Asymptotic Behavior of Solution to the Incompressible Nematic Liquid Crystal Flows in R^3

TL;DR

This paper studies the long-term behavior of solutions to the 3D incompressible nematic liquid crystal flow equations, establishing global existence, decay rates, and convergence properties under small initial data.

Contribution

It provides new results on global existence and decay rates for solutions with small initial data in the 3D nematic liquid crystal flow model.

Findings

Global existence of solutions under small initial data

Decay rates for velocity and director fields

Convergence rates for mixed space-time derivatives

Abstract

In this paper, we investigate the Cauchy problem for the incompressible nematic liquid crystal flows in three-dimensional whole space. First of all, we establish the global existence of solution by energy method under assumption of small initial data. Furthermore, the time decay rates of velocity and director are built when the initial data belongs to $L^1(\mathbb{R}^3)$ additionally. Finally, one also constructs the time convergence rates for the mixed space-time derivatives of velocity and director.