Asymptotic Behavior for a Nematic Liquid Crystal Model with Different Kinematic Transport Properties

TL;DR

This paper analyzes the long-term behavior of solutions in a nematic liquid crystal flow model, demonstrating convergence to steady states with specific rates in 2D and 3D cases.

Contribution

It establishes the asymptotic convergence of global solutions to steady states for a coupled Navier-Stokes and kinematic transport system, including convergence rates.

Findings

Global strong solutions converge to steady states over time.

Convergence rates are provided for 2D and specific 3D cases.

Results apply to models with different molecular shapes and kinematic transports.

Abstract

We study the asymptotic behavior of global solutions to hydrodynamical systems modeling the nematic liquid crystal flows under kinematic transports for molecules of different shapes. The coupling system consists of Navier-Stokes equations and kinematic transport equations for the molecular orientations. We prove the convergence of global strong solutions to single steady states as time tends to infinity as well as estimates on the convergence rate both in 2D for arbitrary regular initial data and in 3D for certain particular cases.