REGULARITY AND UNIQUENESS OF SOME GEOMETRIC HEAT FLOWS AND IT\u27S APPLICATIONS

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\u27s condition, the regularity and uniqueness is also established. As an application, the hydrodynamic flow of nematic liquid crystals in Serrin\u27s 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