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

We consider Leray solutions of the three–dimensional incompressible Navier–Stokes equations on ${\mathbb{R}}^3$ with smooth, rapidly decaying initial data. The analysis is based on a frequency decomposition into low and high modes via the cutoffs ${\text{A}}_R=\phi (\mathrm{\mid }D\mathrm{\mid }\mathrm{/}R)$ and ${\text{A}}^R=I-{\text{A}}_R$. 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 $L^{\mathrm{\infty }}$ (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 $L^{\mathrm{\infty }}$ estimate of Giga–Inui–Matsui type (with the cutoff ${\text{A}}^R$) 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).