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 $$B^{-1}_{\infty ,\infty }$$ 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.