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

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.