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 $N>0$, let $w_N$ be the sum of all spectral components of the velocity fields whose all three wave numbers are greater than $N$ absolutely. Then, we show that for any $N>0$, the finiteness of the Serrin type norm of $w_N$ 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 $w_N$ and corresponding energy spectrum, in some sense, roughly decays slower than $\kappa^{-2}$ 
Namkwon Kim, Minkyu Kwak and Minha Yoo
math.AP