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 $\R^3$ with smooth, rapidly decaying initial data. The analysis is based on a frequency decomposition into low and high modes via the cutoffs $\A_R=ϕ(|D|/R)$ 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).

💡 Research Summary

The paper addresses the long‑standing open problem of finite‑time singularity formation for three‑dimensional incompressible Navier–Stokes equations by introducing a quantitative “high‑frequency tail” condition and a novel diagnostic Picard iteration. The authors consider Leray weak solutions with smooth, rapidly decaying initial data u₀∈𝒮(ℝ³). They split the velocity field u into low‑frequency and high‑frequency parts using Fourier cut‑offs A_R = φ(|D|/R) and A^R = I−A_R, where φ is a smooth compactly supported function equal to one on a unit ball. Standard energy inequality together with Bernstein estimates give a uniform L∞ bound for the low‑frequency component A_R u that grows at most like R^{3/2}.

The central hypothesis, called the turbulence (or high‑frequency tail) condition, asserts that for a sufficiently large scale R, integers k≥2, exponent p∈