Global Regularity to the liquid crystal flows of Q-tensor model

In this paper we investigate a forced incompressible Navier-Stokes equation coupled with a parabolic type equation of Q-tensors in a domain $U\subset {\mathbb{R}}^3\mathrm{.}$ In the case $U$ is bounded, we prove the existence of a global strong solution when the initial data are sufficiently small, improving a result in Xiao’s paper [J. Differ. Equations 2017]. The key tool of the proof is a {maximum principle.} Then, we establish also a result of continuous dependence of solutions on the initial data. Finally, if $U={\mathbb{R}}^3,$ based on a result of Du, Hu and Wang [Arch. Rational Mech. Anal. 2020], we give an interesting regularity criterium just via the ${\dot{B}}_{\mathrm{\infty },\mathrm{\infty }}^{-1}$ norm of $u$ and the $L^{\mathrm{\infty }}$ norm of the initial data $Q_0$.