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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 the two-dimensional case, each global weak solution converges to a single stationary state when h(t) and g(t) converge to a time-independent boundary datum h∞ and 0, respectively. Estimates on the convergence rate are also obtained. In the threedimensional case, we prove that global weak solutions are eventually strong so that results similar to the two-dimensional case can be proved. 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. © 2013 Society for Industrial and Applied Mathematics.

Long-Time Behavior for a Hydrodynamic Model on Nematic Liquid Crystal Flows with Asymptotic Stabilizing Boundary Condition and External Force