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March 30, 2026Open Access

Vorticity Anisotropy and the Navier-Stokes Regularity Problem: An Extended Abstract

Key Points

The research aims to explore the regularity problem of the 3D Navier-Stokes equations using vorticity anisotropy.
Develop a geometric framework to analyze the Navier-Stokes regularity problem.
Partition the flow domain into three zones based on vorticity anisotropy.
Demonstrate the emptiness of the Caffarelli–Kohn–Nirenberg singular set in each zone.
In the isotropic zone, incompressibility suppresses vortex stretching, leading to a parabolic inequality.
Viscosity is shown to control behavior in the boundary zone.
A quasi-1D structure is identified in the deep anisotropic zone.

Abstract

We present a geometric framework for studying the regularity problem of the 3D Navier–Stokes equations. The central object is the vorticity anisotropy A(ω) ∈0,1, measuring how strongly one component of the vorticity dominates the others. The flow domain is partitioned into three zones according to A, and we show that the Caffarelli–Kohn–Nirenberg singular set is empty in each zone by different mechanisms. The key observation is that incompressibility alone (tr S = 0) suppresses vortex stretching in the isotropic zone, yielding a parabolic inequality without source term. Viscosity controls the boundary zone, and a quasi-1D structure governs the deep anisotropic zone.

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