What question did this study set out to answer?

The aim is to propose a new proof route addressing the 3D incompressible Navier–Stokes regularity problem.

June 15, 2026Open Access

Finite CKN Density, Level-Scale Collapse, and a Proposed Compactness Route for the 3D Navier–Stokes Regularity Problem

Key Points

The aim is to propose a new proof route addressing the 3D incompressible Navier–Stokes regularity problem.
Introduced finite Caffarelli–Kohn–Nirenberg density lemma.
Utilized a level-scale selection argument to isolate critical proof components.
Emphasized compactness and pressure decay in the analysis.
Provided a structured proof route for expert scrutiny.
Highlighted the importance of local pressure decomposition in regularity.
Demonstrated harmonic pressure decay as a central mechanism in the analysis.

Abstract

This manuscript records a proposed compactness route toward the three-dimensional incompressible Navier–Stokes regularity problem. The work focuses on a finite Caffarelli–Kohn–Nirenberg density regularity lemma and a level-scale selection argument intended to isolate the remaining critical proof gates. The manuscript is presented as a proposed proof route for expert verification, with particular attention to compactness, local pressure decomposition, harmonic pressure decay, and CKN epsilon regularity.

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