What question did this study set out to answer?

The aim is to explore geometric measures of vorticity and prove global regularity of solutions to Navier-Stokes equations.

June 30, 2026Open Access

Geometric Vorticity Control and Global Regularity of 3D Navier–Stokes

Key Points

The aim is to explore geometric measures of vorticity and prove global regularity of solutions to Navier-Stokes equations.
Introduced coherence tube density and Udjat structure operator for vorticity coherence measurement.
Derived dispersion decomposition separating vortex stretching from viscous dissipation.
Applied the Beale–Kato–Majda criterion to achieve results.
Proved that coherence tube density remains uniformly bounded away from zero for all time.
Established global-in-time existence of smooth solutions under arbitrary smooth divergence-free initial conditions.

Abstract

We introduce the Coherence Tube Density (CTD) and the Udjat Structure Operator, two geometric objects that measure the local directional and magnitude coherence of a vorticity field. Using these tools, we derive a Dispersion Decomposition Equation separating vortex stretching from viscous dissipation, and prove that the coherence tube density remains uniformly bounded away from zero for all time. Combined with the Beale–Kato–Majda criterion, this yields global-in-time existence of smooth solutions to the three-dimensional incompressible Navier–Stokes equations for arbitrary smooth divergence-free initial data. A companion paper resolves three structural gaps identified in the original proof.

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