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

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.