Regularity for axisymmetric Navier-Stokes with an Euler length

Abstract

We prove local regularity for axisymmetric suitable weak solutions of the 3D Navier-Stokes equations, which are smooth before the terminal time $t=0$, and satisfy Type~II pointwise bounds at a vanishing length scale $\ell(t)$. We say $\ell(t)$ is an Euler length if it is non-increasing, satisfies a doubling condition, and if $\ell(t)\to0$ and $(-t)/\ell(t)^2\to0$ as $t\to0^-$. This includes power laws $\ell(t)=(-t)^γ$ with $0<γ<1/2$, and logarithmic enlargements of the parabolic length. Our main result shows that local bounds of the type $|u(\cdot,t)| \leq C\ell(t)/(-t)$ and $|\nabla^2 u(\cdot,t)|\leq C/((-t)\ell(t))$ for all $t\in (-1,0)$ imply regularity. The proof adapts the circulation and potential-vorticity argument of our earlier paper~\cite{CIV26} to ancient limits obtained from the rescaled vorticity system by zooming in. We show that the second derivative a priori assumption may be replaced by a Hölder bound on the azimuthal vorticity and a corresponding bound on its potential vorticity.