The Poiseuille flow of the full Ericksen-Leslie model for nematic liquid crystals: The general Case

TL;DR

This paper investigates the global behavior of the full Ericksen-Leslie model for nematic liquid crystals during Poiseuille flow, extending previous work to more general conditions and handling complex PDEs with irregular coefficients.

Contribution

It extends prior results to the general case of the Ericksen-Leslie model, providing a systematic approach to PDEs with irregular coefficients and establishing global solutions.

Findings

Global solutions exist beyond singularities.

Extended analysis to general physical setup.

Handled PDEs with irregular coefficients.

Abstract

In this work, we study the Cauchy problem of Poiseuille flow of the full Ericksen-Leslie model for nematic liquid crystals. The model is a coupled system of two partial differential equations: One is a quasi-linear wave equation for the director field representing the crystallization of the nematics, and the other is a parabolic PDE for the velocity field characterizing the liquidity of the material. We extend the work in [Chen, et. al. {\em Arch. Ration. Mech. Anal.} {\bf 236} (2020), 839-891] for a special case to the general physical setup. The Cauchy problem is shown to have global solutions beyond singularity formation. Among a number of progresses made in this paper, a particular contribution is a systematic treatment of a parabolic PDE with only H\"older continuous diffusion coefficient and rough (worse than H\"older) nonhomogeneous terms.