On blow-up trees for the harmonic map heat flow from $B^2$ to $S^2$

TL;DR

This paper studies the harmonic map heat flow from a 2D disk to a sphere, proving the uniqueness of bubble formation in finite time and demonstrating solutions that blow up infinitely, using maximum principles.

Contribution

It establishes that only one bubble forms in finite-time blow-up for equivariant solutions and constructs solutions with infinite-time blow-up for any symmetry degree.

Findings

Finite-time blow-up involves only one bubble.

Existence of solutions blowing up in infinite time for all symmetries.

Method relies on maximum and comparison principles.

Abstract

We consider finite-time and $k$-equivariant solutions to the harmonic map heat flow from $B^2$ to $S^2$ under general time-dependent boundary data and prove that the bubble tree decomposition contains only one bubble. The method relies on the Maximum and Comparison Principle. We also exhibit solutions blowing up in infinite time for any $k \geq 1$.