Remarks on local regularity of axisymmetric solutions to the 3D Navier--Stokes equations

In this note, a new local regularity criteria for the axisymmetric solutions to the 3D Navier--Stokes equations is investigated. It is slightly supercritical and implies an upper bound for the oscillation of $\Gamma=r u^{\theta}$: for any $0< \tau<1$, there exists a constant $c>0$, $$ |\Gamma(r,x_{3},t)|\leq N e^{-c\, |\ln r|^{\tau}},\ 0<r\leq \frac{1}{4}. $$