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May 27, 2026Open Access

Global Regularity of the 3d Navier-Stokes Equations: The Nonsingular Finality via Intrinsic Topological Back-Reaction (Stage 3)

Key Points

The research aims to establish absolute global regularity and weak-strong uniqueness for the 3D Navier-Stokes equations.
Introduced a geometric framework using Intrinsic Topological Back-Reaction.
Lifted the fluid domain to a 4D manifold to analyze the equations.
Demonstrated the suppression of finite-time blow-up through BKM criterion and topological constraints.
Established that the finite-time blow-up is analytically suppressed and topologically obstructed.
Showed that viscous dissipation behaves similarly to Ricci flow, leading towards a globally smooth structure.
Confirmed the absence of singularities in the generalized manifold.

Abstract

In this final installment (Stage 3), we establish the absolute global regularity and weak-strong uniqueness of the 3D incompressible Navier-Stokes equations. We bridge the purely analytical time-independent H m bounds established in Stage 2 with a profound geometric framework: the Intrinsic Topological Back-Reaction. By lifting the fluid domain to a 4D manifold, we demon- strate that the finite-time blow-up dictated by the Beale-Kato-Majda (BKM) criterion is not only analytically suppressed but topologically obstructed. The viscous dissipation mimics the Ricci flow, driving the system toward a globally smooth, singularity-free generalized manifold.

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