Coarse-Grained Resolution and Pressure-Flux Work Depletion for Navier-Stokes CKN Badness

Abstract

We prove a finite-scale coarse-grained decomposition for the three-dimensional incompressible Navier-Stokes equations near the Caffarelli-Kohn-Nirenberg local regularity framework. The first part is a local resolution lemma for the scale-critical quantity: for every spatial filter length ell > 0, Psi(r) <= 4 Psi^ell(r) + 4 Omega^ell(r), where Psi^ell(r) is the corresponding coarse-grained velocity-pressure quantity and Omega^ell(r) is the explicitly defined subfilter residual. Thus a CKN-bad scale is either visible at the resolved level or is carried by unresolved velocity-pressure oscillation. The second part is an exact fixed-chain depletion theorem for the combined pressure-flux work distribution G^ell = Pi^ell + div(P^ell U^ell), Pi^ell = -R^ell : grad U^ell, which is the signed work density appearing in the localized resolved-energy balance. For finite-dimensional active test families with common endpoint traces, we obtain a constructive active-work extraction and a weighted telescoping inequality: forward combined work and resolved dissipation are paid by the initial localized kinetic energy, explicit localization leakage, and negative combined work/backscatter.