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A geometric characterization of potential Navier-Stokes singularities

Zhen Lei, Xiao Ren, Gang Tian

TL;DR

This work addresses the global regularity problem for the 3D Navier–Stokes equations by proposing a geometric regularity criterion based on vorticity directions. The authors introduce a cone-based condition on the vorticity directions in regions of large |$\omega$| and develop a mechanism centered on the local absolute vorticity flux to obtain scale-invariant control and Type I bounds. The key contributions include a new geometric regularity criterion, a flux-control framework that yields decay and regularity through De Giorgi–Nash–Moser arguments, and corollaries including angular-separation results and axisymmetric cases. The results provide geometric insight into singularity formation, suggesting that vortex-stretching alone may be insufficient for blow-up and linking potential singularities to flux decay, with potential implications for related fluid models.

Abstract

For a local suitable weak solution to the Navier-Stokes equations, we prove that if the vorticity vectors belong to a double cone in regions of high vorticity magnitude, then the solution is regular. Roughly speaking this implies that, near a potential singularity, the directions of vorticity cannot avoid any great circle on the unit sphere. Our method, based on the control of local vorticity fluxes, is inspired by the classical Kelvin-Helmholtz law for ideal fluids and the Type I regularity theory for axisymmetric Navier-Stokes solutions.

A geometric characterization of potential Navier-Stokes singularities

TL;DR

This work addresses the global regularity problem for the 3D Navier–Stokes equations by proposing a geometric regularity criterion based on vorticity directions. The authors introduce a cone-based condition on the vorticity directions in regions of large || and develop a mechanism centered on the local absolute vorticity flux to obtain scale-invariant control and Type I bounds. The key contributions include a new geometric regularity criterion, a flux-control framework that yields decay and regularity through De Giorgi–Nash–Moser arguments, and corollaries including angular-separation results and axisymmetric cases. The results provide geometric insight into singularity formation, suggesting that vortex-stretching alone may be insufficient for blow-up and linking potential singularities to flux decay, with potential implications for related fluid models.

Abstract

For a local suitable weak solution to the Navier-Stokes equations, we prove that if the vorticity vectors belong to a double cone in regions of high vorticity magnitude, then the solution is regular. Roughly speaking this implies that, near a potential singularity, the directions of vorticity cannot avoid any great circle on the unit sphere. Our method, based on the control of local vorticity fluxes, is inspired by the classical Kelvin-Helmholtz law for ideal fluids and the Type I regularity theory for axisymmetric Navier-Stokes solutions.
Paper Structure (13 sections, 10 theorems, 119 equations)

This paper contains 13 sections, 10 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Main ideas of the proof.
  3. Organization of the paper.
  4. Preliminaries
  5. Suitable weak solutions
  6. Quantitative partial regularity
  7. A lemma from De Giorgi-Nash-Moser theory
  8. Boundedness of the vorticity flux
  9. Proof of Theorem \ref{['thm:main']} via Proposition \ref{['prop:main']}.
  10. Local smallness of the absolute vorticity flux
  11. Proofs of Corollaries \ref{['cor:114']} and \ref{['thm:rot']}
  12. On the direction of velocity
  13. Acknowledgement.

Key Result

Theorem 1.1

Let $v$ be a suitable weak solution to the Navier-Stokes equations in $Q(1)$. Suppose that there exists a unit vector $e \in \mathbb{S}^2$ and some positive constants $\delta, M$ such that, for any regular point We say that $(x,t)$ is a regular point of $v$ if $v$ is bounded in a parabolic cylinder then $v$ is regular in $\overline{Q(1/2)}$.

Theorems & Definitions (22)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Remark 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • ...and 12 more