Variational Analysis and Hydrodynamics of Liquid Crystals

液晶的变分分析和流体动力学

基本信息

批准号：
2101224
负责人：
Changyou Wang
金额：
$ 26.67万
依托单位：
Purdue University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101224&HistoricalAwards=false
关键词：
Variational Analysis Hydrodynamics Liquid Crystals

项目摘要

The questions that will be studied as part of this project are not only extremely challenging mathematically but also have close connections to and important applications in other fields, including complex fluid mechanics, materials science, and engineering (for example, in both the design and control of optical display devices). The rigorous analysis of certain types of solutions to the model equations for liquid crystals in either the static or the dynamic regime can predict the formation and structure of defects, allow researchers to better understand turbulence phenomena, and justify both experimental and computational studies by applied scientists. The questions will also be integrated into the training of graduate students, and the results will be disseminated through a research monograph aimed at researchers in the field.The technical side of the project consists of three parts: 1) Ericksen-Leslie system modeling the hydrodynamics of nematic liquid crystals; 2) Phase transition problems on isotropic and nematic phases and liquid crystal droplets; and 3) Fractional harmonic map heat flows between manifolds. In the first part, the principal investigator and his collaborators will investigate the (Lagrangian-averaged) Ericksen-Leslie system, which is a dissipative system strongly coupling the forced Navier-Stokes equation (with Lagrangian-average) for the underlying fluid and the evolution for the orientation director fields for liquid crystal molecules (a transported harmonic map heat flow). The aim is to establish both existence and partial regularity of global Leray-Hopf type solutions for any arbitrarily large initial data. In the second part of the project, the Gamma-convergence theory will be used to study the sharp interface limit problem of minimizers to a singularly perturbed Ericksen functional of liquid crystals with variable degrees of orientation; another goal is to establish Gamma-limit results for their corresponding gradient flows, that is, the mean curvature flow of interfaces coupled with the generalized harmonic map heat flow for the director fields in the bulk regions. The existence and uniqueness of minimal configurations of energy functionals describing liquid crystal droplets will also be investigated. The third part of the project is concerned with the study of the gradient flow of nonlocal energy functionals of maps between manifolds.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.

作为该项目的一部分将研究的问题不仅在数学上极具挑战性，而且与其他领域有密切的联系和重要的应用，包括复杂的流体力学、材料科学和工程（例如，在设计和控制方面）光学显示设备）。对静态或动态状态下液晶模型方程的某些类型的解进行严格分析可以预测缺陷的形成和结构，使研究人员能够更好地理解湍流现象，并证明应用科学家的实验和计算研究的合理性。这些问题也将被纳入研究生的培训中，其结果将通过针对该领域研究人员的研究专着进行传播。该项目的技术方面由三个部分组成：1）Ericksen-Leslie 系统对流体动力学进行建模向列液晶； 2）各向同性相、向列相和液晶滴的相变问题； 3) 分数调和图表示流形之间的热流。在第一部分中，首席研究员和他的合作者将研究（拉格朗日平均）Ericksen-Leslie 系统，这是一个耗散系统，与底层流体和演化的受迫纳维斯托克斯方程（与拉格朗日平均）强耦合用于液晶分子的取向导向场（传输谐波图热流）。目的是为任意大的初始数据建立全局 Leray-Hopf 型解的存在性和部分正则性。在该项目的第二部分中，伽玛收敛理论将用于研究具有不同取向度的液晶的奇扰动埃里克森泛函的最小化器的锐界面极限问题；另一个目标是为其相应的梯度流建立伽玛极限结果，即界面的平均曲率流与体区域中的导向场的广义谐波映射热流相结合。还将研究描述液晶液滴的能量泛函最小构型的存在性和唯一性。该项目的第三部分涉及流形之间图的非局部能量泛函的梯度流的研究。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。