# Poiseuille flow of nematic liquid crystals via the full Ericksen-Leslie model

## TL;DR

This paper investigates the Poiseuille flow of nematic liquid crystals using the full Ericksen-Leslie model, revealing finite-time cusp singularities and establishing global weak solutions with bounded energy, highlighting the role of flux density in singularity formation.

## Contribution

It introduces a detailed analysis of cusp singularity formation in the full Ericksen-Leslie model and demonstrates global existence of weak solutions with bounded energy.

## Key findings

- Finite-time cusp singularities can form due to wave interactions.
- Flux density remains bounded despite singularity formation.
- Global weak solutions exist with Hölder continuity and finite energy.

## Abstract

In this paper, we study the Cauchy problem of the Poiseuille flow of full Ericksen-Leslie model for nematic liquid crystals. The model is a coupled system of a parabolic equation for the velocity and a quasilinear wave equation for the director. For a particular choice of several physical parameter values, we construct solutions with smooth initial data and finite energy that produce, in finite time, cusp singularities - blowups of gradients. The formation of cusp singularity is due to local interactions of wave-like characteristics of solutions, which is different from the mechanism of finite time singularity formations for the parabolic Ericksen-Leslie system. The finite time singularity formation for the physical model might raise some concerns for purposes of applications. This is, however, resolved satisfactorily; more precisely, we are able to establish the global existence of weak solutions that are H\"older continuous and have bounded energy. One major contribution of this paper is our identification of the effect of the flux density of the velocity on the director and the reveal of a singularity cancellation - the flux density remains uniformly bounded while its two components approach infinity at formations of cusp singularities.