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arXiv · 1610.09497

Stable self-similar blowup in the supercritical heat flow of harmonic maps

Paweł Biernat·Roland Donninger·Birgit Schörkhuber
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Abstract

We consider the heat flow of corotational harmonic maps from $\mathbb R^3$ to the three-sphere and prove the nonlinear asymptotic stability of a particular self-similar shrinker that is not known in closed form. Our method provides a novel, systematic, robust, and constructive approach to the stability analysis of self-similar blowup in parabolic evolution equations. In particular, we completely avoid using delicate Lyapunov functionals, monotonicity formulas, indirect arguments, or fragile parabolic structure like the maximum principle. As a matter of fact, our approach reduces the nonlinear stability analysis of self-similar shrinkers to the spectral analysis of the associated self-adjoint linearized operators.

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Paweł Biernat, Roland Donninger, Birgit Schörkhuber. 2016-10-29. Stable self-similar blowup in the supercritical heat flow of harmonic maps. https://arxiv.org/abs/1610.09497

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