  We study a simplified system of the original Ericksen--Leslie equations for
the flow of nematic liquid crystals. This is a coupled non-parabolic
dissipative dynamic system. We show the convergence of global classical
solutions to single steady states as time goes to infinity (uniqueness of
asymptotic limit) by using the \L ojasiewicz--Simon approach. Moreover, we
provide an estimate on the convergence rate. Finally, we discuss some possible
extensions of the results to certain generalized problems with changing density
or free-slip boundary condition.
