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.