Analysis of Liquid Crystal and Ideal Gas Equations

ZhengDMS-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，如液晶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异性或冲击。 这些方法包括硬，软，渐近分析，数值计算和数学建模技术。 在流体专题中，研究者探讨了对称性在描述多维欧拉方程激波反射问题解的结构中的作用。 这与冯·诺依曼悖论有关。 液晶的问题是提供一个定量和定性的基础，操纵电子器件中的缺陷的影响。 这些理论问题的研究（1）对流体和液晶产生了新的认识，这对许多工程科学的发展至关重要，如航空航天工程、机器人设计和节能装置;（2）为研究生或博士后研究人员提供了高级培训;（3）加强数学、材料科学和物理学系之间的合作和交叉培训，从而为培养这些广泛领域的学生奠定基础。 研究人员研究流体动力学（包括空气和水的运动）和材料科学中液晶物理的一些应用数学问题。 科学家和工程师们已经用某些数学方程，称为偏微分方程，来模拟系统的运动或变化。 流体中的湍流和材料中的缺陷以奇异性和不稳定性的形式出现在模拟系统行为的方程的解中。 即使在方程相当简单的情况下，也正是这些奇异性和不稳定性常常破坏解的精确数值计算。 研究人员使用最先进的分析工具来研究这些溶液的结构。 在可压缩气体的情况下，例如空气，他隔离了典型的奇点（飓风，龙卷风，冲击等）。and investigates调查their其individual个人structures结构. 研究的结果是对最坏的可能解决方案或解决方案的结构有了更清楚的理解。 这样的结果量化了我们的物理知识，并提供指导，在高性能的数值计算的一般解决方案。 这里的结果影响科学领域，如天气预报，流体动力学，和材料科学，并提供关键知识的许多应用领域，如航空航天工程，机器人设计和节能设备的进步。 此外，该项目还为研究生和博士后研究人员提供高级培训，并加强数学，材料科学和物理学之间的合作和交叉培训。

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.