Bounds on Kolmogorov spectra for the Navier - Stokes equations

TL;DR

This paper establishes bounds on the energy spectrum of solutions to the Navier-Stokes equations, connecting mathematical analysis with Kolmogorov's turbulence theory, and provides estimates on the inertial range and spectral regime duration.

Contribution

It offers rigorous bounds on the Kolmogorov spectrum for Navier-Stokes solutions, linking scale invariance with mathematical estimates of turbulence spectra.

Findings

Bounds on the energy spectrum consistent with Kolmogorov's law

Estimates on the inertial range limits

Upper bounds on the duration of Kolmogorov spectral regime

Abstract

Let $u(x,t)$ be a (possibly weak) solution of the Navier - Stokes equations on all of ${\mathbb R}^3$, or on the torus ${\mathbb R}^3/ {\mathbb Z}^3$. The {\it energy spectrum} of $u(\cdot,t)$ is the spherical integral \[ E(\kappa,t) = \int_{|k| = \kappa} |\hat{u}(k,t)|^2 dS(k), \qquad 0 \leq \kappa < \infty, \] or alternatively, a suitable approximate sum. An argument involking scale invariance and dimensional analysis given by Kolmogorov (1941) and Obukhov (1941) predicts that large Reynolds number solutions of the Navier - Stokes equations in three dimensions should obey \[ E(\kappa, t) \sim C_0\varepsilon^{2/3}\kappa^{-5/3} \] over an inertial range $\kappa_1 \leq \kappa \leq \kappa_2$, at least in an average sense. We give a global estimate on weak solutions in the norm $\|{\mathcal F}\partial_x u(\cdot, t)\|_\infty$ which gives bounds on a solution's ability to satisfy the Kolmogorov law. A subsequent result is for rigorous upper and lower bounds on the inertial range, and an upper bound on the time of validity of the Kolmogorov spectral regime.