NSF Award
1236959
Zheng<br/>DMS-0908207<br/><br/> The investigator studies the Euler equations modeling<br/>inviscid fluids, and nonlinear variational wave equations<br/>modeling liquid crystals. His objective is to gain better<br/>understanding of complicated phenomena, such as defects in liquid<br/>crystals and shocks in fluid flows, that show themselves as<br/>singularities or shocks in the solutions of the equations. The<br/>methods include hard, soft, and asymptotic analysis, numerical<br/>computation, and techniques of mathematical modeling. In the<br/>fluids topic the investigator explores the role of symmetry in<br/>describing the structure of solutions to shock reflection<br/>problems for the multi-dimensional Euler equations. This bears<br/>on the von Neumann paradox. The issue in the nematic liquid<br/>crystals topic is to provide a quantitative as well as<br/>qualitative foundation for manipulating the effect of defects in<br/>electronic devices. The investigation of these mathematical<br/>issues (1) yields new understanding regarding fluids and liquid<br/>crystals, which are critical for the advancement of many<br/>engineering sciences such as aerospace engineering, robot<br/>designing, and energy efficient devices; (2) provides advanced<br/>training for graduate students or postdoctoral researchers; (3)<br/>enhances collaboration and cross training of faculties between<br/>mathematics, materials science, and physics, thereby establishing<br/>a foundation for training students in these broad areas.<br/><br/> The investigator studies some applied mathematical problems<br/>in fluid dynamics (which includes the motion of air and water)<br/>and in liquid crystal physics in materials science. Scientists<br/>and engineers have used certain mathematical equations, called<br/>partial differential equations, to model motion or change in a<br/>system. Turbulence in fluids and defects in materials show up in<br/>the form of singularities and instabilities in the solutions of<br/>the equations that model the behavior of the systems. Even in<br/>cases where the equations are quite simple, it is these<br/>singularities and instabilities that often spoil accurate<br/>numerical computations of the solutions. The investigator uses<br/>state of the art analytical tools to study the structures of the<br/>solutions. In the case of a compressible gas such as air, for<br/>example, he isolates typical singularities (hurricanes,<br/>tornadoes, shocks, etc.) and investigates their individual<br/>structures. The result of the investigation is a clearer<br/>understanding of the worst possible solutions, or of the<br/>structure of solutions. Such results quantify our knowledge of<br/>the physics and offer guidance in high-performance numerical<br/>computations of general solutions. Results here influence<br/>scientific areas such as weather forecasting, fluid dynamics, and<br/>materials science, and provide critical knowledge for the<br/>advancement of many application areas such as aerospace<br/>engineering, robot design, and energy efficient devices. In<br/>addition, the project provides advanced training for graduate<br/>students and postdoctoral researchers and enhances collaboration<br/>and cross training of faculties between mathematics, materials<br/>science, and physics.
