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.

将作为该项目的一部分进行研究的问题不仅在数学上是极具挑战性的，而且还与其他领域（包括复杂的流体力学，材料科学和工程学）的联系和重要应用（例如，在光学显示设备的设计和控制中）。对静态或动态状态中液晶模型方程的某些类型的解决方案的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，并通过应用科学家证明实验性和计算研究是合理的。这些问题还将集成到研究生的培训中，结果将通过针对该领域的研究人员的研究专着进行传播。该项目的技术方面包括三个部分：1）Ericksen-Leslie系统对列液晶的流体动力学建模； 2）在各向同性和列中阶段以及液晶液滴上的相变问题； 3）分数谐波地图在歧管之间的热流。在第一部分中，主要研究者及其合作者将调查（拉格朗日平均）Ericksen-LeSlie系统，这是一个耗散系统，与强制性的Navier-Stokes方程（具有Lagrangian-Pravery）的强制性液体和液体液体的进化（用于液体晶体晶体的液体的进化）（用于液体晶体分类的进化）（用于运输的液位）（一种运输型号）（一种运输）（一种运输）（一种运输）的（一种运输）（一种flow）。目的是为任何任意大的初始数据确定全局LERAY-HOPF类型解决方案的存在和部分规律性。在该项目的第二部分中，伽马连接理论将用于研究最小化的尖锐界面限制问题，即具有变化程度的液晶液晶的奇异扰动的埃里克森功能。另一个目标是为其相应的梯度流动建立γ-限值结果，即接口的平均曲率流以及散装区域导演场的广义谐波热流量结合。还将研究描述液晶液滴的能量功能的最小构型的存在和独特性。该项目的第三部分涉及对歧管之间非本地能量功能的梯度流的研究。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响审查标准，被认为值得通过评估来获得支持。