A logarithmic regularity criterion for the axisymmetric Navier-Stokes equations

For axisymmetric solutions of the Navier-Stokes equations in ${\mathbb{R}}^3$, the circulation $\mathrm{\Gamma }=r{u}^{\theta }$ obeys a maximum principle, and the resulting a priori bound on $\mathrm{\Gamma }$ 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 $\mathrm{\mid }\mathrm{\Gamma }\mathrm{\mid }\lesssim (\log r^{-1}{)}^{-k}$ near the symmetry axis; Lei and Zhang proved regularity for $k=2$, and Wei later lowered the exponent to $k=3\mathrm{/}2$. In this note, we strengthen the criterion further to $k=5\mathrm{/}4$. The argument follows the usual route of energy estimates for the quantities ${\omega }^r\mathrm{/}r$ and ${\omega }^{\theta }\mathrm{/}r$, but now with a sharper bound on the vortex stretching term in the equation for ${\omega }^r\mathrm{/}r$, 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.