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arXiv · 1101.0713

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

Paweł Biernat·Piotr Bizoń
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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 $3\leq d\leq 6$. 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.

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Paweł Biernat, Piotr Bizoń. 2011-06-10. Shrinkers, expanders, and the unique continuation beyond generic blowup in the heat flow for harmonic maps between spheres. https://doi.org/10.1088/0951-7715%2F24%2F8%2F005

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