A special Hardy estimate in the axisymmetric swirl problem

We examine the proof of Dongyi Wei’s logarithmic circulation criterion. A trace estimate specific to $\mathrm{\nabla }(u_r\mathrm{/}r)$ replaces a generic squared-swirl form bound by one with one logarithm rather than two. Repeating the coupled $J$–$\mathrm{\Omega }$ energy and comparison argument then gives a proposed threshold of order $(\log D{)}^{-5\mathrm{/}4}$. Under the hypotheses stated below, the proof is intended to establish regularity when $\mathrm{\mid }r{u}_{\theta }\mathrm{\mid }\le C\mathrm{\mid }\log r{\mathrm{\mid }}^{-\alpha }$ uniformly near the axis with $\alpha >5\mathrm{/}4$, and at $\alpha =5\mathrm{/}4$ with an absolute sufficiently small coefficient. Every analytic step, including the cutoff core and exterior terms, is included for independent checking.