Analysis of Equations in the Applied Sciences

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ZhengDMS-0603859 The investigator studies the Euler equations modelinginviscid fluids, Cahn-Hilliard and Ginzburg-Landau equations inphase-field modeling of alloys, and nonlinear variational waveequations modeling liquid crystals. His objectives are to gainboth better understanding and simplifications of complexity(which includes complexity reduction for multi-dimensional shockreflection problems), of the effect of solutes on the enhancementof strength of alloys, and of the mechanism of singularityformation in air, water, and liquid crystals. The methodsinclude hard, soft, and asymptotic analysis, numericalcomputation, and techniques of mathematical modeling. Themathematical issues are fundamental for the understanding of therespective subject areas. For instance, a model is sought inmulti-dimensional shock reflection problems to reduce possibleturbulence and thus bring the complexity to a comprehensiblelevel and settle the von Neumann paradoxes (e.g., the paradoxicalboundary between regular shock reflection and Mach reflection). The issue in the phase-field model of alloys is to provide aquantitative as well as qualitative foundation for manipulatingsolutes to strengthen the alloys. Study of these mathematicalissues (1) yields new understanding regarding alloys, liquids,gases, and liquid crystals, which are critical for theadvancement of many engineering sciences such as solid solutionhardening, aerospace engineering, robot designing, and energyefficient devices; (2) provides advanced training for graduatestudents or postdoctoral researchers; (3) enhances collaborationand cross-training between mathematics, material research, andphysics, thereby establishing a foundation for training studentsin this broad area. The investigator studies some applied mathematical problemsin fluid dynamics (which includes the motion of air and water),modeling of alloys, and liquid crystal physics in materialscience. Scientists and engineers have used mathematicalequations, called partial differential equations, to modelmotions or evolution. The turbulent nature of fluid flows,defects in materials, and the complexity of life show up in theform of singularities and instabilities in the solutions of theequations or in the complexity of the equations themselves. Incases where the equations are quite simple, it is thesesingularities and instabilities that often spoil accuratenumerical computations of the solutions. The investigator usesanalytical mathematical tools to study the structures of thesingular solutions. In the case of a compressible gas such asair, for example, he isolates typical singularities (hurricanes,tornadoes, shocks, etc.) and investigates their individualstructures. The result of the investigation is a clearerunderstanding of the worst possible -- most singular --solutions, or a drastic reduction of complexity, which quantifiesour knowledge of the physics and offers guidance inhigh-performance numerical computations of general solutions. Such results influence scientific areas such as weatherforecasting, alloys, liquid, gases, and liquid crystals, andprovide critical knowledge for the advancement of manyengineering sciences such as solid solution hardening, aerospaceengineering, robot design, and energy-efficient devices. Inaddition, the project provides opportunities for advancedtraining for graduate students and postdoctoral researchers andenhances collaboration and cross-training between mathematics,material research, and physics, creating a foundation fortraining students in this broad area.