Hydrodynamics of Liquid Crystals and Heat Flow of Harmonic Maps

液晶的流体动力学和谐波图的热流

• 批准号: 2247773
• 负责人: Fang-Hua Lin
• 金额: $ 58.32万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-06-01 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2247773&HistoricalAwards=false
• 关键词: Hydrodynamics; Liquid; Crystals; Heat; Flow

A significant part of this project is motivated by questions in physics, differential geometry, and material science. As such, the project is expected to have a positive impact on interdisciplinary research and applications to other fields of science. The list of concrete research projects includes the Ericksen-Leslie system, describing the dynamics of liquid crystals, and its mathematical and physical properties due to the nonlinear coupling between the Navier-Stokes fluid dynamics and the microscopic molecular orientation evolution. In particular, one is interested in the developments of singularities in both fluid flows and topological defects in the liquid crystal orientation as well as its long-time behavior. The project also investigates some geometric variational problems such as the classical problem of energy minimizing maps into spheres among continuous maps, the harmonic Ricci flows into surfaces. A fascinating mathematical fact is that these are somehow all connected with the research on liquid crystal dynamics. The project is an important and integral part of the principal investigator's training program for graduate students and postdoctoral researchers through topics courses, special lectures and thesis projects. The results obtained as part of the project are disseminated through publications in professional journals as well as through the lectures, seminars, and conferences. The principal investigator (PI) also organizes conferences and mentors graduate and undergraduate researchers on topics related to the work of the project.This project is aimed at solving several challenging problems from the theory of liquid crystals. Despite numerous efforts by various researchers and tremendous progress over the past three decades, the fundamental problem concerning the global existence of suitable weak solutions of the Ericksen-Leslie system in three dimensions remains a fascinating open problem. The project studies a new modifed model and explores the subtle underlying coupled nonlinear structure. Of related interest are the long time asymptotics and the partial regularity of suitable global weak solutions and the finite-time blow up in both two and three dimensions of initially smooth solutions for this new model. A main focus of the project is the defect and its dynamics. The PI also studies a list of concrete problems related to the heat flow of harmonic maps. When the target is a sphere, which is relevant in the study of liquid crystals, the PI is interested in the gradient flow of the so-called relaxed energy of maps. By using a minimizing movement scheme, one studies its connection to the generalized Brakke flow. A related problem for such coupled equations in geometry is to understand the blow-up mechanism of the harmonic Ricci flow. When the target is a non-positively curved Alexandroff space, the project studies, through a refined minimizing movement scheme, the existence of better weak solutions that satisfy, in addition to the Struwe monotonicity property, the monotonicity of Almgren's frequency. The latter has important consequences.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.

这个项目的一个重要部分是由物理、微分几何和材料科学的问题所激发的。因此，该项目预计将对跨学科研究和其他科学领域的应用产生积极影响。具体的研究项目包括描述液晶动力学的Ericksen-Leslie系统，以及由于Navier-Stokes流体动力学与微观分子取向演化之间的非线性耦合而产生的液晶的数学和物理性质。特别是，人们对流体流动中的奇点和液晶取向中的拓扑缺陷及其长期行为的发展感兴趣。本课题还研究了一些几何变分问题，如连续映射中能量最小化映射成球的经典问题、调和里奇流进入曲面的经典问题。一个令人着迷的数学事实是，这些都与液晶动力学的研究有关。该项目是通过专题课程、专题讲座和论文项目对研究生和博士后进行培养的重要组成部分。作为该项目的一部分，获得的成果通过专业期刊的出版物以及讲座、研讨会和会议进行传播。首席研究员（PI）还组织会议，并就与项目工作相关的主题指导研究生和本科生研究人员。该项目旨在解决液晶理论中的几个具有挑战性的问题。尽管在过去的三十年中，许多研究者做出了大量的努力并取得了巨大的进展，但关于三维Ericksen-Leslie系统的合适弱解的全局存在性的基本问题仍然是一个引人入胜的开放性问题。该项目研究了一种新的修正模型，并探索了微妙的潜在耦合非线性结构。值得关注的是该模型的全局弱解的长时间渐近性和部分正则性，以及初始光滑解在二维和三维的有限时间爆破性。项目的主要焦点是缺陷及其动态。PI还研究了一系列与谐波图热流有关的具体问题。当目标是一个球体时，这与液晶的研究有关，PI感兴趣的是所谓的地图松弛能量的梯度流动。通过最小化运动方案，研究了其与广义制动流的关系。这类耦合方程在几何上的一个相关问题是理解调和里奇流的爆破机理。当目标为非正弯曲Alexandroff空间时，通过一种改进的最小化运动方案，研究了除Struwe单调性外，还满足Almgren频率单调性的较弱解的存在性。后者具有重要的影响。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。