Analysis of Equations in the Physical, Material, and Life Sciences

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.