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