Global regularity for solutions of the Navier-Stokes equation sufficiently close to being eigenfunctions of the Laplacian

TL;DR

This paper establishes a new regularity criterion for Navier-Stokes solutions close to Laplacian eigenfunctions, linking regularity to spectral properties and turbulence theory, improving previous control conditions.

Contribution

It introduces a scale-critical regularity criterion involving the Laplacian eigenfunction proximity, refining earlier criteria based on Sobolev norms.

Findings

New regularity criterion involving spectral proximity

Improved control over solutions near Laplacian eigenfunctions

Potential connection to turbulence spectra

Abstract

In this paper, we will prove a new, scale critical regularity criterion for solutions of the Navier--Stokes equation that are sufficiently close to being eigenfunctions of the Laplacian. This estimate improves previous regularity criteria requiring control on the $\dot{H}^\alpha$ norm of $u,$ with $2\leq \alpha<\frac{5}{2},$ to a regularity criterion requiring control on the $\dot{H}^\alpha$ norm multiplied by the deficit in the interpolation inequality for the embedding of $\dot{H}^{\alpha-2}\cap\dot{H}^{\alpha} \hookrightarrow \dot{H}^{\alpha-1}.$ This regularity criterion suggests, at least heuristically, the possibility of some relationship between potential blowup solutions of the Navier--Stokes equation and the Kolmogorov-Obhukov spectrum in the theory of turbulence.