Bubbling of the heat flows for harmonic maps from surfaces

In this article we prove that any Palais-Smale sequence of the energy functional on surfaces with uniformly L²-bounded tension fields converges pointwise, by taking a subsequence if necessary, to a map from connected (possibly singular) surfaces, which consist of the original surfaces and finitely many bubble trees. We therefore get the corresponding results about how the solutions of heat flow for harmonic maps from surfaces form singularities at infinite time.