On Energy Cascades in the Forced 3D Navier–Stokes Equations

We show—in the framework of physical scales and \((K_1,K_2)\)-averages—that Kolmogorov’s dissipation law combined with the smallness condition on a Taylor length scale is sufficient to guarantee energy cascades in the forced Navier–Stokes equations. Moreover, in the periodic case we establish restrictive scaling laws—in terms of Grashof number—for kinetic energy, energy flux, and energy dissipation rate. These are used to improve our sufficient condition for forced cascades in physical scales.