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⊂R3. 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 Equ 262:1291–1316, 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=R3, based on a result of Du et al. (Arch Rational Mech Anal 238:749–803, 2020), we give an interesting regularity criterium just via the B˙∞,∞-1 norm of u and the L∞ norm of the initial data Q0.