On singularities of the heat flow for harmonic maps from surfaces into spheres
This is an investigation of the local behavior of solutions of the heat equation for harmonic maps from surfaces into spheres near singular points. The author proves that there is no unaccounted energy loss for blow-ups of solutions. Any energy drop is shown to correspond precisely to the production of bubbles, i.e., nontrivial harmonic maps from 2-sphere into the target. It was shown earlier by \textit{M. Struwe} [Comment. Math. Helv. 60, 558-581 (1985; Zbl 0595.58013)] that global solutions exist and are regular except at possibly finitely many isolated points. More recently, \textit{K. C. Chang, W. Ding} and \textit{R. Ye} [J. Differ. Geom. 36, No. 2, 507-515 (1992; Zbl 0765.53026)] proved that these singular points indeed exist in some cases. The present paper thus offers an explanation what happens to the solutions near these singularities.
scientific article; zbMATH DE number 856918