2023/02/16 by Geng Chen, Chen, Geng, Weishi Liu +3 · 2 citations
Engineering · Materials Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Liquid Crystal Research Advancements
paper · pdf · doi:10.48550/arxiv.2302.08616
openalex publication_date 2023/02/16 · openalex created_date 2023/02/22 · openalex updated_date 2026/07/28
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 Hölder continuous diffusion coefficient and rough (worse than Hölder) nonhomogeneous terms.