Asymptotic Behavior of Solutions to the Liquid Crystal System in $H^m(\mathbb{R}^3)$

TL;DR

This paper investigates the long-term behavior of solutions to a nematic liquid crystal system in three-dimensional Sobolev spaces, establishing optimal decay rates that match those of the linearized system for small initial data.

Contribution

It provides the first derivation of optimal decay rates in Sobolev spaces for the nematic liquid crystal system with constant density and small initial data.

Findings

Established optimal decay rates in Sobolev spaces.

Decay rates coincide with the linear system.

Results apply to solutions with small initial data.

Abstract

In this paper we study the large time behavior of regular solutions to a nematic liquid crystal system in Sobolev spaces $H^m(\R^3)$ for $m\geq 0$.We obtain optimal decay rates in $H^m(R^3)$ spaces, in the sense that the rates coincide with the rates of the underlying linear counterpart. The fluid under consideration has constant density and small initial data.