Global solutions for the critical, higher-degree corotational harmonic map heat flow to $\mathbb{S}^2$

TL;DR

This paper proves the global existence and decay of certain harmonic map heat flow solutions from b2 to b2, extending classical results to higher degrees and energies using concentration-compactness and stability methods.

Contribution

It establishes global solutions for higher-degree harmonic map heat flows with energies above the threshold, extending prior results through new stability and concentration-compactness techniques.

Findings

Global existence for degree b0 b2 maps with low energy

Decay to harmonic maps for solutions with higher initial energy

Extension of classical results to higher degrees and energies

Abstract

We study m-corotational solutions to the Harmonic Map Heat Flow from $\mathbb{R}^2$ to $\mathbb{S}^2$. We first consider maps of zero topological degree, with initial energy below the threshold given by twice the energy of the harmonic map solutions. For $m \geq 2$, we establish the smooth global existence and decay of such solutions via the {\it concentration-compactness} approach of Kenig-Merle, recovering classical results of Struwe by this alternate method. The proof relies on a profile decomposition, and the energy dissipation relation. We then consider maps of degree $m$ and initial energy above the harmonic map threshold energy, but below three times this energy. For $m \geq 4$, we establish the smooth global existence of such solutions, and their decay to a harmonic map (stability), extending results of Gustafson-Nakanishi-Tsai to higher energies. The proof rests on a stability-type argument used to rule out finite-time bubbling.