Singular Solutions to Certain Equations in the Physical Sciences

0071858

Zheng

It is proposed to study some nonlinear partial differential equations from

fluid dynamics and liquid crystal physics. These equations are the laws of

motion of their respective physics. The turbulent nature and/or defects in

the materials make the solutions of the equations singular, unstable, and

hard to calculate. It is planned to use advanced analytical tools to study

the structures of the singular solutions. In the case of the liquid crystal

wave equation, for example, the plan is to study it as one of the simplest

examples of nonlinear generalizations of the basic linear wave equation. It

can be seen from its elegant form that there will be many applications of

the equation in the future. Singularities of its solutions have been shown to

exist recently by the principal investigator and his collaborators. It is these

singularities that block the establishment of a general theory of existence,

uniqueness, and stability of solutions. It is planned to do detailed estimates

and analysis on the singularities and improve current compactness

arguments to form an existence theory of solutions. The result of the

investigation will be a clear understanding of the worst possible solutions,

and thereby quantify our knowledge of the physics and offer guidance in

numerical computations of general solutions.

The research will involve the study of some applied mathematical

problems in the fields of fluid dynamics (which includes motion of the air

and water) and liquid crystal physics in material science. Scientists and

engineers have used mathematical equations, called partial differential

equations, to model the motions. The turbulent nature and/or defects in the

materials show up in the form of singularities and instabilities in the

solutions of the equations. It is these singularities and instabilities that

often spoil accurate numerical computations of the solutions. It is planned

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, the principal investigator 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, and thereby quantify our knowledge of the

physics and offer guidance in high-performance numerical computations

of general solutions.