Bubble decomposition for the harmonic map heat flow in the equivariant case

TL;DR

This paper proves the unique and continuous bubble decomposition for the harmonic map heat flow from the plane to the sphere under equivariant symmetry, advancing understanding of solution behavior and energy concentration.

Contribution

It establishes the uniqueness and continuity of bubble decomposition in the harmonic map heat flow, introducing the concept of collision intervals inspired by soliton resolution research.

Findings

Bubble decomposition is unique for equivariant harmonic map heat flow.

Decomposition occurs continuously in time.

Introduction of collision intervals concept.

Abstract

We consider the harmonic map heat flow for maps from the plane taking values in the sphere, under equivariant symmetry. It is known that solutions to the initial value problem can exhibit bubbling along a sequence of times -- the solution decouples into a superposition of harmonic maps concentrating at different scales and a body map that accounts for the rest of the energy. We prove that this bubble decomposition is unique and occurs continuously in time. The main new ingredient in the proof is the notion of a collision interval motivated by the authors' recent work on the soliton resolution problem for equivariant wave maps.