Controlled harmonic map heat flow and blow-up from the disk to the sphere
Abstract
We study the one-corotational harmonic map heat flow from the disk to the sphere with a Dirichlet boundary control. We introduce a control-adapted gluing strategy to prevent blow-up. More precisely, we construct explicit upper profiles and well-prepared boundary traces such that every nonnegative datum below one of these profiles and having boundary value above π blows up under the corresponding constant trace, whereas the corresponding solution is global under every control below the associated well-prepared boundary trace. Controls with uniformly bounded time derivative also prevent infinite-time concentration. We also show that boundary control can drive any nonnegative datum to blow-up before any prescribed time, whereas bounded controls cannot prevent suitably fast concentration.
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BibTeX
@misc{xiang2026controlled,
title = {{Controlled harmonic map heat flow and blow-up from the disk to the sphere}},
author = {Shengquan Xiang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10129},
eprint = {2609.10129},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10129},
}
RIS
TY - UNPB
TI - Controlled harmonic map heat flow and blow-up from the disk to the sphere
AU - Shengquan Xiang
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10129
UR - https://arxiv.org/abs/2609.10129
AB - We study the one-corotational harmonic map heat flow from the disk to the sphere with a Dirichlet boundary control. We introduce a control-adapted gluing strategy to prevent blow-up. More precisely, we construct explicit upper profiles and well-prepared boundary traces such that every nonnegative datum below one of these profiles and having boundary value above π blows up under the corresponding constant trace, whereas the corresponding solution is global under every control below the associated well-prepared boundary trace. Controls with uniformly bounded time derivative also prevent infinite-time concentration. We also show that boundary control can drive any nonnegative datum to blow-up before any prescribed time, whereas bounded controls cannot prevent suitably fast concentration.
ER -