  In this paper, we consider a simplified Ericksen-Leslie model for the nematic
liquid crystal flow. The evolution system consists of the Navier-Stokes
equations coupled with a convective Ginzburg-Landau type equation for the
averaged molecular orientation. We suppose that the Navier-Stokes equations are
characterized by a no-slip boundary condition and a time-dependent external
force g(t), while the equation for the molecular director is subject to a
time-dependent Dirichlet boundary condition h(t). We show that, in 2D, each
global weak solution converges to a single stationary state when h(t) and g(t)
converge to a time-independent boundary datum h_\infty and 0, respectively.
Estimates on the convergence rate are also obtained. In the 3D case, we prove
that global weak solutions are eventually strong so that results similar to the
2D case can be proven. We also show the existence of global strong solutions,
provided that either the viscosity is large enough or the initial datum is
close to a given equilibrium.
