Global heat-seeded Picard convergence near a nonlinear periodic Navier–Stokes flow

We consider the unforced three-dimensional incompressible Navier–Stokes equations on the normalized 2π-periodic torus at viscosity one. For W=(sin z,sin x,sin y) and every smooth solenoidal, real, mean-zero perturbation satisfying ∑*{k≠0}|k|⁻¹|ĥ(k)|≤10⁻⁴, we certify convergence of the ordinary Picard sequence starting from the complete heat flow e^{tΔ}(W+h). Convergence holds on the whole half-line in ‖f‖*𝓧=(∫₀^∞(∑*{k≠0}|f̂(k)|)² dt)^{1/2}, with error at iteration n≥3 at most 6(9/10)^{n−3} and ∫₀^∞A₀(u(t))² dt<9, where A₀(f)=∑*{k≠0}|f̂(k)|. The proof retains the full causal history in a finite block norm and verifies entry of the actual heat-seeded third iterate into one invariant set. Separate critical-Fourier comparison estimates establish larger explicit global-regularity neighborhoods. A validated finite-reference certificate also gives the fixed-W all-cutoff bound sup_N ∫₀^∞‖∇u_N‖₂⁴ dt<21/2. The stated quantitative estimates are certified and are not claimed optimal. Exact finite certificates and their analytic connection to the full PDE are specified in the accompanying reproducibility material.

Version 1.1 clarifies the relation to Lei–Lin and periodic small-data results, adds the reference-free baseline implied by the existing comparison lemma, defines the solution action at first use, and explains that the quantitative estimates are not claimed optimal. The existing theorem statements, assumptions, constants, scientific programs and archived certificate data are unchanged. The original version 1.0 replay receipts are retained; this editorial revision does not claim a new scientific replay or independent human peer review. A detailed change note is included in Code.zip.

This record contains the paper PDF and a companion reproducibility archive. Paper.pdf and the non-software manuscript, documentation and certificate data in Code.zip are licensed under Creative Commons Attribution 4.0 International (CC BY 4.0), to the extent applicable rights exist. The software source files in Code.zip are licensed under the MIT License. The archive's license and notices files specify this division and preserve any separately identified third-party terms. The two licenses apply to different materials; they are not a choice of alternative licenses for every file.