  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.
