What question did this study set out to answer?

The main aim is to establish a critical norm closure theorem within the TEBAC Navier-Stokes program.

May 9, 2026Open Access

TEBAC Navier–Stokes Program IV: Critical Norm Closure and Galerkin Limit Assembly

Key Points

The main aim is to establish a critical norm closure theorem within the TEBAC Navier-Stokes program.
Developed a normalization bridge between vorticity/enstrophy dissipation and critical H^{1/2} estimation.
Provided cutoff-uniform critical Stokes estimate independent of the Galerkin cutoff N.
Utilized dyadic level relations to encode critical dissipation.
Proved the module-level critical norm closure theorem related to NS-I–NS-IIIa absorption.
Established that NS-IV achieves the critical H^{1/2} estimate for Galerkin approximants.
NS-IV does not complete the solution to the Clay Millennium problem; this remains for NS-V.

Abstract

This preprint is the fourth module of a modular TEBAC approach to the three-dimensional incompressible Navier--Stokes regularity problem on the periodic torus\ T³= R³/ (2 Z) ³. \ The preceding modules establish the periodic Leray--Stokes--Galerkin foundation, the vorticity-to-shell decomposition, the resonance classification, and the terminal axial-envelope packing closure needed for critical resonance absorption. The purpose of the present module is to convert the upstream NS-III/NS-IIIa critical-resonance absorption export into the cutoff-uniform critical Stokes estimate\₀ ₓ ₓ\|A^1/4uN (t) \|₋ℂ²+₀T\|A^3/4uN (t) \|₋ℂ²\, dt (T, , u₀), \ (C (T, , u₀) \) independent of the Galerkin cutoff \ (N\). The central technical step of NS-IV is the normalization bridge between the vorticity/enstrophy shell dissipation used in the resonance modules and the critical \ (H^1/2\) -Stokes scale needed for the final regularity assembly. At the dyadic level, this is encoded by the weighted relation\2^-q\|q\|₋ℂ²2^3q\|uq\|₋ℂ², identifies the correctly reweighted NS-III flux absorption with the NS-IV critical dissipation ledger. The manuscript proves the module-level critical norm closure theorem relative to the upstream NS-I--NS-IIIa absorption package. In this precise sense, NS-IV closes the critical \ (H^1/2\) -estimate for Galerkin approximants. It does not, by itself, claim a complete solution of the Clay Millennium problem. The final global regularity conclusion is reserved for the subsequent NS-V assembly module, which must carry out the Galerkin limit passage, weak-to-strong upgrade, continuation criterion, uniqueness, and final non-circularity audit. Status statement: NS-IV is a theorem-bearing closure module inside the TEBAC Navier--Stokes program. It proves the cutoff-uniform critical norm estimate needed for the final assembly, assuming the previously established upstream resonance-absorption exports. The full Clay-compatible global regularity theorem remains assigned to NS-V.

TEBAC Navier–Stokes Program IV: Critical Norm Closure and Galerkin Limit Assembly

Key Points

Abstract

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