The $\mathcal{R}$-boundedness of solution operators for the $Q$-tensor model of nematic liquid crystals

TL;DR

This paper proves the $\ ext{\mathcal{R}}$-boundedness of solution operators for a resolvent problem associated with the $Q$-tensor model of nematic liquid crystals in the half-space, aiding in understanding the linear stability and regularity of solutions.

Contribution

It establishes the $\mathcal{R}$-boundedness of solution operators for the resolvent problem of the $Q$-tensor model near the origin, which was previously unaddressed.

Findings

Proves $\mathcal{R}$-boundedness of solution operators.

Provides resolvent estimates for the linear system.

Enhances understanding of linear stability in liquid crystal flows.

Abstract

In this paper, we consider a resolvent problem arising from the $Q$-tensor model for liquid crystal flows in the half-space. Our purpose is to show the $\mathcal{R}$-boundedness for the solution operator families of the resolvent problem when the resolvent parameter lies near the origin. The definition of the $\mathcal{R}$-solvability implies the uniform boundedness of the operator and, consequently, the resolvent estimates for the linear system.