Regularity criteria for 3D Navier–Stokes equations in terms of a mid frequency part of velocity in B˙-1 ∞,∞

We prove that a Leray–Hopf weak solution u to 3D Navier–Stokes equations is regular in ( 0 , T ] {(0,T]} if the L 1 ⁢ ( 0 , T ; B ˙ ∞ , ∞ - 1 ) {L^{1}(0,T;\dot{B}^{-1}_{\infty,\infty})} - or L 2 ⁢ ( 0 , T ; B ˙ ∞ , ∞ - 1 ) {L^{2}(0,T;\dot{B}^{-1}_{\infty,\infty})} -norm of u k / 2 , k {u_{k/2,k}} , the mid frequency part of Fourier modes k 2 ≤ | ξ | < k {\frac{k}{2}\leq|\xi|<k} , is small depending on the kinematic viscosity ν, initial value u 0 {u_{0}} and the maximum of an averaged energy dissipation rate A ≡ sup t ∈ ( 0 , T ) ⁡ ( ν ⁢ t - 1 ⁢ ∫ 0 t ∥ ∇ ⁡ u ∥ 2 ⁢ 𝑑 τ ) A\equiv\sup_{t\in(0,T)}\biggl{(}\nu t^{-1}\int_{0}^{t}\lVert\nabla u\rVert^{2}% \,d\tau\biggr{)} for some k ≥ k 0 ⁢ ( ν , u 0 , A ) {k\geq k_{0}(\nu,u_{0},A)} . In particular, when a sufficiently high frequency part of u 0 {u_{0}} decays fast at an exponential rate, then we obtain regularity conditions in terms of smallness of the L 1 ⁢ ( 0 , T ; B ˙ ∞ , ∞ - 1 ) {L^{1}(0,T;\dot{B}^{-1}_{\infty,\infty})} - or L 2 ⁢ ( 0 , T ; B ˙ ∞ , ∞ - 1 ) {L^{2}(0,T;\dot{B}^{-1}_{\infty,\infty})} -norm of u k / 2 , k {u_{k/2,k}} , which involve only the known data ν and u 0 {u_{0}} .