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 $d=2\pi /\mu$. Here $\mu >0$ is the twist rate. $d$ is the distance between two planes which are parallel to the $x_1{x}_2$-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 $t\to \infty$. An optimal blow up rate is also obtained.

24 pages

References in corpus (2)

• Recent developments of analysis for hydrodynamic flow of nematic liquid crystals
• Stable blow up dynamics for the 1-corotational energy critical harmonic heat flow

Cited by in corpus (1)

• Nonuniqueness of nematic liquid crystal flows in dimension three