Frequency separation approaches in the study of Navier-Stokes equation

We present two frequency separation approaches in the study of Navier-Stokes equation. One is designed by accounting the competition between the nonlinear term and the linear dissipation term. It leads to improved regularity criterion which only poses condition on the low modes part of the solution. The other one arises from the parabolic structure of the linear part of the Navier-Stokes equation, which can be used to study the long time behavior of the solution.