A Beale-Kato-Majda criterion with optimal frequency and temporal localization

TL;DR

This paper establishes a new Beale-Kato-Majda criterion for 3D Navier-Stokes equations that uses optimal frequency and time localization, improving understanding of conditions leading to solution regularity or blowup.

Contribution

It introduces a frequency-localized criterion based on controlling Fourier modes below a critical frequency, with explicit dependence on time scales, advancing the analysis of Navier-Stokes regularity.

Findings

Provides a frequency-localized regularity criterion in $B^{-1}_{ abla, abla}$ space.

Derives a lower bound on the decay rate of $L^p$ norms for potential blowup solutions.

Introduces new estimates for cutoff dissipation and energy at small time scales.

Abstract

We obtain a Beale-Kato-Majda-type criterion with optimal frequency and temporal localization for the 3D Navier-Stokes equations. Compared to previous results our condition only requires the control of Fourier modes below a critical frequency, whose value is explicit in terms of time scales. As applications it yields a strongly frequency-localized condition for regularity in the space $B^{-1}_{\infty,\infty}$ and also a lower bound on the decaying rate of $L^p$ norms $2\leq p <3$ for possible blowup solutions. The proof relies on new estimates for the cutoff dissipation and energy at small time scales which might be of independent interest.