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

Abstract

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 $c>0$, $$|\Gamma (r,x_3,t)|\le N{e}^{-c|\ln r{|}^{\tau }},0<r\le \frac{1}{4}.$$