A logarithmic regularity criterion for the axisymmetric Navier-Stokes equations

Abstract

For axisymmetric solutions of the Navier-Stokes equations in $\mathbb R^3$, 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 $|Γ|\lesssim(\log r^{-1})^{-k}$ near the symmetry axis; Lei and Zhang proved regularity for $k=2$, and Wei later lowered the exponent to $k=3/2$. 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.