  In this paper, we prove the global well posedness and the decay estimates for
a $\mathbb Q$-tensor model of nematic liquid crystals in $\mathbb R^N$, $N \geq
3$. This system is 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.
