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

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Abstract

In this talk, we will present the recent progress on axisymmetric solution to the 3D Navier Stokes equations. Moreover,  we will give  a new local regularity criteria. 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 $c>0$,
$|\Gamma(r, x_{3},t)|\leq N e^{-c\, |\ln r|^{\tau}},\ 0<r\leq \frac{1}{4}.$

(Based on work with Hui Chen and Taipeng Tsai).