Optimal decay of global strong solutions to nematic liquid crystal flows in the half-space

TL;DR

This paper investigates the decay rates of strong solutions to nematic liquid crystal flows in a half-space, revealing faster decay under certain initial data conditions using advanced mathematical tools.

Contribution

It provides new decay rate results for derivatives of solutions to nematic liquid crystal flows in a half-space, extending understanding of their asymptotic behavior.

Findings

Higher-order derivatives decay faster than heat kernel predictions.

Decay rates depend on initial data in weighted Sobolev spaces.

Utilizes $L^p-L^q$ estimates and steady Stokes system analysis.

Abstract

We study asymptotic behaviors of the higher-order spatial derivatives and the first-order time derivatives for the strong solution to nematic liquid crystal flows in the half-space $\mathbb{R}_+^3$. Furthermore, when the initial data lie in an appropriately weighted Sobolev space, we obtain the decay rates that are faster than the heat kernel. The main tools employed in this paper are the $L^p-L^q$ estimates of the Stokes semigroup, the a priori estimates of the steady Stokes system in $\mathbb{R}_+^3$, and the representation formula of the Leray projection operator.