Global Regularity for the Three‑Dimensional Incompressible Navier–Stokes Equations at the Critical Scale
Summary
The manuscript presents a scale‑invariant, PDE‑internal route to global regularity for 3D incompressible Navier–Stokes: an \(\varepsilon\)‑regularity lemma at the critical vorticity \(L^{3/2}\) scale, a quantitative bridge to a small \(BMO^{-1}\) velocity slice, extraction of a minimal ancient critical element, and its elimination via small‑data well‑posedness in \(BMO^{-1}\) combined with backward uniqueness.
Abstract
We prove global regularity for the three‑dimensional incompressible Navier–Stokes equations on \(\mathbb R^3\) with smooth, divergence‑free initial data. The argument is strictly scale‑invariant and PDE‑internal. It has four components: (i) a vorticity‑based \(\varepsilon\)‑regularity lemma at the critical \(L^{3/2}\) Morrey scale; (ii) a quantitative bridge from critical vorticity control on parabolic cylinders to a small \(BMO^{-1}\) time slice for the velocity; (iii) compactness extraction of a minimal ancient critical element together with a De~Giorgi density‑drop that pins the threshold; and (iv) elimination of the critical element by combining small‑data global well‑posedness in \(BMO^{-1}\) with backward uniqueness. All estimates respect the parabolic scaling and no extraneous structural hypotheses are imposed.
Key Points
- Critical‑scale \(\varepsilon\)‑regularity for vorticity yielding local \(L^{\infty}\) bounds.
- Carleson/tent‑space bridge: vorticity \(L^{3/2}\) control implies a small \(BMO^{-1}\) slice.
- Critical element method with De~Giorgi density‑drop to pin the threshold.
- Rigidity via small‑data \(BMO^{-1}\) global theory and backward uniqueness.