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arXiv · 2608.22447

No bubble trees for the $1$-equivariant harmonic map heat flow and the radial energy-critical nonlinear heat equation in low dimensions

Taegyu Kim
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Abstract

We consider the $1$-equivariant harmonic map heat flow (HMHF) from $\mathbb R^2$ to $\mathbb S^2$ and the radial energy-critical nonlinear heat equation (NLH) in dimensions $d=3,4,5$. We prove that every finite-energy solution of (HMHF) and every $\dot H^1$-bounded solution of (NLH) has at most one bubble: every finite-time blow-up has exactly one bubble, whereas every global solution has either no bubble or one bubble at infinite time. Starting from the soliton resolution, we exclude bubble trees by modulation analysis, deriving a contradiction from the relative dynamics of the two innermost scales. This energy method does not rely on maximum principle and, in particular, applies without restriction on the bubble signs.

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Taegyu Kim. 2026-08-23. No bubble trees for the $1$-equivariant harmonic map heat flow and the radial energy-critical nonlinear heat equation in low dimensions. https://arxiv.org/abs/2608.22447

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