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 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.