Classification for the $2$-equivariant harmonic map heat flow and the radial energy-critical nonlinear heat equation in dimension $6$
Abstract
We consider the global dynamics of finite-energy solutions to the $2$-equivariant harmonic map heat flow from bbR2 to bbS2 and Ḣ1-bounded radial solutions to the energy-critical nonlinear heat equation in dimension $6$. Building upon soliton resolution, we obtain a classification of their multi-bubble dynamics. First, we prove that all such solutions are global. For the harmonic map heat flow, the number and sign of the bubbles are determined by the energy class of the initial data, while for the nonlinear heat equation the bubble signs necessarily alternate. In both models, the largest scale is determined, up to a positive multiplicative constant, by the spatial tail of the initial datum. The remaining scales concentrate according to exponential and iterated-exponential laws determined by the largest scale. Finally, global bubble trees with an arbitrary number of bubbles exist for both models. For the harmonic map heat flow, this follows directly from the classification; for the nonlinear heat equation, we construct alternating bubble trees with any prescribed radial Ḣ1 tail, thereby realizing the scale laws arising in the classification.
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BibTeX
@misc{jeong2026classification,
title = {{Classification for the $2$-equivariant harmonic map heat flow and the radial energy-critical nonlinear heat equation in dimension $6$}},
author = {Uihyeon Jeong and Taegyu Kim},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27683},
eprint = {2609.27683},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27683},
}
RIS
TY - UNPB
TI - Classification for the $2$-equivariant harmonic map heat flow and the radial energy-critical nonlinear heat equation in dimension $6$
AU - Uihyeon Jeong
AU - Taegyu Kim
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27683
UR - https://arxiv.org/abs/2609.27683
AB - We consider the global dynamics of finite-energy solutions to the $2$-equivariant harmonic map heat flow from bbR^2 to bbS^2 and Ḣ^1-bounded radial solutions to the energy-critical nonlinear heat equation in dimension $6$. Building upon soliton resolution, we obtain a classification of their multi-bubble dynamics. First, we prove that all such solutions are global. For the harmonic map heat flow, the number and sign of the bubbles are determined by the energy class of the initial data, while for the nonlinear heat equation the bubble signs necessarily alternate. In both models, the largest scale is determined, up to a positive multiplicative constant, by the spatial tail of the initial datum. The remaining scales concentrate according to exponential and iterated-exponential laws determined by the largest scale. Finally, global bubble trees with an arbitrary number of bubbles exist for both models. For the harmonic map heat flow, this follows directly from the classification; for the nonlinear heat equation, we construct alternating bubble trees with any prescribed radial Ḣ^1 tail, thereby realizing the scale laws arising in the classification.
ER -