Date Available

6-30-2013

Year of Publication

2013

Degree Name

Doctor of Philosophy (PhD)

Document Type

Doctoral Dissertation

College

Arts and Sciences

Department/School/Program

Mathematics

First Advisor

Changyou Wang

Abstract

This manuscript demonstrates the regularity and uniqueness of some geometric heat flows with critical nonlinearity.

First, under the assumption of smallness of renormalized energy, several issues of the regularity and uniqueness of heat flow of harmonic maps into a unit sphere or a compact Riemannian homogeneous manifold without boundary are established.

For a class of heat flow of harmonic maps to any compact Riemannian manifold without boundary, satisfying the Serrin's condition,

the regularity and uniqueness is also established.

As an application, the hydrodynamic flow of nematic liquid crystals in Serrin's class is proved to be regular and unique.

The natural extension of all the results to the heat flow of biharmonic maps is also presented in this manuscript.

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