Shrinkers, expanders, and the unique continuation beyond generic blowup in the heat flow for harmonic maps between spheres

TL;DR

This paper combines analytical and numerical techniques to study singularity formation and continuation in the heat flow for harmonic maps between spheres, revealing unique continuation and topological degree changes at blow-up points.

Contribution

It constructs global weak solutions through gluing asymptotically self-similar solutions, demonstrating unique continuation beyond blow-up and topological degree changes.

Findings

Constructed solutions with finite-time blow-up at poles.

Proved uniqueness of continuation beyond blow-up.

Showed the degree decreases by one at each blow-up.

Abstract

Using mixed analytical and numerical methods we investigate the development of singularities in the heat flow for corotational harmonic maps from the $d$-dimensional sphere to itself for $3\leq d\leq 6$. By gluing together shrinking and expanding asymptotically self-similar solutions we construct global weak solutions which are smooth everywhere except for a sequence of times $T_1<T_2<...<T_k<\infty$ at which there occurs the type I blow-up at one of the poles of the sphere. We show that in the generic case the continuation beyond blow-up is unique, the topological degree of the map changes by one at each blow-up time $T_i$, and eventually the solution comes to rest at the zero energy constant map.