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

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 $B_{\infty ,\infty }^{-1}$ and also a lower bound on the decaying rate of $L^p$ norms $2\le p<3$ 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.

various minor improvements and added acknowledgments