Global solutions for the critical, higher-degree corotational harmonic map heat flow to S^2

We study m-corotational solutions to the Harmonic Map Heat Flow from R2 to S2. We first consider maps of zero topological degree, with initial energy below the threshold given by twice the energy of the harmonic map solutions. For m≥2, we establish the smooth global existence and decay of such solutions via the concentration-compactness approach of Kenig-Merle, recovering classical results of Struwe by this alternate method. The proof relies on a profile decomposition, and the energy dissipation relation. We then consider maps of degree m and initial energy above the harmonic map threshold energy, but below three times this energy. For m≥4, we establish the smooth global existence of such solutions, and their decay to a harmonic map (stability), extending results of Gustafson-Nakanishi-Tsai to higher energies. The proof rests on a stability-type argument used to rule out finite-time bubbling.