Abstract for NSF Proposal DMS-0305114 PI: Yuxi Zheng (Title: Analysis of Equations in the Physical, Material, and Life Sciences)
Yuxi Zheng proposes to study the Euler equations modeling inviscid fluids, Cahn-Hilliard and Ginzburg-Landau equations in phase-field modeling of alloys, nonlinear variational wave equations modeling liquid crystals, Schrodinger-Poisson and Vlasov-Poisson equations in plasma physics, and the protein folding problem in molecular biology. His objectives are to gain both better understanding and simplifications of complexity, which include complexity reduction for the protein folding problem, effect of solutes on the enhancement of strength of alloys, and 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 proposed to study are all fundamental for the understanding of the respective subject areas. For instance, the search for a measurement of distance between three points in the protein folding problem is to reduce drastically the huge number of local minima of the energy potential and thus bring the complexity to a comprehensible level. The issues proposed in the phase-field model of alloys is to provide the quantitative as well as qualitative foundation for manipulating the effect of solutes in strengthening the alloys. The theoretical issue regarding the limit from Schrodinger-Poisson to Vlasov-Poisson equations is a consistency issue of great importance in the overall understanding of matter and mathematical modeling. The investigation of these mathematical issues will (1) yield new understanding regarding alloys, liquid, gases, plasmas, liquid crystals, and bio-materials, which are critical for the advancement of many engineering sciences such as protein-engineering, drug designing, solid solution hardening, aerospace engineering, robot designing, energy efficient devices, etc.; (2) provide advanced training for graduate students or postdoctoral researchers; (3) enhance collaboration and cross training of faculties between mathematics, material research, physics, biochemistry, molecular biology, and other life sciences, thereby establish a foundation for training students in this broad area while promoting research.
Yuxi Zheng proposes to study some applied mathematical problems in fluid dynamics (which includes the motion of air and water), modeling of alloys, plasma physics, protein folding in molecular biology, 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 and/or defects in the 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 the protein folding problem, the equations themselves need to be mathematically simplified for a computer to do real time numerical simulation. In all the other cases, where the equations are quite simple, it is these singularities and instabilities that often spoil accurate numerical computations of the solutions. Yuxi Zheng plans to use the state of the art analytical tools to study the structures of the singular solutions. In the case of a compressible gas such as air, for example, Yuxi Zheng plans to isolate typical singularities (hurricanes, tornadoes, shocks, etc.) and investigate their individual structures. The result of the investigation will be a clear understanding of the worst possible solutions, or drastic reduction of complexity, and thereby quantify our knowledge of the physics and offer guidance in high-performance numerical computations of general solutions. The success here will influence scientific areas such as alloys, liquid, gases, plasmas, liquid crystals, and bio-materials, and provide critical knowledge for the advancement of many engineering sciences such as protein-engineering, drug designing, solid solution hardening, aerospace engineering, robot designing, energy efficient devices, etc. In addition, the success here will provide advanced training for graduate students and postdoctoral researchers and enhance collaboration and cross training of faculties between mathematics, material research, physics, biochemistry, molecular biology, and other life sciences, thereby establish a foundation for training students in this broad area while promoting research.
