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We study m-corotational solutions to the Harmonic Map Heat Flow from$\mathbb{R}^2$ to $\mathbb{S}^2$. We first consider maps of zero topologicaldegree, with initial energy below the threshold given by twice the energy ofthe harmonic map solutions. For $m \geq 2$, we establish the smooth globalexistence and decay of such solutions via the {\it concentration-compactness}approach of Kenig-Merle, recovering classical results of Struwe by thisalternate method. The proof relies on a profile decomposition, and the energydissipation relation. We then consider maps of degree $m$ and initial energyabove the harmonic map threshold energy, but below three times this energy. For$m \geq 4$, we establish the smooth global existence of such solutions, andtheir decay to a harmonic map (stability), extending results ofGustafson-Nakanishi-Tsai to higher energies. The proof rests on astability-type argument used to rule out finite-time bubbling.

Global solutions for the critical, higher-degree corotational harmonic map heat flow to $\mathbb{S}^2$