{"ID":23673045,"CreatedAt":"2026-09-18T08:32:08.518820066Z","UpdatedAt":"2026-09-18T08:32:08.518820066Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.20762","arxiv_id":"2609.20762","title":"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 $0\u003cγ\u003c1/2$, 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.","short_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,...","url_abs":"https://arxiv.org/abs/2609.20762","url_pdf":"https://arxiv.org/pdf/2609.20762v1","authors":"[\"Peter Constantin\",\"Mihaela Ignatova\",\"Vlad Vicol\"]","published":"2026-09-17T17:44:06Z","proceeding":"math.AP","tasks":"[\"math.AP\"]","methods":"[]","has_code":false}