A Geometric–Multiscale–Temporal Conjecture of Depletion through Localized Enstrophy for the 3D Navier–Stokes Equations

Abstract

Abstract (English) We study geometric and temporal mechanisms of localized vorticity amplification in the threedimensional incompressible Navier–Stokes equations. The exact localized-enstrophy balance separates vortex stretching, transport and diffusion. A finite core/multiscale/far-field decomposition retains directional coherence, local filling, signed cancellation and localization costs as distinct quantities. A conditional estimate under a fixed cutoff, radius and positive vorticity threshold controls localizedenstrophy and dissipation when a temporal coefficient is integrable. That coefficient selects between a weighted directional modulus and a mixed pairwise moment. Its integrability has not been deduced from the energy inequality. We give an explicit partial inclusion from a family of Chae–Kang–Lee direction–amplitude hypotheses and identify two sufficient amplitude-based reductions that already imply known local vorticity criteria. These inclusions do not establish a converse or redundancyof the general pairwise hypothesis. Application at a possible terminal singular time requires the appropriate suitable-solution setting or a separate continuation argument. The numerical component distinguishes curvature under pair-count controls on 15 fixed events from cross-event regressions on that sample and on two prospective scales with 78 and 79 valid events. The radius-normalized response uses each event’s actual radius. Its prospective slopes are not distinguishable from zero by the procedures examined; this is not evidence of equivalence. Calibration of seven inferential procedures is model-dependent. The curvature ranges describe pair-subsampling variability, not uncertainty across turbulent realizations. The note does not establish a universal numerical law, a new regularity class, or global regularity. The adaptive conjecture still requires persistence, a controlled dissipative margin and a noncircular continuation step.