An explicit family, with an explicit convergence rate.
The manuscript concerns the unforced Navier–Stokes equation at viscosity one on the normalized 2π-periodic three-dimensional torus. It takes u₀ = W + h, where W = (sin z, sin x, sin y) and h is smooth, real, mean-zero and divergence-free.
Within the stated radius, Theorem 2.1 gives a computer-assisted certificate that the ordinary Picard iteration, started from the complete heat flow etΔ(W+h), enters a common invariant set at its third iterate and converges on the whole time half-line, without restarting the iteration.
हा हस्तलिखितातील प्रमेयाचा दावा आहे. त्याची वैधता संपूर्ण गणिती युक्तिवाद आणि प्रमाणपत्रांवर अवलंबून आहे. येथील तुलना त्या संपूर्ण पुराव्याची स्वतंत्र पुनर्पडताळणी नाही.
STATED HYPOTHESIS & BOUND
A−1(h) ≤ 10−4
‖vn − u‖X ≤ 6(9/10)n−3
n ≥ 3 · X = L²((0,∞); A₀)
As(f) = Σk≠0 |k|s |f̂(k)|, using the Euclidean norm of each Fourier coefficient.
Smooth h may have infinitely many Fourier modes. The statement retains its smoothness assumption.
02
THE CHECKED COMPARISON
Where the contribution sits.
Picard iteration, invariant tubes and a posteriori control are established methods. The specific contribution described here is the combination of the W-centred perturbation class, entry of the actual third iterate, global convergence and explicit uniform constants.
Global regularity for a specified cyclic-shear class, alongside other families.
A separate manuscript regularity class strictly contains that cyclic class after a coordinate permutation. See the exact scope below.
The full comparison gives proposition-level details for these and three further primary sources, exact witnesses and reproducible elementary calculations. This targeted comparison does not establish worldwide priority.
SEPARATE RESULT · THEOREM 3.1
A larger regularity class
The manuscript includes u₀ = aW+h with |a| ≤ 8/7 and A−1(h) ≤ 1/100, keeping the same structural and smoothness assumptions.
Swapping x and y maps Silva's cyclic field U to W. Since A−1(h) ≤ A₂(h), his class 0 ≤ a ≤ 7/10, A₂(h) ≤ 1/100 is included. The inclusion is strict: W is in the larger class, while its minimum A₂ distance from the earlier amplitude segment is 9/10.
This is a global-regularity comparison for one specified cyclic class. The main Picard convergence conclusion is not extended to this larger region.
The later finite-time error-checking work uses an established energy-estimate approach. These demonstrations have their own assumptions and scope; they do not establish the original Picard theorem or its novelty.
मुख्य प्रमेय, स्वतंत्र regularity निष्कर्ष आणि नंतरची प्रयोग-साधने यांचे दावे वेगवेगळे आहेत. प्रत्येकाचा पुरावा त्याच्या दिलेल्या व्याप्तीत पाहावा.
03 · AN EXACT EXAMPLE
One input you can inspect now.
h = (1/5000) sin(2y) ex
Its A−1 norm is exactly 1/10000, so it lies on the main theorem's stated boundary. For the manuscript's elementary origin-centred contraction estimate, the squared sufficient-test factor is 9.00048001 > 1; that estimate cannot certify this example.
This concerns one sufficient estimate. It does not show divergence or failure of every earlier method.
Walke, R. M. (2026). Global heat-seeded Picard convergence near a nonlinear periodic Navier–Stokes flow (Version 1.1) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.23429477
When discussing or building on a particular result, identify the theorem and version used. Credit prior methods and third-party software separately.
या नोंदीमुळे नाव, दावा आणि पुरावे अचूकपणे जोडले जातात. संशोधनाची मान्यता, कायदेशीर हक्क किंवा आर्थिक परतावा यांची हमी यातून मिळत नाही.
Research assistance and responsibility
Manuscript §5 identifies ChatGPT (OpenAI) as the primary AI research assistant and Claude (Anthropic) as the secondary assistant. Assistance included mathematical investigation, argument and code development or review, and manuscript preparation. Neither system is an author or an external referee. The human author remains responsible for the research claims and disclosure. This contribution page was also prepared with AI assistance.