Analysis of Liquid Crystal and Ideal Gas Equations

点击翻译按钮获取中文摘要

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.