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、微分几何和材料科学中的问题推动的。因此，预计该项目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，它描述了液晶的动力学，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产生的数学和物理性质。特别是，人们感兴趣的是流体流动中的奇异性和液晶取向中的拓扑缺陷的发展及其长期行为。该项目还研究了一些几何变分问题，如经典的能量最小化映射到连续映射之间的球面，调和Ricci流到曲面。一个引人入胜的数学事实是，这些都在某种程度上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研究人员进行的首席研究员培训计划的重要组成部分。作为该项目的一部分，所取得的成果通过专业期刊上的出版物以及讲座、研讨会和会议进行传播。首席研究员(PI)还组织与项目工作相关的主题的会议和导师，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里，各种研究人员做了大量的努力并取得了巨大的进展，但关于三维Ericksen-Leslie系统的整体适当弱解的存在这一根本问题仍然是一个迷人的开放问题。该项目研究了一种新的修正模型，并探索了微妙的耦合非线性结构。我们感兴趣的是新模型的适当的整体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上的有限时间爆破。该项目的一个主要焦点是缺陷及其动态。PI还研究了与调和映射的热流有关的一系列具体问题。当目标是一个球体时，这在液晶研究中是相关的，PI感兴趣的是映射的所谓松弛能量的梯度流。通过使用最小化运动方案，研究了它与广义Brakke流之间的联系。对于几何中的这种耦合方程，一个相关的问题是理解谐和Ricci流的爆破机制。当目标是非正弯曲的Alexandroff空间时，该项目通过一种改进的最小化运动方案，研究了除了满足Struwe单调性外，还满足Almgren频率的单调性的更好的弱解的存在性。后者具有重要的影响。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.