Mathematics Colloquium

I will discuss a joint work with Jacek Jendrej and Wilhelm Schlag about the two dimensional harmonic map heat flow for maps taking values in the sphere. It has been known since the 90's that solutions can exhibit bubbling along a well-chosen sequence of times -- the solution decouples into a superposition of concentrating harmonic maps and a body map accounting for the rest of the energy. We prove that every sequence of times contains a subsequence along which such bubbling occurs. This is deduced as a corollary of our main theorem, which shows that the solution approaches the family of multi-bubbles in continuous time. The proof uses the notion of "minimal collision energy" developed in the context of the soliton resolution problem for nonlinear waves.