Regularity conditions of 3D Navier–Stokes flow in terms of large spectral components

We develop Ladyzhenskaya–Prodi–Serrin type spectral regularity criteria for 3D incompressible Navier–Stokes equations in a torus. Concretely, for any N>0N>0, let wNwN be the sum of all spectral components of the velocity fields whose wave numbers |ki|>N|ki|>N for all i=1,2,3i=1,2,3. Then, we show that for any N>0N>0, the finiteness of the Serrin type norm of wNwN implies the regularity of the flow. It implies that if the flow breaks down in a finite time, the energy of the velocity fields cascades down to the arbitrarily large spectral components of wNwN and corresponding energy spectrum, in some sense, roughly decays slower than κ−2κ−2.