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In this paper, we will prove a new, scale critical regularity criterion forsolutions of the Navier--Stokes equation that are sufficiently close to beingeigenfunctions of the Laplacian. This estimate improves previous regularitycriteria 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 inequalityfor the embedding of $\dot{H}^{\alpha-2}\cap\dot{H}^{\alpha} \hookrightarrow\dot{H}^{\alpha-1}.$ This regularity criterion suggests, at leastheuristically, the possibility of some relationship between potential blowupsolutions of the Navier--Stokes equation and the Kolmogorov-Obhukov spectrum inthe theory of turbulence.

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