Attainment of a Scale-Critical Vortex-Stretching Functional in Three Dimensions

Abstract

This preprint studies a scale-critical variational functional associated with the nonlinear production of vortex stretching in three-dimensional incompressible Navier–Stokes flow. For divergence-free vorticity fields ω ∈ H1(ℝ3; ℝ3), the functional is defined by F(ω) = N(ω) / (ℰ(ω)1/2 Q(ω)3/2) where ℰ is the enstrophy, Q is the palinstrophy, and N contains both the strain contribution and the nonlocal pressure-Hessian contribution arising from the nonlinear derivative of vortex-stretching production. The main result proves that the corresponding supremum Cℝ3 is finite, strictly positive, and attained. The proof uses a concentration–compactness argument, exact normalization by the critical amplitude and spatial scalings, exclusion of vanishing and dichotomy, divergence-free splitting via the Bogovskiĭ operator, off-diagonal decay for Calderón–Zygmund pressure operators, and strong H1 compactness modulo translations. Strict positivity is established constructively using an explicit periodic Fourier seed followed by a slowly modulated whole-space construction. The periodic seed is verified exactly with rational arithmetic by the supplementary Python script included in the deposit. Version 1.1.0 update. A numerical exploration of localized Gaussian–polynomial vector-potential ansatzes is included as a reproducible starting point for future study of the extremizer and the value of Cℝ3. These numerical results are presented only as lower-bound candidates and are not claimed to determine the sharp constant or the global maximizer. The work is related to, but distinct from, the established literature on extremal enstrophy growth in three-dimensional Navier–Stokes flows, including the variational programs of Lu–Doering, Ayala–Protas, and Kang–Yun–Protas. To the author's knowledge, no prior attainment theorem has been identified for the specific scale-critical quotient studied here on ℝ3; this novelty assessment remains subject to independent specialist review. This is a static variational result. It does not claim global regularity, finite-time blow-up, or a solution of the Navier–Stokes Millennium Prize Problem. AI assistance. Grok was used in an earlier research handoff. OpenAI ChatGPT/Codex assisted the subsequent mathematical audit, code preparation, execution of calculations, literature checking, editing, translation, and document preparation. AI tools are not listed as authors or contributors. Results are marked as computations on specified synthetic models. Raw AI outputs are not included as primary research artifacts.