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Mathematical Analysis of Nematic Liquid Crystals and L-infinity Variational Problems

向列液晶与L-无穷变分问题的数学分析

基本信息

批准号：
1764417
负责人：
Changyou Wang
金额：
$ 24万
依托单位：
Purdue University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1764417&HistoricalAwards=false
关键词：
Mathematical Analysis Nematic Liquid Crystals

项目摘要

The problems to be solved in this project are not only very challenging mathematically but also have strong connections and profound applications to other fields such as fluid mechanics and complex fluids, applied physics and material sciences, and control engineering. Rigorous analysis of both the existence and the smoothness of certain kinds solutions to our models can predict the formation of singularities, allow researchers to gain insights into the turbulence phenomena, and justify both computational and experimental studies made by applied scientists and engineers. The proposed problems in the project will also serve as tools to train graduate students, and constitute as main parts of future research monographs aimed at both advanced graduate students and researchers.The technical side of this project is to study analytic issues in the three parts: (i) the hydrodynamic flow of nematic liquid crystals, (ii) variational problems on both liquid crystal droplets and the isotropic-nematic phase transitions in liquid crystals, and (iii) the L-infinity variational problems. The first part deals with the Ericksen-Leslie system modeling the hydrodynamics of nematic liquid crystals, which is a strongly nonlinear-coupled system between the incompressible Navier-Stokes equation of the underlying fluid velocity field and the transported heat flow of harmonic maps for the orientation director field of the nematic liquid crystal molecules. The objective is to establish existence and partial regularity for Leray-Hopf type weak solutions in dimension three for arbitrary large initial data. The second part investigates both existence and classification of possible optimal configuration in the liquid crystal droplets and the formation of sharp interface between the isotropic and nematic phases by employing the Ericksen's model of variable degree of orientations for uniaxial nematic liquid crystals. The third part is to study the uniqueness of Aronsson's equations or absolute minimizers of L-infinity functionals that involve Hamiltonian functions with spatial dependence, the regularity of viscosity solutions to general Aronsson's equations, and the Liouville property of infinity harmonic functions in any dimension. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

本项目所要解决的问题不仅在数学上具有很大的挑战性，而且与流体力学和复杂流体、应用物理和材料科学以及控制工程等领域有着密切的联系和深远的应用，对我们模型的某些解的存在性和光滑性的严格分析可以预测奇点的形成，使研究人员能够深入了解湍流现象，并证明应用科学家和工程师所做的计算和实验研究。该项目中提出的问题也将作为培养研究生的工具，并构成未来研究专着的主要部分，针对高级研究生和研究人员。本项目的技术方面是研究分析问题的三个部分：（i）双折射液晶的流体力学流动，（ii）液晶液滴和液晶各向同性-双折射相变的变分问题，（iii）L-无穷大变分问题。第一部分研究了Ericksen-Leslie系统对液晶流体力学的模拟，该系统是一个由不可压缩Navier-Stokes方程和液晶分子取向指向矢场的谐波映射热流组成的强非线性耦合系统。目的是建立任意大初值下三维Leray-Hopf型弱解的存在性和部分正则性。第二部分采用Ericksen的单轴双相液晶变取向度模型，研究了液晶液滴中可能的最佳构型的存在和分类，以及各向同性相和双相之间尖锐界面的形成。第三部分研究了具有空间依赖性的Hamilton函数的Aronsson方程或L-无穷泛函的绝对极小元的唯一性，一般Aronsson方程粘性解的正则性，和刘维尔性质的无穷谐波函数在任何方面。这个奖项反映了NSF的法定使命，并已被认为是值得支持，通过评估使用该基金会的知识优点和更广泛的影响审查标准。