Restricted-Carleson VACM for 3D NSE]{Unconditional Dyadic--Conic Control for 3D Navier--Stokes via \\ Restricted, NSE-Native Carleson at the Active Scale

Unconditional Dyadic–Conic Control for the 3D Navier–Stokes Equations

Description:
This paper develops a formal framework intended to resolve the global regularity problem for the three-dimensional incompressible Navier–Stokes equations on R3\mathbb{R}^3R3, one of the Clay Millennium Problems.

The method constructs a dyadic and angular decomposition of the velocity field using Variable-Axis Conic Multipliers (VACM), which allow for localized directional control of nonlinear interactions. A combination of scale-sensitive commutator bounds, diffusion balance identities, and a stop-time ledger structure produces a fully nonlinear inequality that prevents singularity formation.

Key mechanisms include:

• Frequency–directional localization via VACM projectors

• A mollified strain tensor ensuring stable commutator estimates

• A diffusion commutator bound that absorbs nonlinear terms into dissipation

• A Łojasiewicz-type inequality controlling sublevel sets of the spectral gap

• A ledger tracking slabwise vorticity growth, ensuring summable badness

• A critical Lyapunov inequality that closes globally via recursive energy contraction

The main result is a global-in-time estimate on the enstrophy and a bound on the integral of the maximum vorticity norm. This satisfies the Beale–Kato–Majda blow-up criterion without assuming a priori bounds, Carleson structure, or symmetry reductions.

This is a formal and explicit solution strategy intended to be scrutinized by the community. The methods introduced may also be adaptable to a broader class of nonlinear PDEs involving scale interactions and dissipation.