Kolmogorov Theory via Finite-time Averages

Several relations from the Kolmogorov theory of fully-developed threedimensional turbulence are rigorously established for finite-time averages over LerayHopf weak solutions of the Navier-Stokes equations. The Navier-Stokes equations are considered with periodic or no-slip boundary conditions and an external forcing term. The main parameter is the Grashof number associated with the forcing term. The relations rigorously proved in this article include estimates for the energy dissipation rate, the Kolmogorov wavenumber, the Taylor wavenumber, the Reynolds number, and the energy cascade process. For some estimates the averaging time depends on the macroscale wavenumber and the kinematic viscosity alone, while for others such as the Kolmogorov energy dissipation law and the energy cascade, the estimates depend also on the Grashof number. As compared with earlier works by some of the authors the more physical concept of finite time average is replacing the concept of infinite time average used before.