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 ${\mathbb{R}}^3$, or on the torus ${\mathbb{R}}^3\mathrm{/}{\mathbb{Z}}^3$. The {\it energy spectrum} of $u(\cdot ,t)$ is the spherical integral $$E(\kappa ,t)={\int }_{\mathrm{\mid }k\mathrm{\mid }=\kappa }\mathrm{\mid }\widehat{u}(k,t){\mathrm{\mid }}^{2}dS(k),0\le \kappa <\mathrm{\infty },$$ 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 $$E(\kappa ,t)\sim C_0{\epsilon }^{2\mathrm{/}3}{\kappa }^{-5\mathrm{/}3}$$ over an inertial range ${\kappa }_1\le \kappa \le {\kappa }_2$, at least in an average sense. We give a global estimate on weak solutions in the norm $\mathrm{\parallel }\mathcal{F}{\mathrm{\partial }}_{x}u(\cdot ,t){\mathrm{\parallel }}_{\mathrm{\infty }}$ 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.