液晶中相关数学问题的研究

There are three different physical theories to model the liquid crystals:molecular kinetci(Doi-Onsager)theory、 Oseen-Frank-Ericksen-Leslie theory (vector model) and Q-tensor theory. The first theory is the microscopic theory based on Statistical mechanics, while the later two theories are the macroscopic theory based on Continuum mechanics. In this project, we will study some mathematical problems in three kinds of theories for the liquid crystal, which includes:(1)study the properties of the stationary(equilibrium) solutions;(2) study the well-posedness, the global existence and singularity of weak solutions for the dynamical models;(3) study the relations among three physical theories;(4)study the stability of stationary solutions and the long time behaviour of the molecular dynamical equations. These research will help us to understand the relations, the range of application, the physical meanings of various parameters for three kinds of theories more deeply.

现有的液晶模型可以分为三类：分子模型（Doi-Onsager理论）、向量模型(Oseen-Frank-Ericksen-Leslie理论)和张量模型(Q-张量理论)。分子模型是基于统计力学的微观理论，而向量模型和张量模型是基于连续介质力学的宏观理论。本项目将对这三类液晶理论中的数学问题进行深入研究，主要包括：(1)静力学理论的平衡态解的性态分析；(2)动力学方程的适定性、弱解的整体存在性和奇性；(3)三种不同理论之间的关系；(4)分子动力学方程平衡态解的稳定性以及长时间性态。这些问题的研究有助于人们更深入地理解三个模型以及它们之间的关系、适用范围以及各参数的物理意义。

本项目主要研究液晶中的数学问题，主要取得了如下重要成果：（1）在液晶无缺陷情形，严格地从Doi-Onsager方程以及张量模型推导出了Ericksen-Leslie方程，系统的研究了液晶微观模型到宏观模型的推导；（2）证明了液晶二维度为1/2的点缺陷的稳定性；（3）从静力学和动力学两个方面，系统地研究了液晶的Isotropic-Nematic相变问题。