A logarithmic regularity criterion for the axisymmetric Navier-Stokes equations
Abstract
For axisymmetric solutions of the Navier-Stokes equations in ℝ3, the circulation Γ=ruθ obeys a maximum principle, and the resulting a priori bound on Γ is scale-invariant. Nonetheless, the global regularity problem remains completely open. The sharpest known regularity criteria in terms of the circulation are slightly subcritical, namely |Γ|≲(log r−1)−k near the symmetry axis; Lei and Zhang proved regularity for k=2, and Wei later lowered the exponent to k=3/2. In this note, we strengthen the criterion further to k=5/4. The argument follows the usual route of energy estimates for the quantities ωr/r and ωθ/r, but now with a sharper bound on the vortex stretching term in the equation for ωr/r, in which the loss is proportional only to the square root of the number of scales between the inner vortex core and the far field.
Related
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BibTeX
@misc{palasek2026logarithmic,
title = {{A logarithmic regularity criterion for the axisymmetric Navier-Stokes equations}},
author = {Stan Palasek},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29012},
eprint = {2610.29012},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29012},
}
RIS
TY - UNPB
TI - A logarithmic regularity criterion for the axisymmetric Navier-Stokes equations
AU - Stan Palasek
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29012
UR - https://arxiv.org/abs/2610.29012
AB - For axisymmetric solutions of the Navier-Stokes equations in ℝ^3, the circulation Γ=ru^θ obeys a maximum principle, and the resulting a priori bound on Γ is scale-invariant. Nonetheless, the global regularity problem remains completely open. The sharpest known regularity criteria in terms of the circulation are slightly subcritical, namely |Γ|≲(log r^−1)^−k near the symmetry axis; Lei and Zhang proved regularity for k=2, and Wei later lowered the exponent to k=3/2. In this note, we strengthen the criterion further to k=5/4. The argument follows the usual route of energy estimates for the quantities ω^r/r and ω^θ/r, but now with a sharper bound on the vortex stretching term in the equation for ω^r/r, in which the loss is proportional only to the square root of the number of scales between the inner vortex core and the far field.
ER -