Long-time behavior for a hydrodynamic model on nematic liquid crystal flows with asymptotic stabilizing boundary condition and external force

TL;DR

This paper studies the long-term behavior of a simplified nematic liquid crystal flow model, showing convergence to equilibrium states in 2D and eventual regularity in 3D under certain conditions.

Contribution

It establishes convergence results for weak solutions in 2D and proves eventual regularity and existence of strong solutions in 3D for a coupled Navier-Stokes and Ginzburg-Landau model.

Findings

Weak solutions in 2D converge to stationary states.

In 3D, weak solutions become strong over time.

Existence of global strong solutions under high viscosity or near equilibrium.

Abstract

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.