Papers · Preprint · v4
A Conditional Spectral-Closure Theorem for Three-Dimensional Navier-Stokes
A conditional route to smooth fluid flow forever, with the strength of its assumption named.
In plain words
Does a smooth three dimensional fluid stay smooth for all time? This paper proves a conditional answer. Split the flow into low and high frequencies. The spectral closure assumes that on each finite stretch of time one frequency cutoff and one fraction below one keep the high part at most that fraction of the whole. Kinetic energy already controls the low part, so the whole stays bounded and the flow cannot break down. It is then smooth in space at positive times and solves the equations pointwise, with a pressure.
The paper is plain about what this buys. Together with the energy bound, the assumption is an a priori bound on the solution, and for flows that never vanish it is equivalent to global existence. So the theorem is the classical continuation principle in spectral form, not a reduction to an easier estimate. A second conditional theorem uses finite Fourier certificates and an assumed terminal form of the Bradshaw Grujic blow up criterion.
A longer route, organized around nine records, tries to derive such bounds from the equations. It comes with no go results and admission rules for shortcuts, but it establishes neither closure and neither theorem uses it. Both conditional theorems are formalized in Lean with Mathlib.
What it shows
- Global existence and pointwise smoothness from a single spectral closure hypothesis.
- For flows that never vanish, that hypothesis is equivalent to global existence.
- A second conditional theorem through finite Fourier certificates and an assumed terminal criterion.
- No go results and admission rules for shortcuts on a record based route.
- Lean proofs of both conditional theorems.
What it does not claim
No unconditional Clay claim. Neither closure is proved for arbitrary data, and the main theorem does not reduce regularity to an easier estimate. The match between the terminal hypothesis and the published Bradshaw Grujic criterion is assumed, not proved.
Cite
Tsiokos, I. (2026). A Conditional Spectral-Closure Theorem for Three-Dimensional Navier-Stokes. Zenodo. https://doi.org/10.5281/zenodo.23086987