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

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