Analysis of Equations in the Applied Sciences

Zheng DMS-0603859 The investigator studies the Euler equations modeling inviscid fluids, Cahn-Hilliard and Ginzburg-Landau equations in phase-field modeling of alloys, and nonlinear variational wave equations modeling liquid crystals. His objectives are to gain both better understanding and simplifications of complexity (which includes complexity reduction for multi-dimensional shock reflection problems), of the effect of solutes on the enhancement of strength of alloys, and of the mechanism of singularity formation in air, water, and liquid crystals. The methods include hard, soft, and asymptotic analysis, numerical computation, and techniques of mathematical modeling. The mathematical issues are fundamental for the understanding of the respective subject areas. For instance, a model is sought in multi-dimensional shock reflection problems to reduce possible turbulence and thus bring the complexity to a comprehensible level and settle the von Neumann paradoxes (e.g., the paradoxical boundary between regular shock reflection and Mach reflection). The issue in the phase-field model of alloys is to provide a quantitative as well as qualitative foundation for manipulating solutes to strengthen the alloys. Study of these mathematical issues (1) yields new understanding regarding alloys, liquids, gases, and liquid crystals, which are critical for the advancement of many engineering sciences such as solid solution hardening, aerospace engineering, robot designing, and energy efficient devices; (2) provides advanced training for graduate students or postdoctoral researchers; (3) enhances collaboration and cross-training between mathematics, material research, and physics, thereby establishing a foundation for training students in this broad area. The investigator studies some applied mathematical problems in fluid dynamics (which includes the motion of air and water), modeling of alloys, and liquid crystal physics in material science. Scientists and engineers have used mathematical equations, called partial differential equations, to model motions or evolution. The turbulent nature of fluid flows, defects in materials, and the complexity of life show up in the form of singularities and instabilities in the solutions of the equations or in the complexity of the equations themselves. 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 analytical mathematical tools to study the structures of the singular 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 -- most singular -- solutions, or a drastic reduction of complexity, which quantifies our knowledge of the physics and offers guidance in high-performance numerical computations of general solutions. Such results influence scientific areas such as weather forecasting, alloys, liquid, gases, and liquid crystals, and provide critical knowledge for the advancement of many engineering sciences such as solid solution hardening, aerospace engineering, robot design, and energy-efficient devices. In addition, the project provides opportunities for advanced training for graduate students and postdoctoral researchers and enhances collaboration and cross-training between mathematics, material research, and physics, creating a foundation for training students in this broad area.