{3}{p}-1}_{p,1} (\mathbb{R}^{3}) \times \dot{B}^{\frac{3}{q}}_{q,1} (\mathbb{R}^{3})$ with some suitable conditions imposed on $p, q\in(1,\infty)$, there exists $T^{*}>0$ depending only on initial data, such that the nematic liquid crystal flows admits a unique solution $(u,d)$ on $\mathbb{R}^{3} \times (0,T^{*})$, and satisfies \begin{align*} \|e^{\sqrt{t} \Lambda_{1}}{u}(t) \|_{\widetilde{L}^{\infty}_{T^{*}} (\dot{B}^{\frac{3}{p}-1}_{p,1}) \cap \widetilde{L}^{1}_{T^{*}} (\dot{B}^{\frac{3}{p}+1}_{p,1})} + \|e^{\sqrt{t}\Lambda_{1}} ({d}(t)- \overline{d}_{0}) \|_{\widetilde{L}^{\infty}_{T^{*}} (\dot{B}^{\frac{3}{q}}_{q,1}) \cap \widetilde{L}^{1}_{T^{*}}(\dot{B}^{\frac{3}{q}+2}_{q,1})} < \infty. \end{align*} Here, $\overline{{d}}_{0} \in \mathbb{S}^{2}$ is a constant unit vector, and $\Lambda_{1}$ is the Fourier multiplier whose symbol is given by $|\xi|_{{1}}=|\xi_{1}|+|\xi_{2}|+|\xi_{3}|$. Moreover, if the initial data is sufficiently small enough, then $T^{*}=\infty$. As a consequence of the method, decay estimates of higher-order derivatives of solutions in Besov spaces are deduced.