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.

本课题旨在对一类复杂流体进行深入的理论研究，这些复杂流体既包括在显示器件中广泛应用的液晶，也包括在现代医学和生命科学中极为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态现象，以及模式和奇点的形成。对这些复杂流体的建模、分析和模拟在理论上是非常重要和具有挑战性的。它需要新的思想和方法，因此它将提高我们的知识，这也将适用于许多其他科学问题。更具体地说，该建议由两部分组成。第一部分是研究描述液晶流体力学的偏微分方程和相关的复杂流体模型。本部分的研究重点是研究Ericksen-Leslie理论中液晶流动的合适弱解的整体存在性；不可压缩粘弹性流体的Oldroyd b模型解的整体存在性；无粘不可压缩磁流体动力系统，一般来说，流体与其他几何物体的耦合非线性动力学。第二部分研究了一类椭圆型特征值极值问题。这类问题也出现在材料科学和凝聚态物理的优化设计、模式形成和其他应用中。PI计划研究的偏微分方程（PDE）涉及描述输运、相场、映射或几何物体演化的方程与Navier Stokes方程之间的非线性耦合。它们可以是抛物线型和双曲型，并具有奇点或多重尺度。特征值和特征函数的变分问题是经典和基本的。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PDE分析和变分学的研究或程序的新方向。该研究活动是学院本科生、研究生和博士后培养计划的重要组成部分。