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JOURNAL OF MATHEMATICAL PHYSICS 48, 065202 共2007兲 Estimates for the two-dimensional Navier–Stokes equations in terms of the Reynolds number J. D. Gibbon and G. A. Pavliotisa兲 Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom 共Received 17 May 2006; accepted 25 August 2006; published online 4 June 2007兲 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 ␯, 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 Doering and Foias 关C. R. Doering and C. Foias, J. Fluid Mech. 467, 289 共2002兲兴 to the twodimensional Navier–Stokes equations on a periodic domain 关0 , L兴2 by estimating quantities of physical relevance, particularly long-time averages 具·典, in terms of the Reynolds number Re= U ᐉ / ␯, where U2 = L−2具储u储22典 and ᐉ is the forcing scale. In particular, the Constantin–Foias–Temam upper bound 关P. Constantin, C. Foias, and R. Temam, Physica D 30, 284 共1988兲兴 on the attractor dimension converts to a2ᐉRe共1 + ln Re兲1/3, while the estimate for the inverse Kraichnan length is 共a2ᐉRe兲1/2, where aᐉ 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. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2356912兴 I. INTRODUCTION A. General introduction In the last two decades the notion of global attractors in parabolic partial differential equations has become a well-established concept.1–4 The general nature of the dynamics on the attractor A, in a time averaged sense, can roughly be captured by identifying sharp estimates of the Lyapunov 共or fractal or Hausdorff兲 dimension of A, or the number of determining modes,5 with the number of degrees of freedom. Introduced by Landau,6 this latter idea says that in a dynamical system of spatial dimension d of scale L, the number of degrees of freedom N is roughly defined to be that number of smallest eddies or features of scale ␭ and volume ␭d that fit into the system volume Ld, N⬃ 冉冊 L ␭ d . 共1.1兲 This is the origin of the much-quoted N ⬃ Re9/4 result associated with the three-dimensional Navier–Stokes equations which rests on taking ␭ ⬃ ␭k ⬃ L Re−3/4, where ␭k is the Kolmogorov length scale. In the absence of a proof of existence and uniqueness of solutions of the threedimensional Navier–Stokes equations, at best this is no more than a rule of thumb result. It rests on a more solid and rigorous foundation, however, for the closely related three-dimensional a兲 Electronic mail: g.pavliotis@imperial.ac.uk 0022-2488/2007/48共6兲/065202/14/$23.00 48, 065202-1 © 2007 American Institute of Physics Downloaded 07 Jun 2007 to 155.198.190.103. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 065202-2 J. Math. Phys. 48, 065202 共2007兲 J. D. Gibbon and G. A. Pavliotis LANS-␣ equations for which Foias, Holm, and Titi7 have proved existence and uniqueness of solutions. Following on from this, Gibbon and Holm8 have demonstrated that the dimension of the global attractor for this system has an upper bound proportional to Re9/4. An important milestone has been passed recently in another closely related problem with the establishment by Cao and Titi9 of an existence and uniqueness proof for Richardson’s three-dimensional primitive equations for the atmosphere. For the Navier–Stokes equations the idea sits more naturally in studies in the two-dimensional context. The existence and uniqueness of solutions has been a closed problem for many decades and the nature of the global attractor has been well-established.1–5,10–14 While the two- and threedimensional equations have the same velocity formulation, in reality, the former have a tenuous connection with the latter because of the absence of the drastic property of vortex stretching. As a result, the presence of vortex stretching in three dimensions, and perhaps other more subtle properties, have set up seemingly unsurmountable hurdles even on periodic boundary conditions. For problems on non-periodic boundaries, such as lid-driven flow, solving the two-dimensional Navier–Stokes equations is a technically more demanding problem—see some references in Refs. 10, 15, and 16. The sharp estimate found by Constantin, Foias, and Temam1 for the