Bounds on moments of the energy spectrum for weak solutions of the three-dimensional Navier–Stokes equations

We consider weak solutions of the incompressible Navier–Stokes equations in a finite three-dimensional volume without boundaries. For flows driven by a time-independent body force, we focus on some long-time-averaged ratios of norms of derivatives of the velocity field corresponding to moments of the Fourier energy spectrum of the solutions. The rigorous bounds derived here—without making any additional assumptions on regularity of solutions—are consistent with an asymptotic k−8/3 energy spectrum. In contrast, if spatial fluctuations in velocity gradients are suppressed by assumption in the analysis, then the bounds are consistent with the k−5/3 turbulent Kolmogorov energy spectrum. These results are interpreted and discussed in the context of spatial intermittency and the energy cascade picture of turbulent dynamics.