PreprintCC BY 4.0arXiv
A special Hardy estimate in the axisymmetric swirl problem
Abstract
We examine the proof of Dongyi Wei's logarithmic circulation criterion. A trace estimate specific to ∇(ur/r) replaces a generic squared-swirl form bound by one with one logarithm rather than two. Repeating the coupled J--Ω energy and comparison argument then gives a proposed threshold of order (log D)−5/4. Under the hypotheses stated below, the proof is intended to establish regularity when |r uθ|≤ C|log r|−α uniformly near the axis with α>5/4, and at α=5/4 with an absolute sufficiently small coefficient. Every analytic step, including the cutoff core and exterior terms, is included for independent checking.
Related
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BibTeX
@misc{zhou2026special,
title = {{A special Hardy estimate in the axisymmetric swirl problem}},
author = {Daoguo Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31435},
eprint = {2610.31435},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31435},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A special Hardy estimate in the axisymmetric swirl problem
AU - Daoguo Zhou
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31435
UR - https://arxiv.org/abs/2610.31435
AB - We examine the proof of Dongyi Wei's logarithmic circulation criterion. A trace estimate specific to ∇(u_r/r) replaces a generic squared-swirl form bound by one with one logarithm rather than two. Repeating the coupled J--Ω energy and comparison argument then gives a proposed threshold of order (log D)^−5/4. Under the hypotheses stated below, the proof is intended to establish regularity when |r u_θ|≤ C|log r|^−α uniformly near the axis with α>5/4, and at α=5/4 with an absolute sufficiently small coefficient. Every analytic step, including the cutoff core and exterior terms, is included for independent checking.
ER -