LPS's Criterion for Incompressible Nematic Liquid Crystal Flows

TL;DR

This paper establishes a criterion based on Ladyzhenskaya-Prodi-Serrin conditions for the breakdown of smooth solutions in incompressible nematic liquid crystal flows, linking solution regularity to integrability conditions of velocity and director field gradients.

Contribution

It derives a new blow-up criterion for classical solutions to the nematic liquid crystal flow, extending the understanding of solution regularity and breakdown conditions in this model.

Findings

Breakdown occurs if velocity and gradient of director field do not satisfy Ladyzhenskaya-Prodi-Serrin conditions.

Provides a precise mathematical criterion for the maximal existence time of smooth solutions.

Extends classical fluid dynamics criteria to nematic liquid crystal flows.

Abstract

In this paper we derive LPS's criterion for the breakdown of classical solutions to the incompressible nematic liquid crystal flow, a simplified version of Ericksen-Leslie system modeling the hydrodynamic evolution of nematic liquid crystals in $\mathbb R^3$. We show that if $0<T<+\infty$} is the maximal time interval for the unique smooth solution $u\in C^\infty([0,T),\mathbb R^3)$, then $|u|+|\nabla d|\notin L^q([0,T],L^p(\mathbb R^3))$, where $p$ and $q$ safisfy the Ladyzhenskaya-Prodi-Serrin's condition: $\frac{3}{p}+\frac{2}{q}=1$ and $p\in(3,+\infty]$