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

TL;DR

This paper introduces a turbulence condition and a diagnostic iteration for the Navier--Stokes equations, demonstrating that solutions satisfying this turbulence regime cannot develop finite-time singularities, thus ensuring smoothness.

Contribution

It proposes a new turbulence tail condition and an iterative method to control high-frequency components, preventing finite-time blow-up in solutions.

Findings

Turbulence condition implies solutions are globally bounded.

The diagnostic iteration converges under the turbulence hypothesis.

Solutions with turbulence tail are proven to be smooth for all times.

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).