Hydrodynamics of Liquid Crystals and Extremum Problems for Eigenvalues

液晶流体动力学和特征值极值问题

• 批准号: 1501000
• 负责人: Fang-Hua Lin
• 金额: $ 62.5万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-06-01 至 2020-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1501000&HistoricalAwards=false
• 关键词: Hydrodynamics; Liquid; Crystals; Extremum; Problems

This proposal is for a deep theoretical study on a class of complex fluids including both the liquid crystals which have been widely used in display devices and charged biological fluids which are extremely important in modern medicine and life sciences. Of particular interests are their intriguing and complex dynamical phenomena as well as formations of patterns and singularities. The modeling, analysis and simulations involved in understanding these complex fluids are theoretically very important and challenging. It needs new ideas and methods, and hence it would advance our knowledge which would be applicable to many other scientific problems as well.More specifically, the proposal consists of two parts. The first part is to study partial differential equations that describe the hydrodynamics of liquid crystals and related complex fluid models. The main focus of this part of the research will be to study global existence of suitable weak solutions of liquid crystal flows in the Ericksen-Leslie theory; the global existence of solutions of Oldroyd B-model of incompressible visco-elastic fluids; the inviscid incompressible magneto-hydrodynamic system and, in general, coupled nonlinear dynamics of fluids with other geometric objects. The second part is to study a large class of extremum problems of elliptic eigenvalues. Such problems also arise in optimal designs, pattern formations and other applications in material sciences and condense matter physics. The partial differential equations (PDE) that the PI plans to study involve nonlinear couplings between equations that describe transport, phase-field, mapping or geometric object's evolutions and that of Navier Stokes equations. They may be of both parabolic and hyperbolic nature and possess singularities or multiple scales. The variational problems for eigenvalues and eigenfunctions that linked with underlying domains are classical and fundamental. These are fascinating and challenging problems that require new ideas and methods, which could lead to new directions of research or programs in the analysis of PDE and calculus of variations. The proposed research activity is an important and integral part of the PI's training program of under-graduate, graduate, and post-doctoral students.

该提案旨在对一类复杂流体进行深入的理论研究，包括广泛用于显示设备的液晶和在现代医学和生命科学中极其重要的带电生物流体。特别令人感兴趣的是它们有趣而复杂的动力学现象以及模式和奇点的形成。理解这些复杂流体所涉及的建模、分析和模拟在理论上非常重要且具有挑战性。它需要新的想法和方法，因此它将提高我们的知识，这也适用于许多其他科学问题。更具体地说，该提案由两部分组成。第一部分是研究描述液晶流体动力学的偏微分方程和相关的复杂流体模型。这部分研究的重点将是研究Ericksen-Leslie理论中液晶流动合适的弱解的全局存在性；不可压缩粘弹性流体 Oldroyd B 模型解的全局存在性；无粘不可压缩磁流体动力系统，并且通常将流体的非线性动力学与其他几何对象耦合。第二部分是研究一大类椭圆特征值极值问题。此类问题也出现在材料科学和凝聚态物理的优化设计、图案形成和其他应用中。 PI 计划研究的偏微分方程 (PDE) 涉及描述输运、相场、映射或几何对象演化的方程与纳维斯托克斯方程之间的非线性耦合。它们可能具有抛物线和双曲线性质，并且具有奇点或多尺度。与基础域相关的特征值和特征函数的变分问题是经典且基本的。这些都是令人着迷且具有挑战性的问题，需要新的想法和方法，这可能会导致偏微分方程和变分法分析的新研究方向或项目。拟议的研究活动是PI本科生、研究生和博士后培养计划的重要组成部分。