Lyapunov dimension of the global attractor A expressed in terms of the Grashof number Gr dL共A兲 艋 c1Gr2/3共1 + ln Gr兲1/3 , 共1.2兲 has been one of the most significant results in two-dimensional Navier–Stokes analysis on a 2 periodic domain ⍀ = 关0 , L兴per . The traditional length scale in the two-dimensional Navier–Stokes equations is the Kraichnan length, ␩k, which plays an equivalent role in two dimensions to that of the Kolmogorov length, ␭k, which is more important in three dimensions. In two dimensions, ␩k and ␭k are defined respectively in terms of the enstrophy and energy dissipation rates ⑀ens and ⑀, ⑀ens = ␯L−2 冓冕 ⍀ 冔 ⑀ = ␯L−2 兩ⵜ␻兩2dV , 冓冕 冔 ⍀ 兩␻兩2dV , 共1.3兲 where the pair of brackets 具·典 denote a long-time average defined as2,3,10–13 具g共·兲典 = lim lim sup t→⬁ g共0兲 1 t 冕 t g共␶兲d␶ . 共1.4兲 0 −1 The inverse Kraichnan length ␩−1 k and the inverse Kolmogorov length ␭k are defined in terms of ⑀ens and ⑀ as ␩−1 k = 冉 冊 ⑀ens ␯3 1/6 , ␭−1 k = 冉冊 ⑀ ␯3 1/4 . 共1.5兲 It has been shown by Constantin, Foias, and Temam1 that instead of using an estimate for ⑀ens in terms of Gr, the upper bound for dL can be re-expressed in terms of L␩−1 k 共see other literature on this topic17–19兲 −1 1/3 2 dL 艋 c2共L␩−1 k 兲 兵1 + ln共L␩k 兲其 . 共1.6兲 If dL is identified with the number of degrees of freedom N, this result is consistent with the idea expressed in Eq. 共1.1兲 that in a two-dimensional domain, the average length scale of the smallest vortical feature ␭ can be identified with the Kraichnan length ␩k, to within log corrections. The result in Eq. 共1.2兲 has also been improved by Foias et al.20,21 to an estimate proportional to Gr1/2 共to within logarithmic corrections兲 provided Kraichnan’s theory of fully developed turbulence is implemented.22 While these results display a pleasing convergence between rigorous estimates and scaling methods in the two-dimensional case, the tradition in Navier–Stokes analysis of finding estimates Downloaded 07 Jun 2007 to 155.198.190.103. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 065202-3 Estimates for the 2D Navier-Stokes equations J. Math. Phys. 48, 065202 共2007兲 in terms of the Grashof number Gr, whose character depends on the ratio of the forcing to the viscosity ␯, means that it is difficult to compare with the results of scaling theories whose results are expressed in terms of Reynolds number. One of the tasks of this paper is to estimate quantities of physical relevance, particularly long-time averages, in terms of the Reynolds number, whose character depends on the fluid response to the forcing, and which is intrinsically a property of Navier–Stokes solutions. Doering and Foias23 have addressed this problem and have shown that in the limit Gr→ ⬁, solutions of the d-dimensional Navier–Stokes equations must satisfy 关This result is not advertised in Ref. 23 but follows immediately from their Eq. 共48兲.兴 Gr 艋 c共Re2 + Re兲, 共1.7兲 while the energy dissipation rate ⑀ has a lower bound proportional to Gr. The problem, however, is not as simple as replacing standard estimates in terms of Gr by Re2 from Eq. 共1.7兲. Estimates such as that for dL in Eq. 共1.2兲 and the inverse Kraichnan and Kolmogorov lengths defined in Eq. 共1.5兲, depend upon long time averages of the enstrophy and energy dissipation rates defined in Eq. 共1.3兲. Other estimates of inverse length scales 共to be discussed in Sec. I B兲 also depend upon long time averages. When estimated in terms of Re all these turn out to be better than straight substitution using Eq. 共1.7兲. These results are summarized in Sec. I B and worked out in detail in Sec. II. The second topic to be addressed in this paper is that of intermittency. Originally this important effect was considered to be a high Reynolds number phenomenon associated with threedimensional Navier–Stokes flows. First discovered by Batchelor and Townsend,24 it manifests itself in violent fluctuations of very short duration in the energy dissipation rate ⑀. These violent fluctuations away from the average are interspersed by quieter, longer periods in the dynamics. This is a well established, experimentally observable phenomenon;25–27 its appearance