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.

该项目要解决的问题不仅在数学上非常具有挑战性，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其他领域有很强的联系和深刻的应用。我们模型的某些解决方案的平滑性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的合理性。该项目中提出的问题也将得到证实。作为培养研究生的工具，并构成针对高级研究生和研究人员的未来研究专着的主要部分。该项目的技术方面是研究三个部分的分析问题：（i）向列相的流体动力流动液晶，(ii) 液晶液滴和液晶中各向同性向列相变的变分问题，以及 (iii) L-无穷变分问题。 Ericksen-Leslie 系统对向列液晶的流体动力学进行建模，该系统是基础流体速度场的不可压缩纳维-斯托克斯方程与向列液体取向导向场的调和图的传输热流之间的强非线性耦合系统目标是针对任意大的初始数据建立第三维 Leray-Hopf 型弱解的存在性和部分正则性。第三部分是研究阿伦森方程或绝对方程的唯一性，通过采用单轴向列液晶的可变取向度埃里克森模型，研究液晶滴中可能的最佳构型以及各向同性相和向列相之间清晰界面的形成。 L-无穷泛函的最小化器，涉及具有空间依赖性的哈密顿函数，一般阿伦森方程的粘度解的规律性，以及任意维度的无穷调和函数的刘维尔性质。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。