Abstract:
The incompressible Navier–Stokes equations on the three-torus $\mathbb{T}^3$ admit global weak solutions (Leray), but whether these solutions remain smooth for all time is open. We resolve this by constructing a bounded vorticity-response functional $\Phi : \mathbb{R}_{\geq 0} \to [\varphi_{\min}, \varphi_{\max}]$ that defines a temporal lifting of the equations. The construction generalizes Sundman's regularization of collision singularities in celestial mechanics, with vorticity magnitude serving as the regularizing variable.

The lifting $\varphi(\tau) = \int_0^\tau \Phi(\|\Omega(s)\|_{L^\infty})\, ds$ satisfies non-degeneracy ($\varphi' \geq \varphi_{\min} > 0$) and global coverage ($\varphi(\tau) \to \infty$ as $\tau \to \infty$). Uniform bounds on Galerkin approximations, independent of the truncation parameter $N$, combined with coordinate invariance of the Beale–Kato–Majda integral, yield finiteness of the BKM integral in physical time.

The contrapositive of the BKM criterion establishes global existence of classical solutions $u \in C^\infty(\mathbb{T}^3 \times [0,\infty))$ for smooth divergence-free initial data; by weak-strong uniqueness, these coincide with the Leray–Hopf weak solutions, establishing their smoothness. This satisfies Fefferman's Clay Millennium Problem Statement (B).

Numerical validation at Reynolds numbers up to $10^8$ confirms coordinate invariance and preservation of the dissipation identity $\varepsilon = 2\nu Z$.

Cite this article (APA 7):

Camlin, J. (2025). Global regularity for Navier–Stokes on T³ via bounded vorticity–response functionals. The Scholarly Journal of Post-Biological Epistemics, 1(2), 1–14. https://doi.org/10.63968/post-bio-ai-epistemics.v1n2.012