Analysis of Liquid Crystal and Ideal Gas Equations

液晶和理想气体方程的分析

• 批准号: 1236959
• 负责人: Yuxi Zheng
• 金额: $ 9.43万
• 依托单位: Yeshiva University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2012-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1236959&HistoricalAwards=false
• 关键词: Analysis; Liquid; Crystal; Ideal; Gas

ZhengDMS-0908207 The investigator studies the Euler equations modelinginviscid fluids, and nonlinear variational wave equationsmodeling liquid crystals. His objective is to gain betterunderstanding of complicated phenomena, such as defects in liquidcrystals and shocks in fluid flows, that show themselves assingularities or shocks in the solutions of the equations. Themethods include hard, soft, and asymptotic analysis, numericalcomputation, and techniques of mathematical modeling. In thefluids topic the investigator explores the role of symmetry indescribing the structure of solutions to shock reflectionproblems for the multi-dimensional Euler equations. This bearson the von Neumann paradox. The issue in the nematic liquidcrystals topic is to provide a quantitative as well asqualitative foundation for manipulating the effect of defects inelectronic devices. The investigation of these mathematicalissues (1) yields new understanding regarding fluids and liquidcrystals, which are critical for the advancement of manyengineering sciences such as aerospace engineering, robotdesigning, and energy efficient devices; (2) provides advancedtraining for graduate students or postdoctoral researchers; (3)enhances collaboration and cross training of faculties betweenmathematics, materials science, and physics, thereby establishinga foundation for training students in these broad areas. The investigator studies some applied mathematical problemsin fluid dynamics (which includes the motion of air and water)and in liquid crystal physics in materials science. Scientistsand engineers have used certain mathematical equations, calledpartial differential equations, to model motion or change in asystem. Turbulence in fluids and defects in materials show up inthe form of singularities and instabilities in the solutions ofthe equations that model the behavior of the systems. Even incases where the equations are quite simple, it is thesesingularities and instabilities that often spoil accuratenumerical computations of the solutions. The investigator usesstate of the art analytical tools to study the structures of thesolutions. In the case of a compressible gas such as air, forexample, he isolates typical singularities (hurricanes,tornadoes, shocks, etc.) and investigates their individualstructures. The result of the investigation is a clearerunderstanding of the worst possible solutions, or of thestructure of solutions. Such results quantify our knowledge ofthe physics and offer guidance in high-performance numericalcomputations of general solutions. Results here influencescientific areas such as weather forecasting, fluid dynamics, andmaterials science, and provide critical knowledge for theadvancement of many application areas such as aerospaceengineering, robot design, and energy efficient devices. Inaddition, the project provides advanced training for graduatestudents and postdoctoral researchers and enhances collaborationand cross training of faculties between mathematics, materialsscience, and physics.

郑DMS-0908207研究了模拟无粘流体的欧拉方程和模拟液晶的非线性变分波动方程。他的目标是更好地理解复杂的现象，如液体晶体中的缺陷和流体流动中的冲击，这些现象在方程的解中表现出奇性或冲击。方法包括硬分析、软分析、渐近分析、数值计算和数学建模技术。在流体主题中，研究人员探索了对称性在描述多维欧拉方程激波反射问题的解的结构中的作用。这是冯·诺伊曼悖论的写照。向列型液晶的主题是为控制电子器件中缺陷的影响提供一个定量和定性的基础。对这些数学问题的研究：(1)对流体和液体晶体有了新的认识，它们对许多工程科学的发展至关重要，如航空航天工程、机器人设计和节能设备；(2)为研究生或博士后研究人员提供高级培训；(3)加强数学、材料科学和物理学之间的合作和交叉培训，从而为在这些广泛领域培养学生奠定基础。研究人员研究了流体力学(包括空气和水的运动)和材料科学中的液晶物理中的一些应用数学问题。科学家和工程师使用某些数学方程，称为偏微分方程式，来模拟系统中的运动或变化。流体中的湍流和材料中的缺陷在模拟系统行为的方程的解中以奇点和不稳定性的形式表现出来。即使在方程非常简单的情况下，正是这些奇性和不稳定性经常破坏解的精确数值计算。调查者使用最先进的分析工具来研究溶液的结构。在可压缩气体的情况下，例如，他分离出典型的奇点(飓风、龙卷风、冲击波等)。并研究了它们的个体结构。调查的结果是对最糟糕的可能解决方案或解决方案的结构有了更清晰的理解。这样的结果量化了我们的物理知识，并为一般解的高性能数值计算提供了指导。这里的成果影响到天气预报、流体力学和材料科学等科学领域，并为许多应用领域的进步提供关键知识，如航空航天工程、机器人设计和节能设备。此外，该项目为研究生和博士后研究人员提供高级培训，并加强数学、材料科学和物理之间的合作和交叉培训。