Long-time Behavior for Nonlinear Hydrodynamic System Modeling the Nematic Liquid Crystal Flows

TL;DR

This paper investigates the long-term behavior of a simplified nematic liquid crystal flow model, proving solutions converge to steady states and estimating convergence rates using advanced mathematical techniques.

Contribution

It establishes the convergence of solutions to steady states for a simplified liquid crystal model and provides convergence rate estimates using the ojasiewicz--Simon approach.

Findings

Global classical solutions converge to unique steady states

Convergence rate estimates are provided

Results extend to generalized problems with boundary condition variations

Abstract

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.