Regularity for 3D NSE and related problems: the method of wavenumber splitting

The problems of blow-up for 3D Navier-Stokes equations (NSE) have been extensively studied for decades using dierent techniques. By using a method of wavenumber splitting which originated from Kolmogorov's theory of turbulence, we obtained a new regularity criterion for the 3D NSE. The new criterion improves the classical Prodi-Serrin, Beale-Kato-Majda criteria and their extensions. We also discuss the notion of degrees of freedom for fluid fows and connect is with the determining wavenumber whose average is bounded by Kolmogorov's dissipation wavenumber.