二维纳维-斯托克斯方程的Reynolds number估计
Estimates for the two-dimensional Navier-Stokes equations in terms of the Reynolds number
摘要：
传统的Navier-Stokes分析发现估计的Grashof数$\bG$，其性质取决于强迫的比例粘度$ν$，这意味着它是很难进行比较与其他结果表示的雷诺数$\Rey$，其性质取决于流体响应的强制。本文的第一个任务是将Doering和Foias的\cite{DF}方法应用于周期域$[0,L]^{2}$上的二维纳维-斯托克斯方程，通过估计物理相关量，特别是长时间平均$\left<\cdot\right>$，用雷诺数$\Rey = U\ell/ν$表示，其中$U^{2}= L^{-2}\left<\|\bu\|_{2}^{2}\right>$和$\ell$是强迫尺度。特别是，Constantin-Foias-Temam上界\cite{CFT}的吸引子维度转换为$a_{\ell}^{2}\Rey(1 + \ln\Rey)^{1/3}$，而逆Kraichnan长度的估计是$(a_{\ell}^{2}\Rey)^{1/2}$，其中$a_{\ell}$是强迫的纵横比。基于时间平均值并与更高导数相关联的其他逆长度尺度以类似的方式估计。第二个任务是解决不稳定性的问题：它显示了时间轴是如何被分解成非常短的间隔，在这些间隔上，各种量都有下限，大于长时间平均值，这些平均值本身被更长，更安静的时间间隔所散布。
英文摘要：
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 $ν$, 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 $\Rey = U\ell/ν$, 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.
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