向列型液晶动力学方程若干数学问题的研究

The nematic liquid crystal dynamics equation is an important mathematical model which describles the physical phenomenon of liquid crystal and is a typical system in complex fluid studies. Strong nonlinearity of liquid crystal model makes it become a difficulty in the field of partial differential equations. In particular, there are many unresolved mathematical problems in the hydrodynamic theories of liquid crystal. It is an important scientific significance for deep insight into liquid crystal in investigating related mathematical problems. Employing the analysis tools such as harmonic analysis and micro-local analysis in the project, we will devote to considering the several mathematical problems of the nematic liquid crystal dynamics equations:(1)the asymptotic decay of strong solutions for the 3D incompressible nematic liquid crystal dynamics equation in critical Besov spaces; (2)the golbal well-posedness of strong solutions for the 3D compressible nematic liquid crystal dynamics equation with large energy and large ocscillation for the density; (3)the zero Mach number limit for the 3D compressible nematic liquid crystal dynamics equation in critical Besov spaces. We hope that we can establish some new mathematical theories of the liquid crystal dynamics equations and reveal the kinetics laws of liquid crystal flows in order to provide the necessary support for the better applications of the liquid crystal technology by the implementation of this project .

向列型液晶动力学方程是刻画液晶物理现象的一类重要数学模型，是复杂流体研究的典型体系。液晶模型的强非线性性使它成为偏微分方程研究领域的难点，尤其是在液晶的动力学理论方面仍有许多未解决的数学问题。研究这些问题对于深层次了解和认识液晶具有重要的科学意义。本项目利用调和分析和微局部分析等工具拟研究向列型液晶模型如下数学问题：(1) 研究临界Besov空间中三维不可压缩液晶动力学方程强解的渐近衰减性。(2) 研究大能量高振荡初始密度条件下三维可压缩液晶动力学方程的整体适定性。(3) 研究临界Besov空间中三维可压缩液晶动力学方程的零马赫数极限问题。希望通过本项目的实施，建立液晶动力学方程新的数学理论，揭示液晶流的动力学规律，为液晶技术更好地应用提供必要支持。

复杂流体数学问题的研究是当前偏微分方程研究领域的热点。 借助于能量方法和调和分析等工具， 本项目主要在包括液晶流体在内的几类重要流体动力学方程数学问题，涉及解的适定性，正则性和衰减性(长时间行为)等领域开展研究工作。本课题的主要内容包括：（1）维液晶动力学方程强解的适定性和渐近衰减性研究；（2）三维可压缩Hall-MHD强解的整体适定性和大时间行为研究；（3）临界Besov空间中多维非等熵完全可压缩MHD方程的小初值问题强解整体适定性的研究；（4） 临界Besov空间中多维可压缩弹性流体，Chemotaxis 模型强解的衰减性问题研究；（5） 三维不可压缩MHD模型的爆破准则的刻画；（6）多孔介质模型在临界Besov空间的局部适定性研究。在本项目的支持下，课题组已发表SCIE检索论文9篇，参加学术会议15次。徐夫义教授指导6名硕士研究生。