Shrinkers, expanders, and the unique continuation beyond generic blowup in the heat flow for harmonic maps between spheres

Abstract

Using mixed analytical and numerical methods we investigate the development of singularities in the heat flow for corotational harmonic maps from the $d$-dimensional sphere to itself for $3\le d\le 6$. By gluing together shrinking and expanding asymptotically self-similar solutions we construct global weak solutions which are smooth everywhere except for a sequence of times $T_1<T_2<...<T_k<\infty$ at which there occurs the type I blow-up at one of the poles of the sphere. We show that in the generic case the continuation beyond blow-up is unique, the topological degree of the map changes by one at each blow-up time $T_i$, and eventually the solution comes to rest at the zero energy constant map.

24 pages, 8 figures, minor corrections, matches published version

References in corpus (1)

• Selfsimilar expanders of the harmonic map flow

Cited by in corpus (6)

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• Self-similar solutions of the one-dimensional Landau-Lifshitz-Gilbert equation
• Recent results for the Landau-Lifshitz equation
• Nonlinear stability of homothetically shrinking Yang-Mills solitons in the equivariant case