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
H
˙
α
\dot {H}^\alpha
norm of
u
,
u,
with
2
≤
α
>
5
2
,
2\leq \alpha >\frac {5}{2},
to a regularity criterion requiring control on the
H
˙
α
\dot {H}^\alpha
norm multiplied by the deficit in the interpolation inequality for the embedding of
H
˙
α
−
2
∩
H
˙
α
↪
H
˙
α
−
1
.
\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.