向列型液晶材料的缺陷分布，相变机制及动力学规律

本项目主要研究向列相液晶材料数学理论的相关问题。缺陷的位置分布以及几何结构，材料的相变过程，模型的动力学行为等问题的研究是人们长期关注的核心课题。对于稳态系统，本项目主要考虑经典的Landau-de Gennes理论框架相关的几类数学问题，包括零点集的几何结构，弹性消失极限，相变界面的几何刻画等课题。该理论清晰描述双轴向列相液晶材料的物理特性，具有鲜明的工业背景。对于动态系统，本项目主要研究两类基本框架：向量层面的Ericksen-Leslie理论与张量层面的Beris-Edwards理论中的若干问题，包括解的整体适定性，不可压缩极限和无界区域的最优衰减估计等问题。系统带有强非线性与强耦合性，具有重要的理论研究价值。本项目致力于整合已有的研究方法，不断挖掘新的研究思路和理论工具，进一步补充向列相液晶材料数学模型的理论体系，更好地解释向列相液晶材料的物理现象。

This programme mainly investigates several problems in mathematical theory of nematic liquid crystals. The topics which focus on the distribution and geometry structure of defects, the phase transition and the dynamical behavior are essential problems that people concern about. For the steady case, this programme studies some problems on the classical Landau-de Gennes theory, such as, the geometry structure of the nodal sets, the vanishing elasticity limit, phase transition, etc. The Landau-de Gennes theory clearly describes the physical properties of biaxial nematic liquid crystals and admits the specific industrial background. For the dynamical case, this programme studies some problems on the Ericksen-Leslie theory and the Beris-Edwards theory, including the well-posedness of solutions, the incompressible limit and optimal decay estimates in unbounded domain, etc. The systems admit strong nonlinearity and coupling, which provide important theoretic research value. This programme is devoted to integrating the methods available, developing new ideas, supplementing the theoretic system in nematic liquid crystals, and explaining the physical phenomenon of nematic liquid crystals.