2D turbulence in physical scales of the Navier-Stokes equations

0.27 MB·English

by  R. Dascaliuc

2D TURBULENCE IN PHYSICAL SCALES OF THE
NAVIER-STOKES EQUATIONS
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1 R.DASCALIUCANDZ.GRUJIC´
0
2
Abstract. Local analysis of the two dimensional Navier-Stokes equations is

n usedtoobtainestimates ontheenergy andenstrophy fluxes involvingTaylor
a and Kraichnan length scales and the size of the domain. In the framework
J of zero driving force and non-increasing global energy, these bounds produce

1 sufficient conditions for existence of the direct enstrophy and inverse energy
1 cascades. Severalmanifestationsoflocalityofthefluxesundertheseconditions
are obtained. All the scales involved are actual physical scales in R2 and no

] homogeneity assumptionsaremade.
P
A
.
h
at 1. introduction
m
FollowingthegroundbreakingideasofKolmogorov[15,16,14],Batchelor,Kraich-
[
nan and Leith [2, 3, 17, 18, 20] established the foundations of empirical theory of

1 2D turbulence (BKL theory). One of the main features of the BKL theory is the
v existenceofenstrophy cascade overawide rangeoflengthscales,calledthe inertial
9 range, where the dissipation effects are dominated by the transport of enstrophy
0
from higher to lower scales. In contrast to the 3D turbulence, the energy in 2D
2
case is cascading towardthe larger scales, a phenomenon referredto as the inverse
2
. energy cascade. Direct enstrophy and inverse energy cascades have been observed
1
in physical experiments (albeit certain difficulties exist in generating a purely 2D
0
1 turbulent flow), but theoretical justification of these phenomena using equations
1 of fluid motion, and in particular, the Navier-Stokes equations (NSE), remains far
: from being settled. Technical complexity of the NSE makes it difficult to estab-
v
i lish the conditions under which such cascades can occur. In the 2D case, the NSE
X
possess a number of useful regularity properties (unlike the 3D case for which the
r global regularity is an open problem). However, the dynamical complexity of the
a
NSEmakesadetailedstudyoftheirlongtimebehavioradifficultenterprise. Under
certain conditions, existence of the global attractors of high fractal and Hausdorff
dimensionshasbeenestablishedforthe2DNSE;moreover,itisbelievedthatthese
attractorsbecomechaotic(althoughtheproofiselusive). Foranoverviewofvarious
mathematicalmodelsofturbulenceandthetheoryoftheNSE,see,e.g.,[12,13,10]
and [21, 6, 25], respectively.
Mostrigorousstudiesof2DNSEturbulencehavebeenmadeinFouriersettings.
In particular, in [11] the framework of space-periodic solutions and infinite-time
averageswasusedtostudymainaspectsoftheBKLtheory,includingestablishinga
sufficientconditionfor the enstrophycascade. This condition,involvingKraichnan
length scale, is akin to our condition (4.11) obtained in section 4. In contrast to
Date:January13,2011.
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2 R.DASCALIUCANDZ.GRUJIC´
[11], our goal was to work in physical space and with finite-time averages, dealing
with actual length scales in R2 rather than the Fourier wave numbers.
Inthispaperweextendtothe2Dcasetheideasintroducedin[8]toestablishthe
existenceoftheenergycascadeandspacelocalityofthefluxforthe3DNSE.There,
one of the difficulties was the possible lack of regularity, which led us to using the
frameworkofsuitableweaksolutions([24,4]). In2D,the difficulties lie inthe need
to work with higher-order derivatives in the case of the enstrophy cascade, as well
as in dealing with a rather complex phenomenology of the 2D turbulence.
Despite these differences, the basic setting for studying energy and enstrophy
transfer in physical scales remains the same in both 3D and 2D case. We utilize
the refined cut-off functions to localize the relevant physical quantities in physical
space and then employ ensemble averages satisfying certain optimality conditions
together with dynamics of NSE to link local quantities to global ones (see [8] for a
detailed discussion of our physical scales framework).
We restrict our study to a bounded region, a ball, in R2, and consider the case
of short-time or decaying turbulence by setting the driving force to zero. Thus, in
contrast to infinite-time averages used in [11], we use averages over finite times.
The time intervals considered here depend on the size of the domain as well as the
viscosity (see (4.2)). The spatial ensemble averageis taken by considering optimal
coverings of the spatial domain with balls at various scales. Also, to exclude the
situations of the uniform growth of kinetic energy without any movement between
