液晶各动力学模型解之间的关系

There are three kinds of dynamic theories for liquid crystals: molecular theory(Doi-Onsager model), tensor theory(Landau-de Gennes model) and vector theory(Ericksen-Leslie model). The first one is a microscopic theory which is derived from statistical mechanics, and the latter two are derived from continuum mechanics. In this project, we will study the well-posedness of these systems and the connection between the solutions to them in the bounded domain case, which include: (1) the well-posedness of various dynamic equations for liquid crystals; (2)the connection between the solutions of dynamic Landau-de Gennes equation and Ericksen-Leslie equation; (3)the connection between the solutions of Doi-Onsager equation and Ericksen-Leslie equation. These research will help us to understand their connection more deeply and give us a better knowledge on the differences and commonalities between those models and the range of their application. In addition, it can provide microscopic interpretation to various parameters in macroscopic models and therefore is important.

液晶的动力学理论可以分为三类：分子理论(Doi-Onsager模型)，张量理论(Landau-de Gennes模型)和向量理论(Ericksen-Leslie模型)。前者是从统计力学观点出发得到的微观理论，后两者是从连续介质力学观点出发得到的宏观模型。本项目围绕它们的适定性理论以及解之间的关系展开研究，内容包括：(1) 液晶各动力学方程强解的适定性；(2)动力学Landau-de Gennes模型的解与Ericksen-Leslie方程的解之间的关系；(3)Doi-Onsager方程的解与Ericksen-Leslie方程的解之间的关系。这些问题的解决将有助于人们进一步理解它们之间的联系，从而认识各个模型的区别、共性以及各自适用范围，还能给出宏观模型各参数的微观解释，因而具有重要的意义。

本项目围绕复杂流体的数学理论，主要研究了液晶动力学方程解之间的关系和复杂流体自由边界问题，取得了一系列研究成果，主要包括：.1、动力学Q-张量理论在有界区域上的适定性；.2、弱解框架下，微观分子理论与宏观向量理论解之间的联系；.3、两相共存情形，动力学Q-张量理论与向量理论之间的联系；.4、磁流体电流涡片问题的稳定性；