Analysis of Liquid Crystal and Ideal Gas Equations

Zheng

DMS-0908207

The investigator studies the Euler equations modeling

inviscid fluids, and nonlinear variational wave equations

modeling liquid crystals. His objective is to gain better

understanding of complicated phenomena, such as defects in liquid

crystals and shocks in fluid flows, that show themselves as

singularities or shocks in the solutions of the equations. The

methods include hard, soft, and asymptotic analysis, numerical

computation, and techniques of mathematical modeling. In the

fluids topic the investigator explores the role of symmetry in

describing the structure of solutions to shock reflection

problems for the multi-dimensional Euler equations. This bears

on the von Neumann paradox. The issue in the nematic liquid

crystals topic is to provide a quantitative as well as

qualitative foundation for manipulating the effect of defects in

electronic devices. The investigation of these mathematical

issues (1) yields new understanding regarding fluids and liquid

crystals, which are critical for the advancement of many

engineering sciences such as aerospace engineering, robot

designing, and energy efficient devices; (2) provides advanced

training for graduate students or postdoctoral researchers; (3)

enhances collaboration and cross training of faculties between

mathematics, materials science, and physics, thereby establishing

a foundation for training students in these broad areas.

The investigator studies some applied mathematical problems

in fluid dynamics (which includes the motion of air and water)

and in liquid crystal physics in materials science. Scientists

and engineers have used certain mathematical equations, called

partial differential equations, to model motion or change in a

system. Turbulence in fluids and defects in materials show up in

the form of singularities and instabilities in the solutions of

the equations that model the behavior of the systems. Even 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

state of the art analytical tools to study the structures of the

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 solutions, or of the

structure of solutions. Such results quantify our knowledge of

the physics and offer guidance in high-performance numerical

computations of general solutions. Results here influence

scientific areas such as weather forecasting, fluid dynamics, and

materials science, and provide critical knowledge for the

advancement of many application areas such as aerospace

engineering, robot design, and energy efficient devices. In

addition, the project provides advanced training for graduate

students and postdoctoral researchers and enhances collaboration

and cross training of faculties between mathematics, materials

science, and physics.