Global weak solution and blow-up criterion of the general Ericksen-Leslie system for nematic liquid crystal flows

TL;DR

This paper establishes the existence of global weak solutions and a blow-up criterion for the three-dimensional Ericksen-Leslie system modeling nematic liquid crystal flows, addressing mathematical challenges from nonlinearities and boundary conditions.

Contribution

It proves global weak solutions and a blow-up criterion for the general Ericksen-Leslie system, advancing the mathematical understanding of nematic liquid crystal flow models.

Findings

Existence of global-in-time weak solutions under certain conditions.

A blow-up criterion based on the curl of velocity and gradient of director field.

Local well-posedness of classical solutions with Parodi's relation.

Abstract

In this paper we investigate the three dimensional general Ericksen-Leslie (E--L) system with Ginzburg-Landau type approximation modeling nematic liquid crystal flows. First, by overcoming the difficulties from lack of maximum principle for the director equation and high order nonlinearities for the stress tensor, we prove existence of global-in-time weak solutions under physically meaningful boundary conditions and suitable assumptions on the Leslie coefficients, which ensures that the total energy of the E--L system is dissipated. Moreover, for the E--L system with periodic boundary conditions, we prove the local well-posedness of classical solutions under the so-called Parodi's relation and establish a blow-up criterion in terms of the temporal integral of both the maximum norm of the curl of the velocity field and the maximum norm of the gradient of the liquid crystal director field.