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