Twisted solutions to a simplified Ericksen-Leslie equation

TL;DR

This paper constructs global, twisted, and periodic solutions to a simplified Ericksen-Leslie liquid crystal model in three dimensions, revealing singularities along the axis and exponential escape into the third dimension over time.

Contribution

It introduces a method to construct global solutions with specific twisting and periodicity, and analyzes their singularity formation and blow-up behavior.

Findings

Solutions are globally existing and classical for all time.

Solutions develop singularities along the axis and escape exponentially.

An optimal blow-up rate for the solutions is established.

Abstract

In this article we construct global solutions to a simplified Ericksen-Leslie system on $\mathbb{R}^3$. The constructed solutions are twisted and periodic along the $x_3$-axis with period $d = 2\pi \big/ \mu$. Here $\mu > 0$ is the twist rate. $d$ is the distance between two planes which are parallel to the $x_1x_2$-plane. Liquid crystal material is placed in the region enclosed by these two planes. Given a well-prepared initial data, our solutions exist classically for all $t \in [0, \infty)$. However these solutions become singular at all points on the $x_3$-axis and escape into third dimension exponentially while $t \rightarrow \infty$. An optimal blow up rate is also obtained.