Blow up criterion for nematic liquid crystal flows

TL;DR

This paper establishes criteria for finite-time blow-up of classical solutions to nematic liquid crystal flows in two and three dimensions, based on specific integrability conditions of vorticity and director field gradients.

Contribution

It provides a new blow-up criterion for nematic liquid crystal flows, extending understanding of singularity formation in these complex fluid systems.

Findings

Blow-up occurs if certain integrability conditions are violated.

Criteria differ between 2D and 3D cases.

Conditions involve vorticity and director field gradients.

Abstract

In this paper, we establish a blow up criterion for the short time classical solution of the nematic liquid crystal ow, a simplified version of Ericksen-Leslie system modeling the hydrodynamic evolution of nematic liquid crystals, in dimensions two and three. More precisely, $0<T_*<+\infty$ is the maximal time interval iff (i) for $n=3$, $|\omega|+|\nabla d|^2\notin L^1_tL^\infty_x(\mathbb R^3\times [0,T_*])$, and (ii) for $n=2$, $|\nabla|^2\notin L^1_tL^\infty_x(\mathbb R^2\times [0,T_*])$.