in systems other than the Navier–Stokes equations has been discussed in an early and easily accessible paper by Frisch and Morf.28 One symptom of its occurrence is the deviation of the “flatness” of a velocity signal 共the ratio of the fourth order moment to the square of the second order moment兲 from the value of 3 that holds for Gaussian statistics. Recent analysis discussing intermittency in three-dimensional Navier–Stokes flows shows that while it may be connected with loss of regularity, the two are subtly different issues.29 This is reinforced by the fact that although solutions of the two-dimensional Navier–Stokes equations remain regular for arbitrarily long times, nevertheless many of its solutions at high Re are known to be intermittent.30–34 While three-dimensional analysis of the problem is based on the assumption that a solution exists,29,35 so that the higher norms can be differentiated, no such assumption is necessary in the two-dimensional case where existence and uniqueness are guaranteed. The result in both dimensions is such that the time-axis is broken up into good and bad intervals: on the latter there exist large lower bounds on certain quantities, necessarily resulting in their extreme narrowness and thus manifesting themselves as spikes in the data. This is summarized in Sec. I B and worked out in detail in Sec. IV. B. Summary and interpretation of results For simplicity the forcing f共x兲 in the two-dimensional Navier–Stokes equations 共div u = 0兲 ut + u · ⵱u = ␯⌬u − ⵜp + f共x兲 共1.8兲 is taken to be divergence-free and smooth of narrow-band type, with a characteristic single lengthscale ᐉ such that23,29,35 储⵱nf储2 ⬇ ᐉ−n储f储2 . 共1.9兲 Moreover, the aspect ratio of the forcing length scale to the box scale is defined as Downloaded 07 Jun 2007 to 155.198.190.103. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 065202-4 J. Math. Phys. 48, 065202 共2007兲 J. D. Gibbon and G. A. Pavliotis aᐉ = L/ ᐉ . 共1.10兲 With f rms = L−d/2 储 f储2, the usual definition of the Grashof number Gr appearing in Eq. 共1.7兲 in d dimensions is Gr = ᐉ3 f rms . ␯2 共1.11兲 The Reynolds number Re in Eq. 共1.7兲 is defined as Re = Uᐉ , ␯ U2 = L−d具储u储22典, 共1.12兲 where 具·典 is the long-time average defined in Eq. 共1.4兲. One of the main results of this paper is the following theorem whose proof is given in Sec. II A. All generic constants are designated as c. Theorem 1.1: Let u共x , t兲 be a solution of the two-dimensional Navier–Stokes equations 共1.8兲 on a periodic domain 关0 , L兴2, and subject to smooth, divergence-free, narrow-band forcing f共x兲. Then estimates in terms of the Reynolds number Re and the aspect ratio aᐉ for the inverse −1 Kraichnan length ␩−1 k , the attractor dimension dL, and the inverse Kolmogorov length ␭k are given by 2 1/2 L␩−1 k 艋 c共aᐉRe兲 , 共1.13兲 dL 艋 ca2ᐉRe关1 + ln Re兴1/3 , 共1.14兲 5/8 L␭−1 k 艋 caᐉRe . 共1.15兲 In the short proof of this theorem in Sec. II A, the estimate for dL in Eq. 共1.14兲 is not reworked from first principles but is derived from a combination of Eqs. 共1.13兲 and 共1.14兲. The result in Eq. 共1.15兲 comes from a Re5/2 bound on 具H1典 and has also recently been found by Alexakis and Doering.36 It implies that L⑀ 艋 caᐉRe−1/2 , U3 共1.16兲 whereas in three dimensions the right hand side is O共1兲. The estimate in Eq. 共1.14兲 is also consistent with the result of Foias et al.20 when their Gr1/2 estimate is converted to one proportioanl to Re. Their estimate, however, was based on the implementation of certain features of the Kraichnan model,22 while Eq. 共1.14兲 is true for all solutions and requires no assumption of fully developed turbulence. 1/2 1/2 The estimates for ␩−1 k and dL are consistent with the long-standing belief that Re ⫻ Re grid points are needed to numerically resolve a flow; indeed, when the aspect ratio is taken into account, Theorem 1.1 is consistent with aᐉRe1/2 ⫻ aᐉRe1/2. However, both these estimates are dependent upon only the time average of low moments of the velocity field. For non-Gaussian flows, low-order moments are not sufficient to uniquely determine the statistics of a flow. Thus it is necessary to find ways of estimating small length scales associated with higher-order moments. In Sec. II B we follow the way of defining inverse length scales associated with derivatives higher than 2, introduced elsewhere,18 by combining the forcing with higher derivatives of the velocity field such that Fn = 冕 ⍀ 共兩⵱nu兩2 + ␶2兩⵱nf兩2兲dV, 共1.17兲 where ␶ = ᐉ2␯−1关Gr共1 + ln Gr兲兴−1/2 is a characteristic time: this choice of ␶ is discussed in Appendix A. The gradient symbol ⵜn within Eq. 