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We consider a full Navier-Stokes and $Q$-tensor system for incompressibleliquid crystal flows of nematic type. In the two dimensional periodic case, weprove the existence and uniqueness of global strong solutions that areuniformly bounded in time. This result is obtained without any smallnessassumption on the physical parameter $\xi$ that measures the ratio betweentumbling and aligning effects of a shear flow exerting over the liquid crystaldirectors. Moreover, we show the uniqueness of asymptotic limit for each globalstrong solution as time goes to infinity and provide an uniform estimate on theconvergence rate.

Global strong solutions of the full Navier-Stokes and $Q$-tensor system for nematic liquid crystal flows in $2D$: existence and long-time behavior