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.
Categories: math.AP