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

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 (PDEs): 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. (2020) [11] 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 Hölder continuous diffusion coefficient and rough (maybe unbounded) nonhomogeneous terms.