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 of R3, or on the torus R3/Z3. Denoting the Fourier transform by u ̂ = Fu, the energy spectrum of u(·, t) is the spherical integral E(κ, t) = |k|=κ |û(k, t)|2dS(k), 0 ≤ κ <∞, or alternatively, a suitable approximate sum. An argument involking scale invariance and dimensional analysis given by Kolmogorov [7, 9] and Obukhov [14] predicts that in three dimensions large Reynolds number solutions of the Navier- Stokes equations should obey E(κ, t) ∼ C0ε2/3κ−5/3 over an inertial range κ1 ≤ κ ≤ κ2, at least in an average sense. We derive a global estimate on weak solutions in the norm ‖F∂xu(·, t)‖ ∞ which gives bounds on a solution’s ability to satisfy the Kolmogorov law. 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