共1.17兲 refers to all derivatives of every component of u of Downloaded 07 Jun 2007 to 155.198.190.103. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 065202-5 J. Math. Phys. 48, 065202 共2007兲 Estimates for the 2D Navier-Stokes equations order n in L2共⍀兲. The Fn are used to define a set of time-dependent inverse length scales ␬n,r共t兲 = 冉 冊 Fn Fr 1/2共n−r兲 共1.18兲 . 2n Actually, ␬n,0 behaves as the 2nth moment of the energy spectrum as shown by 冕 冕 ⬁ 2n ␬n,0 = 2␲/L ⬁ k2n共兩û兩2 + ␶2兩f̂兩2兲dVk 2␲/L . 共1.19兲 共兩û兩 + ␶ 兩f̂兩 兲dVk 2 2 2 2共n−1兲 behaves as the 2共n − 1兲th moment of the enMore relevant to the two-dimensional case, ␬n,1 strophy spectrum. Using Landau’s argument the dimension of the global attractor dL共A兲 was identified with the number of degrees of freedom N. In Ref. 19 a definition was introduced to represent the number of degrees of freedom associated with all higher derivatives of the velocity field represented by ␬n,r, which is itself an inverse length. This naturally leads to the definition of the infinite set 2 典. Nn,r = L2具␬n,r 共1.20兲 Using the definition of the quantities ⌳n,0 and ⌳n,1 共n 艌 2兲, ⌳n,0 = 3n − 2 , 2n ⌳n,1 = 3n − 4 , 2共n − 1兲 共1.21兲 the second main result of the paper is a theorem whose proof is given in Sec. II B. Theorem 1.2: Let ␬n,r be the moments of a two-dimensional Navier–Stokes velocity field defined in Eq. 共1.18兲. Then in a two-dimensional periodic box of side L the numbers of degrees of freedom Nn,1 and Nn,0 defined in Eq. 共1.20兲 are estimated as 共n 艌 2兲, Nn,1 艋 cn,1共a2ᐉRe兲⌳n,1共1 + ln Re兲1/2 , 共1.22兲 Nn,0 艋 cn,0共a2ᐉRe兲⌳n,0共1 + ln Re兲1/2 , 共1.23兲 where ⌳n,0 and ⌳n,1 are defined in Eq. 共1.21兲. Note that ⌳2,0 = ⌳2,1 = 1. Thus the estimate for the first in each sequence, N2,1 and N1,0, are of the same order as the estimate for dL, namely a2ᐉRe共1 + ln Re兲1/3 except in the exponent of the logarithm. The exponents in Eqs. 共1.22兲 and 共1.23兲 provide an estimate of the extra resolution that is needed to take account of energy at sub-Kraichnan scales. Notice that in the limit n → ⬁ both exponents converge to 3 / 2. The intermittency results of Sec. IV show that there can exist small intervals of time where 2 that are much larger than the upper bound on the long-time there are large lower bounds on ␬n,1 2 average for 具␬n,1典. Translated into pictorial terms, Fig. 1 in Sec. IV is consistent with the existence of spiky data whose duration must be very short. Estimates are found for the width of these spikes which turn out to be in terms of a negative exponent of Re. II. TIME AVERAGE ESTIMATES IN TERMS OF Re A. Proof of Theorem 1.1 The first step in the proof of Theorem 1.1, which has been expressed in Sec. I B, is to find an upper bound on 具H2典 in terms of Re. Consider the equation for the two-dimensional Navier–Stokes vorticity ␻ = ␻k̂, Downloaded 07 Jun 2007 to 155.198.190.103. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 065202-6 J. Math. Phys. 48, 065202 共2007兲 J. D. Gibbon and G. A. Pavliotis ⳵␻ + u · ⵱␻ = ␯⌬␻ + curl f, ⳵t 共2.1兲 and let Hn be defined by 共n 艌 0兲, Hn = 冕 ⍀ 兩⵱nu兩2dV. 共2.2兲 For a periodic, divergence-free velocity field u, H1 = 冕 ⍀ 兩⵱u兩2dV = 冕 兩␻兩2dV. 共2.3兲 ␻ · curl fdV 共2.4兲 ⍀ Then the evolution equation for H1 is 1 Ḣ1 = − ␯H2 + 2 冕 ⍀ 艋− ␯H2 + 储u储2储⵱2f储2 共2.5兲 艋− ␯H2 + ᐉ−2储u储2储f储2 , 共2.6兲 where the forcing term has been integrated by parts in Eq. 共2.4兲 and the narrow-band property has been used to move from Eq. 共2.5兲 to Eq. 共2.6兲. Using the definitions of Re, Gr, and aᐉ in Eqs. 共1.12兲, 共1.11兲, and 共1.10兲, the long-time average of H2 is estimated as 具H2典 艋 L2ᐉ−6␯2Re Gr 艋ca2ᐉᐉ−4␯2Re3 + O共Re2兲. 共2.7兲 共2.8兲 This holds the key to the three results in Theorem 1.1. 