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 x3-axis with period $${d = 2\pi \big/ \mu}$$ . Here $${\mu > 0}$$ is the twist rate and d is the distance between two planes which are parallel to the x1x2-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 x3-axis and escape into third dimension exponentially while $${t \rightarrow \infty}$$ . An optimal blow up rate is also obtained.