Hydrodynamics of Liquid Crystals and Heat Flow of Harmonic Maps

这个项目的一个重要部分是由物理学，微分几何和材料科学的问题。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Ericksen-Leslie系统，描述液晶的动力学，以及由于Navier-Stokes流体动力学和微观分子取向演化之间的非线性耦合而导致的数学和物理性质。特别是，人们对流体流动中的奇异性和液晶取向中的拓扑缺陷及其长期行为的发展感兴趣。本项目还研究了一些几何变分问题，如经典的能量极小化映射到球面的问题，调和Ricci流到曲面的问题。一个有趣的数学事实是，这些都与液晶动力学的研究有关。该项目是主要研究者通过主题课程，专题讲座和论文项目为研究生和博士后研究人员提供培训计划的重要组成部分。作为该项目的一部分，所取得的成果通过在专业期刊上发表文章以及通过讲座、研讨会和会议进行传播。主要研究者（PI）还组织会议，并指导研究生和本科生研究人员有关项目工作的主题。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里，许多研究者付出了大量的努力，取得了巨大的进展，但三维Ericksen-Leslie系统的整体弱解的存在性问题仍然是一个令人着迷的开放问题。本计画研究一种新的修正模型，并探讨其中微妙的耦合非线性结构。相关的兴趣是长时间的渐近性和适当的整体弱解的部分正则性和有限时间爆破在二维和三维的初始光滑的解决方案，这个新的模型。该项目的一个主要焦点是缺陷及其动态。PI还研究了一系列与调和映射热流相关的具体问题。当目标是一个球，这是相关的液晶的研究，PI感兴趣的梯度流的所谓的弛豫能量的地图。通过使用一个最小化的运动计划，研究其连接到广义Brakke流。在几何学中，这类耦合方程的一个相关问题是理解调和Ricci流的爆破机制。当目标是一个非正弯曲的Alexandroff空间，该项目的研究，通过一个细化的最小化运动计划，存在更好的弱解，满足，除了Struwe单调性属性，单调的Almgren的频率。该奖项反映了NSF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。

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.