Three Pillars of 3D Incompressible Navier–Stokes Theory: Kato Local Existence, Fujita–Kato Small-Data Globality, and the Beale–Kato–Majda Continuation Criterion (While the Roof Remains Open)
Year: 2026
Type: preprint
Abstract: Expository classical compilation presenting, with expanded proof sketches and references to the original works, the three standard pillars of 3D incompressible Navier–Stokes theory on the torus: (i) Kato local existence and lifespan bound for arbitrary data in Hs (s > 5/2), with full Kato–Ponce both terms + absorb, mollified/Galerkin + Aubin–Lions, and L² uniqueness with ∇v; (ii) Fujita–Kato global regularity for small data in the critical space Ḣ1/2 with smallness linear in ν (‖u0‖Ḣ1/2 ≤ ε ν); (iii) the Beale–Kato–Majda continuation criterion in one-directional form (∫ ‖ω‖∞ dt < ∞ ⇒ T* > T), with log bound including +‖ω‖L². Smoothness stated for t > 0. Status matrix: Pillar I Unconditional Local / Pillar II Unconditional Small-Data Global / Pillar III One-directional / Clay Statement (B) Open. No unconditional large-data claim. Supplementary note (Littlewood_Paley_Conditional_and_SmallData.tex/.pdf): the critical-Besov small-data global regularity theorem and Kozono–Taniuchi continuation in matching one-directional / C/ν form. Companion geometric draft: 10.5281/zenodo.24079623. License: Apache-2.0. Creator: Frohman, Benjamin Stanley (ORCID 0009-0006-7068-3718). Keep unpublished until the author explicitly confirms.
Author: Frohman, Benjamin Stanley
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225644900
Cites:
Cited by:
Related to: