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.

在该项目中要解决的问题不仅在数学上是非常挑战的，而且还与其他领域（如流体力学和复杂烟道，应用物理和材料科学以及控制工程）具有牢固的联系和深刻的应用。对某些解决方案的存在和平滑度的严格分析可以预测奇异性的形成，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进行的计算和实验研究是合理的。 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 crystals Droplets and the isotropic-nematic phase transitions in liquid crystals, and (iii) the l内分问题。第一部分介绍了埃里克森 - 莱斯利系统对列液晶的流体动力学进行建模，这是一种强烈的非线性耦合系统，在不可压缩的Navier-Stokes方程之间是基础液体速度场的不可压缩的Navier-Stokes方程与谐波液体液晶的谐波热量流的运输热量流量。目的是为任意大的初始数据建立Leray-Hopf型弱解决方案的存在和部分规律性。第二部分通过采用埃里克森（Ericksen）的单位夜间夜间液体晶体的可变程度的方向模型来研究液晶液滴中可能的最佳构型的现有和分类。第三部分是研究Aronsson方程的唯一性或L-内度功能的绝对最小化，涉及具有空间依赖性的Hamiltonian功能，对一般Aronsson方程的粘度解决方案的规律性以及在任何维度中无限谐波功能的Liouville属性。该奖项反映了NSF的法定任务，并通过使用基金会的知识分子优点和更广泛的影响审查标准来评估被认为是宝贵的支持。