Global Well Posedness for a Q-tensor Model of Nematic Liquid Crystals

In this paper, we prove the global well posedness and the decay estimates for a $${\mathbb {Q}}$$ -tensor model of nematic liquid crystals in $$\mathbb {R}^N$$ , $$N \ge 3$$ . This system is a coupled system by the Navier–Stokes equations with a parabolic-type equation describing the evolution of the director fields $${\mathbb {Q}}$$ . The proof is based on the maximal $$L_p$$ – $$L_q$$ regularity and the $$L_p$$ – $$L_q$$ decay estimates to the linearized problem.