Bounds on Kolmogorov spectra for the Navier–Stokes equations

Let u ( x , t ) be a (possibly weak) solution of the Navier–Stokes equations on all R 3 , or on the torus R 3 / Z 3 . Denoting the Fourier transform by u ˆ = F u , the energy spectral function of u ( ⋅ , t ) is the spherical integral E ( κ , t ) = ∫ | k | = κ | u ˆ ( k , t ) | 2 d S ( k ) , 0 ≤ κ < ∞ , or alternatively, a suitable approximate sum. An argument invoking scale invariance and dimensional analysis given by Kolmogorov (1941) [1,3] and Obukhov (1941) [4] predicts that, in three dimensions, large Reynolds number solutions of the Navier–Stokes equations should obey E ( κ , t ) ∼ C 0 ε 2 / 3 κ − 5 / 3 over an inertial range κ 1 ≤ κ ≤ κ 2 , at least in an average sense. We derive a global estimate on weak solutions in the norm ‖ F ∂ x u ( ⋅ , t ) ‖ ∞ which gives bounds on a solution’s ability to satisfy this prediction. A subsequent result gives rigorous upper and lower bounds on the inertial range, and an upper bound on the time of validity of the Kolmogorov spectral regime.