Gibbon, J. D.; Pavliotis, G. A. Estimates for the two-dimensional Navier-Stokes equations in terms of the Reynolds number. (English) Zbl 1144.81348 J. Math. Phys. 48, No. 6, 065202, 14 p. (2007). Summary: The tradition in Navier-Stokes analysis of finding estimates in terms of the Grashof number \(Gr\), whose character depends on the ratio of the forcing to the viscosity \(v\), means that it is difficult to make comparisons with other results expressed in terms of Reynolds number \(Re\), whose character depends on the fluid response to the forcing. The first task of this paper is to apply the approach of C. R. Doering and J. D. Foias [Physica D 165, No. 3–4, 163–175 (2002)] 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 \(Re = U\ell/\nu\), where \(U^2= L^{-2}\left<\|u\|_2^2\right>\) and \(\ell\) is the forcing scale. In particular, the Constantin-Foias-Temam upper bound [Physica D 30, No. 3, 284–296 (1988; Zbl 0658.58030)] on the attractor dimension converts to \(a_{\ell}^{2}Re(1 + \ln Re)^{1/3}\), while the estimate for the inverse Kraichnan length is \((a_{\ell}^2 Re)^{1/2}\), 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. Cited in 4 Documents MSC: 76D05 Navier-Stokes equations for incompressible viscous fluids 35Q30 Navier-Stokes equations 37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems 37N10 Dynamical systems in fluid mechanics, oceanography and meteorology 76F20 Dynamical systems approach to turbulence Citations:Zbl 0658.58030 PDFBibTeX XMLCite \textit{J. D. Gibbon} and \textit{G. A. Pavliotis}, J. Math. Phys. 48, No. 6, 065202, 14 p. (2007; Zbl 1144.81348) Full Text: DOI arXiv References: [1] DOI: 10.1016/0167-2789(88)90022-X · Zbl 0658.58030 [2] Constantin P., Navier-Stokes Equations (1988) · Zbl 0687.35071 [3] DOI: 10.1017/CBO9780511546754 [4] DOI: 10.1007/978-1-4684-0313-8 [5] DOI: 10.1512/iumj.1993.42.42039 · Zbl 0796.35128 [6] Landau L. D., Fluid Mechanics (1986) [7] DOI: 10.1023/A:1012984210582 · Zbl 0995.35051 [8] DOI: 10.1016/j.physd.2006.06.012 · Zbl 1331.76051 [9] DOI: 10.1137/1.9781611970050 [10] Foias C., Rend. Sem. Mat. Univ. Padova 48 pp 219– (1972) [11] Foias C., Rend. Sem. Mat. Univ. Padova 49 pp 9– (1973) [12] DOI: 10.1007/BF02411822 · Zbl 0344.76015 [13] Ladyzhenskaya O. A., The Mathematical Theory of Viscous Incompressible Flow (1963) · Zbl 0121.42701 [14] DOI: 10.1016/0045-7825(91)90003-O · Zbl 0760.76044 [15] DOI: 10.1051/m2an:2001117 · Zbl 0987.35122 [16] DOI: 10.1016/0167-2789(91)90098-T · Zbl 0728.76030 [17] DOI: 10.1017/CBO9780511608803 · Zbl 0838.76016 [18] DOI: 10.1016/0167-2789(95)00302-9 · Zbl 1031.76516 [19] DOI: 10.1023/A:1022814702548 · Zbl 1028.76014 [20] DOI: 10.1023/A:1015782025005 · Zbl 1158.76327 [21] DOI: 10.1063/1.1762301 [22] DOI: 10.1017/S0022112002001386 · Zbl 1029.76025 [23] DOI: 10.1098/rspa.1949.0136 · Zbl 0036.25602 [24] DOI: 10.1017/S0022112071002581 [25] DOI: 10.1017/S0022112091001830 · Zbl 0717.76061 [26] Frisch U., Turbulence: The Legacy of A. N. Kolmogorov (1995) · Zbl 0832.76001 [27] DOI: 10.1103/PhysRevA.23.2673 [28] DOI: 10.1007/s00205-005-0382-5 · Zbl 1129.76014 [29] DOI: 10.1103/PhysRevLett.96.084502 [30] DOI: 10.1063/1.869840 [31] DOI: 10.1103/PhysRevE.64.035301 [32] DOI: 10.1103/PhysRevE.64.036302 [33] DOI: 10.1016/S0167-2789(02)00427-X · Zbl 0997.35046 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.