A critical regularity condition on the angular velocity of axially symmetric Navier-Stokes equations

Let $v$ be the velocity of Leray-Hopf solutions to the axially symmetric three-dimensional Navier-Stokes equations. It is shown that $v$ is regular if the angular velocity $v_\theta$ satisfies an integral condition which is critical under the standard scaling. This condition allows functions satisfying $$|v_\theta(x, t)| \le \frac{C}{r |\ln r|^{2+\epsilon}}, \quad r<1/2,$$ where $r$ is the distance from $x$ to the axis, $C$ and $\epsilon$ are any positive constants. Comparing with the critical a priori bound $$|v_\theta(x, t)| \le \frac{C}{r}, \qquad 0< r \le 1/2,$$our condition is off by the log factor $|\ln r|^{2+\epsilon}$ at worst. This is inspired by the recent interesting paper \cite{CFZ:1} where H. Chen, D. Y. Fang and T. Zhang establish, among other things, an almost critical regularity condition on the angular velocity. Previous regularity conditions are off by a factor $r^{-1}$. The proof is based on the new observation that, when viewed differently, all the vortex stretching terms in the 3 dimensional axially symmetric Navier-Stokes equations are critical instead of supercritical as commonly believed.