向列相液晶Ericksen-Leslie方程若干数学问题的研究

Ericksen-Leslie方程是被研究者们普遍接受的用来描述向列相液晶动力学的有效模型，成功诠释了很多实验结果。由于该方程本身所具有的复杂性以及内含参数的多样性，使其相应数学研究进展缓慢。已有的研究大多是对方程做了一些简化或者是在某些特殊参数情形下给出的，并且很少考虑施加外力后方程的动力学行为。本项目从动力系统的角度系统研究强锚定边界条件下完整Ericksen-Leslie方程一些重要的问题：首先结合系统特殊的可积结构，利用摄动方法和相平面分析法等研究方程稳态Poiseuille流动的动力学行为，进一步尝试给出表观黏度的近似函数表达；其次，在电磁场作用下研究完整方程稳态剪切流的动力学行为和稳定性问题，尝试为Fréedericksz转变提供动力学解释。研究成果将揭示流场大小、各类参数变化以及外加电磁场大小对方程解的存在性、多重性及稳定性的影响，为工程技术人员合理使用液晶材料提供理论支持。

The Ericksen-Leslie equation is generally accepted by researchers as an effective model to describe the dynamics of nematic liquid crystals, and has successfully interpreted many experimental results. However, due to the complexity of the equation itself and the diversity of its contained parameters, its mathematical research progresses slowly. Most of the existing studies have considered the simplified Ericksen-Leslie equation or in the case of some special parameters, and rarely considered the dynamic behavior of the Ericksen-Leslie equation after applying external forces. In this project, we will systematically study some important problems of the complete Ericksen-Leslie equation under the strong anchoring boundary conditions from the perspective of dynamic system. Firstly，combining the special integrable structure of the system, we will study the dynamic behavior of steady-state Poiseuille flow of the equation by using perturbation method and phase plane analysis method，and further try to give the approximate function expression of the apparent viscosity. Secondly, we will study the dynamic behavior and stability of steady-state shear flow of the complete equation under the action of electromagnetic field, and try to provide a dynamic explanation for Fréedericksz transition. The research results will reveal the influence of the magnitude of the flow field, the variation of various parameters and the magnitude of the external electromagnetic field on the existence, multiplicity and stability of solutions of the equation, and provide theoretical support for engineers to use liquid crystal materials reasonably.