液晶的Oseen-Frank理论及其相关的数学问题

本项目旨在研究液晶的数学理论，特别是Oseen-Frank理论方面。Hardt-Kinderlehrer-Lin在80年代对Oseen-Frank能量的极小值进行了研究，由于主项算子的非椭圆特征，迄今为止进展不多，这同时影响了与Oseen-Frank能量相关的Ericksen-Leslie，Q-tensor系统的发展。我们将研究：.1. 弱解存在性问题：三维Ericksen-Leslie 系统的弱解是一个公开问题，即便简化的Ericksen-Leslie 系统也未得到解决；高维Oseen-Frank流，Landau-Lifshitz方程的部分正则解的存在性；.2. 缺陷研究（即弱解的奇点）：二维Oseen-Frank映射的bubble，能量恒等式，稳态解的稳定性，爆破速率，高维奇点集的刻画等；.3. 极限问题：建立Q-tensor到Ericksen-Leslie系统的弱解极限

The project is aimed at the mathematical theory of liquid crystals, especially Oseen-Frank theory. In 1980s, minimal solutions of Oseen-Frank energy are studied by Hardt-Kinderlehrer-Lin, but fewer results are made up to now due to the non-elliptic property of the main term, which influences the development of related Ericksen-Leslie and Q-tensor system. We'll research as follows:.1. The existence of weak solution: whether there exist weak solutions for the 3D Ericksen-Leslie system is open, even for the 3D simplified Ericksen-Leslie system it's also unknown; the existence of partial regular solutions of Oseen-Frank flow and Landau-Lifshitz equations in the high dimension.2. Defects (singular points of weak solution): the analysis of bubble, energy identity, stability of stationary solutions and blow-up rate for the 2D Oseen-Frank flow, the measure of singular set in the high dimension;.3. Limit problems: weak limit from Q-tensor system to Ericksen-Leslie system