What question did this study set out to answer?

The aim is to establish a framework that quantitatively analyzes finite-time blow-up in the 3D Navier-Stokes equations.

March 24, 2026Open Access

Navier–Stokes 3D Сoncrete Finite-Time Blow-up Framework

Key Points

The aim is to establish a framework that quantitatively analyzes finite-time blow-up in the 3D Navier-Stokes equations.
Developed a finite-time blow-up framework for the Navier-Stokes equations in 3D.
Combined techniques such as localized enstrophy barriers and geometric-spectral strain lower bounds.
Analyzed conditions for divergence of the BKM integral under specific closed block constraints.
The framework demonstrates divergence of the BKM integral for the constructed data class.
It shows that smooth solutions cannot continue beyond a finite time under certain conditions.

Abstract

This preprint proposes a quantitative finite-time blow-up framework for the 3D incompressible Navier-Stokes equations in R³. The analysis combines localized enstrophy barriers, geometric-spectral strain lower bounds, and a topological coercivity channel. Under the internally closed block conditions stated in the manuscript, the framework yields divergence of the BKM integral and excludes smooth continuation beyond finite time for the constructed data class.

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