Exact Local Unforced Terminal-Core Dynamics in Three-Dimensional Navier–Stokes: Anisotropic Concentration and Swirl Reversals
Year: 2026
Type: preprint
Abstract: Building on OpenAI’s 2026 construction of finite-time velocity blow-up for the three-dimensional incompressible Navier–Stokes equations with smooth forcing, we isolate its anisotropically contracting axisymmetric terminal core and address astep explicitly left open in that construction: convergence of the unmodified formal inner correction hierarchy. A radialvaluation ladder, a doubled analytic coefficient algebra, and an explicit current-channel decomposition yield an O(n−1)large-order feedback bound and an exponential majorant. The resulting series converges normally on a common terminalcore, and its sum satisfies the unforced equations pointwise for every t < 1, with uz(0, 0, t) = j0(1 − t)−1/2−h. The corehas vanishing local kinetic energy while its local L3 and Serrin-critical norms diverge, and the undivided branch admitsany prescribed finite number of simple swirl reversals after parameter adjustment. A Nash–Moser smoothing estimate forΓ = ruθ strengthens the whole-space continuation obstruction to all finite weak-Lp angular-momentum classes. Finally,exact centre-plane identities yield a covariant phase-locking reduction for fixed-parameter radial continuation: favourableshear damps the radial derivative of the locking defect, but its forcing contains second transverse shear and pressurecurvature. This identifies the precise remaining coupled hierarchy. No whole-space finite-energy unforced singular solutionis constructed.
Author: Newton, Tony
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222961759
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