A high-frequency tail condition and a diagnostic iteration for the Navier--Stokes equations

Abstract

We consider Leray solutions of the three--dimensional incompressible Navier--Stokes equations on $\R^3$ with smooth, rapidly decaying initial data. The analysis is based on a frequency decomposition into low and high modes via the cutoffs $\A_R=\phi(|D|/R)$ and $\A^R=I-\A_R$. Combining the energy inequality with Bernstein estimates yields uniform control of the low--frequency component $\A_R\u$. For the high--frequency component we assume a quantitative \emph{turbulence condition}, requiring that the solution possesses a non--negligible high--frequency tail in $L^\infty$ (in fact, it suffices to impose this condition only on a terminal time layer near a putative blow--up time). Under this hypothesis we introduce a time--localized diagnostic Picard iteration adapted to $\A^R\u$. Using a uniform $L^\infty$ estimate of Giga--Inui--Matsui type (with the cutoff $\A^R$) together with high--frequency heat--flow decay, we show that the iteration is contractive and converges to $\A^R\u$, providing a uniform bound for $\A^R\u$ up to the maximal time of boundedness. Consequently, the turbulence regime is incompatible with finite--time blow--up: any Leray solution satisfying the turbulence condition is bounded, and hence smooth, for all times (equivalently, it cannot blow up in finite time).