若干向列型液晶连续介质模型的数学理论研究

随着液晶材料在高新科技领域广泛和成功应用，液晶已成为一门与多学科交叉的新型学科，其流体动力学规律已成为工业与材料科学等备受重视问题之一。向列型液晶是最常见的液晶类型，在连续介质理论层面上，其对应的动力学模型有Ericksen-Leslie(EL)模型和Beris-Edwards(BE)模型等。本项目研究简化EL模型和BE模型若干数学问题。两模型都是较流体中经典Navier-Stokes方程组更为复杂的非线性偏微分方程组，其数学研究极具挑战性。我们拟利用能量方法、紧性方法、调和分析和几何分析技巧、部分正则性理论和边界正则化技巧等研究EL模型和BE模型相关问题的适定性及解的奇性和大时间行为等性态；利用算子理论和控制理论等研究EL模型初边值问题对应的最优边界控制。本项目研究内容有着多方面的应用背景，在非线性偏微分方程研究领域也具有基本的重要性，研究成果将推动液晶流体动力学相关理论的研究。

With the wide and successful applications of liquid crystal materials in various new-high technologies fields, liquid crystal has become a new interdisciplinary subject, and its hydrodynamic law has become one of the most important issues in industry and material science. Nematic liquid crystal is the most common phase of liquid crystal. The Ericksen-Leslie (EL) model and the Beris-Edwards (BE) model are two comprehensive hydrodynamic models in the continuum level. In this project, we study several mathematical problems of simplified the EL model and the BE model. Both models are more complex nonlinear partial differential equations than the classical Navier-Stokes equations, and their mathematical analysis is full of challenges. We intend to use energy method, compactness method, harmonic analysis and geometric analysis techniques, partial regularization theory and boundary regularization techniques to study the well-posedness of the problems related to the EL model and the BE model, the singularities of solutions and large time behaviors, etc., and use operator theory and control theory to study the optimal boundary control of the boundary system corresponding to the initial boundary value problem of El model. The research contents of this project have many application backgrounds, and it is also of basic importance in the field of nonlinear partial differential equations. The results of the project will promote the research on the hydrodynamic theory of liquid crystals.