3 −2 The inverse Kraichnan length ␩−6 k = ⑀ens / ␯ with ⑀ens = ␯L 具H2典, can now be estimated by noting that L6⑀ens 艋 ca6ᐉ␯3Re3 共2.9兲 2 1/2 L␩−1 k 艋 c共aᐉRe兲 , 共2.10兲 and so which is Eq. 共1.13兲 of Theorem 1.1. The estimate for dL in Eq. 共1.14兲 then follows immediately from the relation between the estimate for dL in Eqs. 共1.6兲 and 共2.10兲. Finally, we turn to proving the estimate for 具H1典 in Eq. 共1.15兲 which turns around the use of the simple inequality H21 艋 H2H0. The next step is to use the fact that 具H1典 艋 具H2典1/2具H0典1/2 = ␯aᐉRe具H2典1/2 . 共2.11兲 共2.12兲 Using the upper bound in Eq. 共2.7兲 gives 具H1典 艋 c␯2a2ᐉᐉ2Re5/2 , 共2.13兲 which then gives Eq. 共1.15兲 in Theorem 1.1. In fact, Eq. 共2.13兲 is an improvement in the bound for 具H1典 from Re3 to Re5/2. This result has also been found recently by Alexakis and Doering.36 Downloaded 07 Jun 2007 to 155.198.190.103. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 065202-7 J. Math. Phys. 48, 065202 共2007兲 Estimates for the 2D Navier-Stokes equations B. Proof of Theorem 1.2 Having introduced the notation for Hn in Eq. 共2.2兲, similar quantities are used that contain the forcing,35,29 namely Fn = 冕 ⍀ 共兩⵱nu兩2 + ␶2兩ⵜnf兩2兲dV, 共2.14兲 defined first in Eq. 共1.17兲, and the moments ␬n,r defined in Eq. 共1.18兲, ␬n,r共t兲 ª 冉 冊 Fn Fr 1/2共n−r兲 . 共2.15兲 The parameter ␶ in Eq. 共2.14兲 is a time scale and needs to be chosen appropriately. The idea is that it should be chosen in such a way that the forcing does not dominate the behavior of the moments of the velocity field. Defining ␻0 = ᐉ−2␯, it is shown in Appendix A that this end is achieved if ␶−1 is chosen as ␶−1 = ␻0关Gr共1 + ln Gr兲兴1/2 共2.16兲 艋c␻0Re共1 + ln Re兲1/2 . 共2.17兲 As a preliminary to the proof of Theorem 1.2, we state the ladder theorem proved in Refs. 35 and 29. Theorem 2.1: The Fn satisfy the differential inequalities 1 2 Ḟ0 艋 − ␯F1 + c␶−1F0 , 共2.18兲 1 2 Ḟ1 艋 − ␯F2 + c␶−1F1 , 共2.19兲 艋 − ␯Fn+1 + cn,1共储⵱u储⬁ + ␶−1兲Fn , 共2.20兲 艋 − 21 ␯Fn+1 + cn,2共␯−1储u储⬁2 + ␶−1兲Fn . 共2.21兲 and, for n 艌 2, either 1 2 Ḟn or 1 2 Ḟn The L⬁ inequalities in Theorem 2.1, particularly 储⵱u储⬁ in Eq. 共2.20兲, can be handled using a modified form of the L⬁ inequality of Brezis and Gallouet that has already been proved in Ref. 18. Lemma 2.1: In terms of the Fn of Eq. 共2.14兲 and ␬3,2 of Eq. 共2.15兲, a modified form of the two-dimensional L⬁ inequality of Brezis and Gallouet is 1/2 储⵱u储⬁ 艋 cF1/2 2 关1 + ln共L␬3.2兲兴 . 共2.22兲 2 典 具␬n,r for r 艌 2. This lemma directly leads to an estimate for Lemma 2.2: For n ⬎ r 艌 2, to leading order in Re, 2 典 艋 c共a2ᐉRe兲3/2共1 + ln Re兲1/2 . L2具␬n,r 共2.23兲 Proof: By dividing Eq. 共2.20兲 by Fn and time averaging, we have 2 ␯具␬n+1,n 典 艋 cn,1具储⵱u储⬁典 + c␻0Re共1 + ln Re兲1/2 . 共2.24兲 However, because ␬n,r 艋 ␬n+1,n for r ⬍ n, for every 2 艋 r ⬍ n, in combination with Lemma 2.1, we have Downloaded 07 Jun 2007 to 155.198.190.103. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 065202-8 J. Math. Phys. 48, 065202 共2007兲 J. D. Gibbon and G. A. Pavliotis 2 1/2 1/2 ␯具␬n,r 典 艋 c具F1/2 2 关1 + ln共L␬3,2兲兴 典 + c␻0Re共1 + ln Re兲 . 共2.25兲 The logarithm is a concave function and ␬3,2 艋 ␬n,r so Jensen’s inequality gives 2 2 典 艋 L2␯−1c具F2典1/2具关1 + ln兵L2具␬n,r 典其兴典1/2 + ca2ᐉRe共1 + ln Re兲1/2 . L2具␬n,r 共2.26兲 is no more than The estimate for 具F2典 can be found from 具H2典 in Eq. 共2.7兲; the extra term ␶ 储 ⵱ 䊏 O共Re2兲. Standard properties of the logarithm turn inequality Eq. 共2.26兲 into Eq. 共2.23兲. 2 典 for r 艌 2. These are used in the following theorem to give Lemma 2.2 gives estimates for 具␬n,r better estimates for the cases r = 0 and r = 1. Prior to this, it is necessary to state the results that immediately derive from Eqs. 共2.18兲 and 共2.19兲 by, respectively, dividing through by F0 and F1 before time averaging 2 2 典 艋 ca2ᐉRe共1 + ln Re兲1/2, N1,0 ⬅ L2具␬1,0 2 f储22 2 N2,1 ⬅ L2具␬2,1 典 艋 ca2ᐉRe共1 + ln Re兲1/2 . 共2.27兲 With the estimates in Eq. 共2.27兲 we are now ready to complete the proof of Theorem 1.2. Proof of Theorem 1.2: Let us return to Eq. 共2.23兲 in Lemma 2.2 and use the fact that 2 具␬n,1 典= 冓冉 冊 冉 冊 冔 Fn F2 1/n−1 F2 F1 1/n−1 2共n−2兲/n−1 2/n−1 = 具␬n,2 ␬2,1 典 , 共2.28兲 and thus 2 2 n−2/n−1 2 1/n−1 典 艋 具␬n,2 典 具␬2,1典 . 具␬n,1 共2.29兲 Using Eq. 