Twisted solutions to a simplified Ericksen-Leslie equation

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.