Non-axisymmetric exclusion of Type I singularities and Euler-length regularity

Establish, without assuming axisymmetry, the exclusion of Type I singularities and the local regularity conclusion under the Euler-length bounds considered for axisymmetric three-dimensional Navier–Stokes solutions, including the power-law range 2/5 ≤ γ < 1/2 and logarithmic enlargements of the parabolic length.

Background

The main theorem proves local regularity only for axisymmetric suitable weak solutions satisfying velocity and second-derivative bounds at an Euler length. The proof relies essentially on the axisymmetric circulation and potential-vorticity equations, which provide maximum-principle and transport mechanisms unavailable in the general three-dimensional setting.

The authors explicitly state that the corresponding exclusion of Type I singularities is not known without axisymmetry. They further state that they do not know how to prove their Euler-length regularity theorem without rotational symmetry in specified parameter regimes, making this a concrete unresolved extension rather than a general aspiration.

References

The exclusion of Type~I singularities is not known without the assumption of axisymmetry, and we do not know how to prove Theorem~\ref{thm:pointwise:variable} without it when $\ell(t)=(-t)\gamma$ with $2/5\leq\gamma<1/2$, or when $\ell$ is a logarithmic enlargement of the parabolic length.

— Regularity for axisymmetric Navier-Stokes with an Euler length  (2609.20762 - Constantin et al., 17 Sep 2026) in Remark 1, “What is special about axisymmetry?” in Section 1, Introduction

We emphasize that the identity theorem uses the exact vanishing in~eq:interior:core, and that we do not know how to replace it by an algebraic rate of smallness.

— Regularity of asymptotically axisymmetric solutions to the 3D Navier-Stokes equations with analytic forcing  (2609.20803 - Constantin et al., 17 Sep 2026) in Remark 2.2(d), subsection “Upper bounds in place of exact vanishing”