共2.23兲 in Lemma 2.2, together with Eq. 共2.27兲, for n 艌 2, 2 典 艋 cn,1共a2ᐉRe兲共3n−4兲/2共n−1兲关1 + ln Re兴1/2 , Nn,1 = L2具␬n,1 共2.30兲 which coincides with a2ᐉRe共1 + ln Re兲1/2 at n = 2 but converges to Re3/2共1 + ln Re兲1/2 as n → ⬁. The exponent ⌳n,1 is defined in Eq. 共1.21兲. Likewise, in the same manner as Eq. 共2.28兲 we have 2 2 共n−1兲/n 2 1/n 典 艋 具␬n,1 典 具␬1,0典 . 具␬n,0 共2.31兲 2 典 艋 cn,0共a2ᐉRe兲共3n−2兲/2n关1 + ln Re兴1/2 . Nn,0 = L2具␬n,0 共2.32兲 Thus we find that for n 艌 2, The exponent ⌳n,0 is defined in Eq. 共1.21兲. 䊏 III. POINT-WISE ESTIMATES Let us consider the differential inequalities for H0 and H1: 1 2 Ḣ0 1 2 Ḣ1 艋 − ␯H1 + 储f储2H1/2 0 , 共3.1兲 艋 − ␯H2 + ᐉ−2储f储2H1/2 0 , 共3.2兲 having used the narrow-band property on Eq. 共3.2兲. Upon combining Poincaré’s inequality with Lemmas B.1 and B.2 in Appendix B we obtain limt→⬁H0 艋 ca6ᐉ␯2Gr2 艋 ca6ᐉ␯2Re4 , 共3.3兲 limt→⬁H1 艋 cᐉ−2a6ᐉ␯2Gr2 艋 cᐉ−2a6ᐉ␯2Re4 . 共3.4兲 and The additive forcing terms in F1 and F0 are of a lower order in Re so we end up with Downloaded 07 Jun 2007 to 155.198.190.103. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 065202-9 J. Math. Phys. 48, 065202 共2007兲 Estimates for the 2D Navier-Stokes equations limt→⬁F0 艋 ca6ᐉ␯2Re4 + O共Re2兲, 共3.5兲 limt→⬁F1 艋 cᐉ−2a6ᐉ␯2Re4 + O共Re2兲. 共3.6兲 The estimate for F1 enables us to obtain point-wise estimates on Fn, n 艌 2 共Ref. 18, Sec. 7.2兲. In fact we have the following lemma. Lemma 3.1: As Gr→ ⬁, 4n limt→⬁Fn 艋 cn␯2ᐉ−2na6n ᐉ Re . 共3.7兲 Proof: Applying a Gagliardo–Nirenberg inequality in two-dimensions to ⵱u we obtain a/2 共1−a兲/2 , 储⵱u储⬁ 艋 c储⵱nu储a2储⵱u储1−a 2 艋 cFn F1 共3.8兲 with a = 1 / 共n − 1兲. Using this in Eq. 共2.20兲 gives 1 2 Ḟn 艋 − ␯Fn+1 + cnF1+a/2 F1−a/2 + c␻0Re共1 + ln Re兲1/2Fn . 1 n 共3.9兲 Moreover, the following inequality can easily be proved using Fourier transforms: q p FN+q , FNp+q 艋 FN−p 共3.10兲 from which, with N = n, p = n − 1, q = 1, it can be deduced that − Fn+1 艋 − Fn/共n−1兲 n F1/共n−1兲 1 共3.11兲 . We now use Eq. 共3.11兲 in Eq. 共3.9兲 to obtain Fn/共n−1兲 1 n F1−a/2 + c␻0Re共1 + ln Re兲1/2Fn , Ḟn 艋 − ␯ 1/共n−1兲 + cnF1+a/2 1 n 2 F1 共3.12兲 with a = 1 / 共n − 1兲. We use now estimate Eq. 共3.6兲 in Eq. 共3.12兲 with the further use of Lemma B.2 to obtain 2n limt→⬁Fn 艋 cn␯2ᐉ−2na6n ᐉ Gr , which leads to the result. The above Lemma enables us to obtain an estimate on the wave numbers ␬n,r. Lemma 3.2: For n ⬎ r 艌 0, as Gr→ ⬁, limt→⬁共L␬n,r兲 艋 cna共4n−r−1兲/共n−r兲 Re共2n−1兲/共n−r兲共1 + ln Re兲1/2共n−r兲 . ᐉ 共3.13兲 䊏 共3.14兲 Proof: Essentially one uses the upper bound on Fn and the lower bound on Fr which can be calculated from the forcing part in terms of Gr, leading to the result 共see also Ref. 18, Chap. 7兲.䊏 IV. INTERMITTENCY: GOOD AND BAD INTERVALS The issue of intermittency in solutions of the two-dimensional Navier–Stokes equations is now addressed. While the Fn and ␬n,r are bounded from above for all time, nevertheless it is possible that their behaviour could be spiky in an erratic manner. To show how this might come about, consider the definition of ␬n,r in Eq. 共1.18兲 from which we find 2 Fn+1 = ␬n,r 冉 冊 ␬n+1,r ␬n,r 2共n+1−r兲 Fn . 共4.1兲 Now consider inequality 共3.9兲 rewritten as Downloaded 07 Jun 2007 to 155.198.190.103. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 065202-10 J. Math. Phys. 48, 065202 共2007兲 J. D. Gibbon and G. A. Pavliotis 冉 冊 冉 冊 1 Ḟn ␬n+1,1 2 艋 − ␯␬n,1 2 Fn ␬n,1 ␬n+1,1 n 1/2 ␬n,1F1/2 1 + c␻0Re共1 + ln Re兲 , ␬n,1 2n + cn 共4.2兲 where we have used Eq. 共4.1兲 and the fact that ␬n,1 艋 ␬n+1,1 in the middle term. Using Young’s inequality on this same term we end up with 冉 冊 1 2 ␬n+1,1 1 Ḟn 艋 − ␯␬n,1 2 Fn 2 ␬n,1 2n + cn␯−1F1 + c␻0Re共1 + ln Re兲1/2 . 共4.3兲 The main question is whether, for Navier–Stokes solutions, the lower bound on ␬n+1,1 艌1 ␬n,1 共4.4兲 can be raised from unity. A variation on the interval theorem proved in Ref. 29 is used. Theorem 4.1: For any value of the parameter ␮ 苸 共0 , 1兲, the ratio ␬n+1,1 / ␬n,1 obeys the long-time averaged inequality 共n 艌 2兲, 冓冋 冉 冊 册 冋 cn − 册 冔 1/␮−1 2 ␮ 共L2␬n,1 兲 2 1/␮−1 ␬n+1,1 ␬n,1 共a2ᐉRe兲⌳n,1共1 + ln Re兲1/2 艌 0, 共4.5兲 where the cn are the same as those in Theorem 1.2. Hence there exists at least one interval of time, designated as a “good interval”, on which the inequality cn 冉 冊 ␬n+1,1 ␬n,1 2 ␮ 共L2␬n,1 兲 2 艌 共4.6兲 共a2ᐉRe兲⌳n,1共1 + ln Re兲1/2 holds. Those other parts of the time-axis on which the reverse inequality cn 冉 冊 ␬n+1,1 ␬n 2 ␮ 共L2␬n,1 兲 2 ⬍ 共4.7兲 共a2ᐉRe兲⌳n,1共1 + ln Re兲1/2 holds are designated as “bad intervals”. Remark: In principle, the whole time-axis could be a good interval, whereas the positive time average in Eq. 