A special Hardy estimate in the axisymmetric swirl problem

Daoguo Zhou

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arXiv · 2610.31435

A special Hardy estimate in the axisymmetric swirl problem

Daoguo Zhou

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Abstract

We examine the proof of Dongyi Wei's logarithmic circulation criterion. A trace estimate specific to $\nabla(u_r/r)$ 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 $|r u_θ|\le C|\log r|^{-α}$ 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.

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Daoguo Zhou. 2026-10-22. A special Hardy estimate in the axisymmetric swirl problem. https://arxiv.org/abs/2610.31435

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