Large-Data Global Regularity for Three-Dimensional Navier--Stokes II: A Direct First-Threshold Continuation Proof for the Full System

Abstract

This is the second paper in a two-part direct-threshold series on large-data global regularity for the three-dimensional incompressible Navier--Stokes equations. It gives the full-system first-threshold continuation argument and uses the companion Part I theorem, which proves the large-data axisymmetric-with-swirl class by the direct full-Dirichlet method. The present paper treats the genuinely three-dimensional front end. A combined critical packet envelope is introduced, and the first time at which this envelope reaches a prescribed level is analyzed by finite-overlap packet selection. The proof uses angular Littlewood--Paley triads, finite-dimensional active-frame rigidity, passive-strain visibility, a quantitative zero-final-defect rigidity theorem, and the companion Part I axisymmetric direct theorem. The main finite-threshold mechanism is that any large leakage, shell, tail, source, passive, phase, or fragmentation error either produces a descendant packet with explicit score lower bound or becomes perturbative. The remaining coherent packet is contracted by the strict local estimates. Thus no large first-threshold packet can occur, and the critical envelope remains bounded on every finite time interval.