共4.5兲 ensures that the complete time axis cannot be “bad”. This paper is based on the worst-case supposition that bad intervals exist, that they could be multiple in number, and that the good and the bad are interspersed. The precise distribution and occurrence of the good/bad intervals and how they depend on n remains an open question. The contrast between the twodimensional and three-dimensional Navier–Stokes equations is prominent; while no singularities can occur in the ␬n,1 in the two-dimensional case, in three dimensions it is within these bad intervals that they can potentially occur. Proof: Take two parameters 0 ⬍ ␮ ⬍ 1 and 0 ⬍ ␣ ⬍ 1 such that ␮ + ␣ = 1. The inverses ␮−1 and −1 ␣ will be used as exponents in the Hölder inequality on the far right-hand side of 2␣ 2␣ 典 艋 具␬n+1,1 典= 具␬n,1 thereby giving 冓冉 冊 冔 冓冉 冊 冔 ␬n+1,1 ␬n,1 2␣ 2␣ ␬n,1 艋 冓冉 冊 冔 冉 冊 ␬n+1,1 ␬n,1 2␣/␮ 艌 2␣ 具␬n,1 典 2 ␣ 具␬n,1 典 1/␮ ␬n+1,1 ␬n,1 2␣ = 具␬n,1 典 2␣/␮ 冉 冊 2␣ 具␬n,1 典 2 具␬n,1 典 ␮ 2 ␣ 具␬n,1 典 , 共4.8兲 ␣/␮ . 共4.9兲 Two-dimensional Navier–Stokes information can be injected into these formal manipulations: the 2 典 from Theorem 1.2 and the lower bound L␬n,1 艌 1 are used in the ratio on the upper bound on 具␬n,1 䊏 far right-hand side of Eq. 共4.9兲 to give Eq. 共4.5兲, with the same cn as in Theorem 1.2. Downloaded 07 Jun 2007 to 155.198.190.103. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 065202-11 Estimates for the 2D Navier-Stokes equations J. Math. Phys. 48, 065202 共2007兲 Now consider what must happen on bad intervals. It is always true that ␬n+1,1 / ␬n,1 艌 1, so Eq. 共4.7兲 implies that on these intervals there is a lower bound 2 L2␬n,1 ⬎ cn共a2ᐉRe兲⌳n,1/␮共1 + ln Re兲1/2␮ . 共4.10兲 This lower bound cannot be greater than the upper point-wise bound in Eq. 共3.14兲, which means that ␮ is restricted by 冉 冊 2n − 1 ⌳n,1 ⬍2 . ␮ n−1 共4.11兲 Moreover, the factor of 1 / ␮ in the exponent makes the lower bound in Eq. 共4.10兲 much larger than 2 the upper bound on the average 具␬n,1 典 given in Theorem 1.2. These intervals must therefore be very short. To estimate how large they can be requires an integration of Eq. 共4.3兲 over short times ⌬t = t − t0 which, in turn, requires the time integral of H1 for short times ⌬t. We use the notation 兰⌬t = 兰tt , with the definition ␻0 = ␯ᐉ−2. 0 Lemma 4.1: To leading order in Re 冕 ⌬t F1dt 艋 ␯a4ᐉ关c1a2ᐉ + c2␻0⌬t兴Re4 . 共4.12兲 Proof: Integrating Eq. 共3.1兲 over a short time ⌬t gives ␯ 冕 1 H1dt 艋 H0共t0兲 + ⌬t关ᐉ−2␯3a4ᐉGr2兴 艋 c1a6ᐉ␯2Re4 + ⌬t关c2ᐉ−2␯3a4ᐉRe4兴, 2 ⌬t 共4.13兲 䊏 having used Eq. 共3.3兲 for the 21 H0共t0兲 term. The forcing term in F1 is only O共Re2兲. Now we wish to estimate ␻0⌬t in terms of Re. Integrating Eq. 共4.3兲, using 共4.13兲 and the lower bound Eq. 共4.10兲 and multiplying by ᐉ2, we have 1 2 2 ᐉ 关ln 2 ⌳n,1/␮ Fn共t兲 − ln Fn共t0兲兴 + 21 cn␯a−2 共1 + ln Re兲1/2␮⌬t 艋 ᐉ2a4ᐉ关c1a2ᐉ + c2␻0⌬t兴Re4 ᐉ 共aᐉRe兲 + cᐉ2␻0⌬tRe共1 + ln Re兲1/2 . 共4.14兲 As Gr→ ⬁, the dominant terms are 2 ⌳n,1/␮ ␻0⌬t兵a−2 共1 + ln Re兲1/2␮ − a6ᐉRe4其 艋 c1a6ᐉRe4 . ᐉ 共aᐉRe兲 共4.15兲 Choosing ␮ in the range, to leading order we have ␮ ⬍ 41 ⌳n,1 , 共4.16兲 ␻0⌬t 艋 c共a2ᐉRe兲4−⌳n,1/␮ . 共4.17兲 then ⌬t must satisfy Because the exponent in Eq. 共4.17兲 is necessarily negative these intervals are very small and decreasing with increasing Re. Combining Eq. 共4.11兲 with Eq. 共4.16兲 we have 1 共n − 1兲 ⌳n,1 ⬍ ␮ ⬍ ⌳n,1 , 4 2共2n − 1兲 共4.18兲 which actually holds for every n 艌 1. Figure 1 is a cartoon-like figure displaying the lower bound on the bad intervals of width 共⌬t兲b and also the maximum of ␬n,1 allowed by Eq. 共3.14兲 in Lemma 3.2. The full dynamics of two-dimensional Navier–Stokes is actually determined by the intersection of all cartoons for every n 艌 3 on the grounds that the position and occurrence of the bad intervals varies with n. Thus we are interested in the limit n → ⬁ which determines that the range of ␮ is squeezed between Downloaded 07 Jun 2007 to 155.198.190.103. