Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Estimates for the two-dimensional Navier-Stokes equations in terms of
  the Reynolds number   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.

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The categories of this paper are: cond-mat
29
The categories of this paper are: funct-an
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The categories of this paper are: hep-th
35
The categories of this paper are: mtrl-th
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The categories of this paper are: q-fin
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The categories of this paper are: nucl-ex
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The categories of this paper are: solv-int
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The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: eess
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The categories of this paper are: bayes-an
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The categories of this paper are: comp-gas
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The categories of this paper are: chem-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: cmp-lg
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The categories of this paper are: stat