A Mixed Radial–Angular Littlewood–Paley Criterion for the 3D Navier–Stokes Equations
Year: 2026
Type: preprint
Abstract: We adapt the angular regularity framework —a joint radial–angular Littlewood–Paley decomposition previously used in dispersive equations— to the Kozono–Taniuchi BMO criterion for the 3D Navier–Stokes equations. The decomposition is refined to project onto individual spherical‑harmonic degrees rather than dyadic angular bands, yielding a Bernstein inequality whose constant is independent of the angular degree \(\ell\). By expressing the nonlinear transfer as a product of angular Gaunt coefficients and radial coupling constants, we prove that the balanced Gaunt coefficient decays only as \(\Theta(\ell^{-1/2})\), far weaker than the \(\beta>3/2\) required for unconditional summability. We further show that the radial coupling is geometrically local only in a restricted corner of the non‑resonant frequency regime; both the resonant window and the high–high separated non‑resonant corner remain uncontrolled by elementary arguments. The remaining analytic difficulty is encapsulated in a summability hypothesis, whose verification would imply unconditional regularity. The work's contribution is the reduction of the regularity problem to two precisely localised obstructions and the explicit demonstration that purely algebraic Gaunt bounds are insufficient to overcome them.
Author: Ozorio Olea, Arnaldo Adrian
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7171528844
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