On the rigidity of nematic liquid crystal flow on $S^2$

TL;DR

This paper proves that nematic liquid crystal flows on the sphere converge uniformly to a steady state over time, under certain conditions related to the director field's behavior at infinity.

Contribution

It establishes the uniform convergence of a simplified Ericksen-Leslie system on $S^2$, extending previous results by incorporating the behavior of the director field and bubbles at infinity.

Findings

Uniform convergence in $L^2$ to steady state as t→∞

Convergence holds under various small initial data

Shared orientation condition at infinity for director and bubbles

Abstract

In this paper we establish the uniformity property of a simplified Ericksen-Leslie system modelling the hydrodynamics of nematic liquid crystals on the two dimensional unit sphere $S^2$, namely the uniform convergence in $L^2$ to a steady state exponentially as t tends to infinity. The main assumption, similar to Topping [15], concerns the equation of liquid crystal director $d$ and states that at infinity time, a weak limit $d_\infty$ and any bubble $\omega_i$($1\le i\le l$) share a common orientation. As consequences, the uniformity property holds under various types of small initial data.