Estimates for the two-dimensional Navier-Stokes equations in terms of the Reynolds number

Abstract

The tradition in Navier-Stokes analysis of finding estimates in terms of the Grashof number $\bG$, whose character depends on the ratio of the forcing to the viscosity $\nu$, means that it is difficult to make comparisons with other results expressed in terms of Reynolds number $\Rey$, whose character depends on the fluid response to the forcing. The first task of this paper is to apply the approach of Doering and Foias \cite{DF} to the two-dimensional Navier-Stokes equations on a periodic domain $[0,L]^{2}$ by estimating quantities of physical relevance, particularly long-time averages $\left<\cdot\right>$, in terms of the Reynolds number $\Rey = U\ell/\nu$, where $U^{2}= L^{-2}\left<\|\bu\|_{2}^{2}\right>$ and $\ell$ is the forcing scale. In particular, the Constantin-Foias-Temam upper bound \cite{CFT} on the attractor dimension converts to $a_{\ell}^{2}\Rey(1 + \ln\Rey)^{1/3}$, while the estimate for the inverse Kraichnan length is $(a_{\ell}^{2}\Rey)^{1/2}$, where $a_{\ell}$ is the aspect ratio of the forcing. Other inverse length scales, based on time averages, and associated with higher derivatives, are estimated in a similar manner. The second task is to address the issue of intermittency : it is shown how the time axis is broken up into very short intervals on which various quantities have lower bounds, larger than long time-averages, which are themselves interspersed by longer, more quiescent, intervals of time.