本项目针对应用中最为广泛的一类宏观动力学液晶模型，即Ericksen-Leslie动力学液晶模型的相关数学问题进行研究，考察该方程组在一定条件下解的存在性、唯一性和破裂准则等。具体地，我们将研究（1）解的存在性和破裂准则，包括解的局部存在性和破裂准则，小初值整体解的存在性，或临界空间整体解的存在性等；（2）解的唯一性，在一定的集合中以及适当的意义下考察Ericksen-Leslie液晶方程组解的唯一性；（3）初值范围对解的整体存在或破裂的影响。通过对这些问题的有关研究将有助于更深刻地揭示和理解Ericksen-Leslie动力学液晶方程组所蕴含的物理特性。

This project is to study the mathematical problems in Ericksen-Leslie liquid crystal system, which is a kind of macroscopic hydrodynamical liquid crystal model extensively used in reality. We will consider problems on the existence, uniqueness and blow up criteria under certain conditions. Explicitly, it is arranged to discuss about (1) the existence and blow up problem, such as local existence and blow up criteria, global existence with small initial data, and global existence in critical spaces; (2) the uniqueness problem, which is to study the uniqueness in some classes under proper meaning; (3) the influence of the set of initial data on the global existence or finite time blow-up. It will help us to disclose and interprete the physical characteristics of the Ericksen-Leslie system rather deeply.