Continuous in time bubble decomposition for the harmonic map heat flow

We consider the harmonic map heat flow for maps R²→S². It is known that solutions to the initial value problem exhibit bubbling along a well-chosen sequence of times. We prove that every sequence of times admits a subsequence along which bubbling occurs. This is deduced as a corollary of our main theorem, which shows that the solution approaches the family of multi-bubble configurations in continuous time.