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 065202-12 J. Math. Phys. 48, 065202 共2007兲 J. D. Gibbon and G. A. Pavliotis FIG. 1. A cartoon, not to scale, of good/bad intervals for some value of n 艌 3. 冉 冊 冉 冊 5 3 1 3 1− ⬍␮⬍ 1− . 8 8 6n 3n 共4.19兲 Thus, in the limit, ␮ takes a value just under 3 / 8. We conclude that the interval theorem 共Theorem 4.1兲 reproduces the effects of intermittency in a two-dimensional flow by manifesting very large lower bounds within bad intervals and suppressing spiky behavior within the good intervals which must be quiescent for long intervals, otherwise the long-time average would be violated. ACKNOWLEDGMENTS The authors would like to thank Matania Ben-Artzi, Charles Doering, Darryl Holm, Haggai Katriel, and Edriss Titi for comments and suggestions. J.D.G. would also like to thank the Mathematics Departments of the Weizmann Institute of Science and the Hebrew University of Jerusalem for their hospitality during December 2005 and January 2006 when some of these ideas were conceived. APPENDIX A: FORCING AND THE FLUID RESPONSE For technical reasons, we must address the possibility that in their evolution the quantities Hn might take small values. Thus we need to circumvent problems that may arise when dividing by these 共squared兲 seminorms. We follow Doering and Gibbon35 who introduced the modified quantities Fn = Hn + ␶2储ⵜnf储22 , 共A1兲 where the “time scale” ␶ is to be chosen for our convenience. So long as ␶ ⫽ 0, the Fn are bounded 2 away from zero by the explicit value ␶2L3ᐉ−2n f rms . Moreover, we may choose ␶ to depend on the parameters of the problem such that 具Fn典 – 具Hn典 as Gr→ ⬁. To see how to achieve this, let us define ␶ = ᐉ2␯−1关Gr共1 + ln Gr兲兴−1/2 . 共A2兲 Then the additional term in Eq. 共A1兲 is 2 ␶2储⵱nf储22 = L3␯−2ᐉ4−2n f rms 关Gr共1 + ln Gr兲兴−1 = ␯2ᐉ−共2n+2兲L3Gr共1 + ln Gr兲−1 . 23 Now Doering and Foias bound of the form 共A3兲 proved that in d dimensions, the energy dissipation rate ⑀ has a lower Downloaded 07 Jun 2007 to 155.198.190.103. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 065202-13 J. Math. Phys. 48, 065202 共2007兲 Estimates for the 2D Navier-Stokes equations ⑀ 艌 c␯3ᐉ−3L−1Gr. 共A4兲 Using this on the far right-hand side of Eq. 共A3兲 we arrive at ␶2储ⵜnf储22 艋 c6⑀ᐉ−共2n−1兲L4␯−1共1 + ln Gr兲−1 = c6 冉冊 L ᐉ 共2n−1兲 L−2共n−1兲具H1典 ⬎ 共1 + ln Gr兲−1 . 共A5兲 Using Poincaré’s inequality in the form H1 艋 共2␲L兲2共n−1兲Hn, as Gr→ ⬁ we have ␶2储ⵜnf储22 艋 c6a共2n−1兲 共1 + ln Gr兲−1 . ᐉ 具Hn典 共A6兲 Hence, the additional forcing term in Eq. 共A1兲 becomes negligible with respect to 具Hn典 as Gr → ⬁, so the forcing does not dominate the response. APPENDIX B: COMPARISON THEOREMS FOR ODE We present a comparison theorem for ODE which is useful for obtaining various estimates. We start with the following classical result. Lemma B.1: Let f : 关0 , T兴 ⫻ R → R be a continuous function which is locally Lipschitz uniformly in t: for all intervals 关a , b兴 傺 R there exists a constant such that 兩f共s , x兲 − f共s , y兲 兩 艋 C 兩 x − y兩 for all x , y 苸 关a , b兴 and all s 苸 关0 , T兴. Furthermore, let x 苸 AC共关0 , T兴 , R兲 be such that ẋ共t兲 艋 f关t,x共t兲兴 for all t 苸 关0 , T兴 and let y共t兲 be the solution of ẏ共t兲 = f关t , y共t兲兴 on 关0 , T兴. Assume further that x共0兲 艋 y共0兲. Then, x共t兲 艋 y共t兲 for all t 苸 关0 , T兴. We can use this Lemma to prove the following useful result. Lemma B.2: Let x : 关0 , T兴 → 关0 , ⬁ 兲 be an absolutely continuous function with x共0兲 ⬎ 0 which satisfies ẋ 艋 ⌬0x + Fxn1 − Exn2 , 共B1兲 where ⌬0 , F , E ⬎ 0 and 1 ⬍ n1 ⬍ n2. Then lim sup x共t兲 艋 共4⌬0E−1兲1/n2−1 + 共2FE−1兲1/n2−n1 . t→⬁ 共B2兲 P. Constantin, C. Foias, and R. Temam, Physica D 30, 284 共1988兲. P. Constantin and C. Foias, Navier-Stokes Equations 共The University of Chicago Press, Chicago, 1988兲. 3 C. Foias, O. Manley, R. Rosa, and R. Temam, Navier-Stokes Equations and Turbulence 共Cambridge University Press, Cambridge, 2001兲. 4 R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, Vol. 68 共Springer-Verlag, New York, 1988兲. 5 D. A. Jones and E. S. 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