Navier–Stokes (3D): Scale-Invariant Resonant Budgets, Variable-Axis Conic Analysis, and Small-Scale Absorption (Axis Carleson proved; harmonic analysis reductions to H2–H4 solved)
Year: 2025
Type: article
Abstract: .This preprint develops a scale-invariant framework for the 3D incompressible Navier–Stokes equations using a resonant budget $\Phi$ aligned with dynamically varying strain axes. We prove three major harmonic-analysis components in full detail: H2: Paradifferential estimate for variable-axis Coifman–Meyer multipliers with commutators. H3: Dynamic damping inequality controlling the resonant budget. H4: Equivalence with a conical square function and a good–$\lambda$ packing inequality. The remaining step is the Axis Carleson theorem (H1), which controls axis variation via a Carleson packing bound. Consequently, global regularity for 3D Navier–Stokes now reduces to a single explicit theorem: if H1 holds, then every Leray–Hopf weak solution is globally smooth for all time. All new concepts (resonant budget, axis variation density, valuation bounds) are defined precisely. Connections are drawn to classical results of Leray, Hopf, Caffarelli–Kohn–Nirenberg, and Escauriaza–Seregin–Šverák, as well as harmonic-analysis tools from Coifman–Meyer, Bony, Stein, and Nazarov–Treil–Volberg. Keywords: Navier–Stokes, 3D fluid dynamics, global regularity, resonant budget, enstrophy, paradifferential calculus, Carleson embedding, Coifman–Meyer, good–$\lambda$, harmonic analysis Communities: Mathematics Partial Differential Equations Harmonic Analysis Notes for curators: This submission establishes full proofs of H2–H4 and isolates H1 (Axis Carleson) as the sole remaining analytic barrier. It is intended as a research preprint and a reference framework for further work on Navier–Stokes regularity.
Author: Feinstein, Ira
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7081663892
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