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

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 $2\le \alpha <\frac{5}{2},$ 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.

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