Regularity criteria for 3D Navier–Stokes equations in terms of finite frequency parts of velocity in Ḃ∞, ∞−1. (January 2020) - Ri, Myong-Hwan
Abstract: In this paper, we prove that a Leray–Hopf weak solution u to 3D Navier–Stokes equations is regular if L ∞ ( 0, T ; B ̇ ∞, ∞ − 1 ) -norm of a suitable low frequency part of u is bounded by a scaling invariant constant depending on the kinematic viscosity ν and initial value u 0 . Moreover, we prove that a Leray–Hopf weak solution is regular if its medium frequency part with Fourier modes between k ∕ 2 and k for a sufficiently high wavenumber k has small B ̇ ∞, ∞ − 1 -norm. Our results imply that energy concentration at sufficiently high wavenumber bands bringing about singularity of the incompressible Navier–Stokes flow can be prevented by an "energy threshold" at a lower wavenumber band.