the scales we restrict our study to physical situations where the kinetic energy on
the(global)spatialdomainΩisnon-increasing,e.g.,aboundeddomainwithno-slip
boundary conditions, or the whole space with either decay at infinity or periodic
boundary conditions.
The paper is structured as follows. In section 2 we provide a brief overview of
the 2D NSE theory, noting the relevant existence and regularity results. We also
pointoutimportantdifferencesbetween2Dand3DNSE,andhowthesedifficulties
are reflected in the differences between 2D and 3D turbulence.
Section 3 introduces the physical quantities of energy, enstrophy, and palinstro-
phy, as well as energy and enstrophy fluxes adopted to our particular settings. We
also define the ensemble averages to be used throughout the paper.
The main result of section 4 is a surprisingly simple sufficient condition for the
enstrophy cascade (4.11), according to which the averaged enstrophy flux toward
the lower scales is nearly constant over a range of scales. This condition, involving
the Kraichnan scale and the size of the domain, is reminiscent of the Poincar´e
inequality on a domain of the corresponding size (see Remark 4.2). Moreover, the
condition in hand would be easy to check in physical experiments as the averages
involved are very straightforward.
Section 5 commences a study of inverse energy cascade in physical space. The
existence of such cascades in the 2D NSE solutions remains an open question.
Several partial results exist; in particular, in the space-periodic setting the energy
flux is oriented towards lower (Fourier) scales in the region below the scales of the
bodyforce([11]),butexistenceofthecascadecouldnotbeestablished. Incontrast,
[1] provides a condition for the inverse energy cascades inside spectral gaps of the
body force. We provethatif the globalTaylorscale isdominatedby the linearsize
ofthedomain,thentheaveragedenergyfluxisconstantoverarangeoflargescales
and is oriented outwards (see Theorem 5.1).
2D TURBULENCE IN NSE 3
Thesecondpartofthepaperconcernslocalityoftheenergyandenstrophyfluxes.
Similarly to the 3D turbulence ([23]), it is believed that the energy and enstrophy
fluxes inside the inertial ranges of the 2D turbulent flows depends strongly on the
flow in nearbyscales,the dependence onlower andmuch higher scales being weak.
The theoretical proof of this conjecture remained elusive. The first quantitative
results on fluxes were obtained by early 70’s (see [19]). Much later, the authors in
[22] used the NSE in the Fourier setting to explore locality of scale interactions for
statisticalaverages,whiletheinvestigationin[9]revealedthelocalityoffiltereden-
ergyflux undertheassumptionthatthesolutionstothevanishingviscosityEuler’s
equations saturate a defining inequality of a suitable Besov space (a weak scaling
assumption). A more recent work [5] provided a proof of the quasi-locality of the
energy flux in the Littelwood-Paley setting.
In section 6 we obtain several manifestations of the locality of both energy and
enstrophyfluxesinthephysicalspacethroughouttheinertialranges. Inparticular,
considering dyadic shells at the scales 2kR (k an integer) in the physical space, we
showthatbothultravioletandinfraredlocalitypropagateexponentially intheshell
number k.
Tothebestofourknowledge,thecondition(4.11)ispresentlytheonlycondition
(in any solution setting) implying both the existence of the inertial range and the
localityoftheenstrophyflux. Thesameistruefortherelation(5.19)whichimplies
both inverse energy cascade and energy flux locality in the physical scales of the
2DNSE. Finally, we point out that our approachis valid for a varietyof boundary
conditions (in particular, the no-slip, periodic, or the whole space with decay at
infinity); moreover,it does notinvolveanyadditionalhomogeneityassumptions on
the solutions to the NSE.
2. preliminaries
We consider two dimensional incompressible Navier-Stokes equations (NSE)
∂
u(t,x) ν∆u(t,x)+(u(t,x) )u(t,x)+ p(t,x)=0
(2.1) ∂t − ·∇ ∇
u(t,x)=0,
∇·
where the space variable x=(x ,x ) is in R2 and the time variable t is in (0, ).
1 2
∞
The vector-valuedfunction u=(u ,u ) and the scalar-valuedfunction p represent
1 2
thefluidvelocityandthepressure,respectively,whiletheconstantν istheviscosity
of the fluid.
Underappropriateboundaryconditionsthissystemadmitsauniquesolution(see
[25], [6]), which is analytic in both space and time. For convenience, we generally
assume no-slip boundary conditions on a bounded domain
(2.2) u =0, Ω bounded in R2
|∂Ω
(although the results hold for the other physical boundary conditions which imply
smoothness and non-increasing global energy u2).
Ω| |
Thus, if φ ((0, ) Ω), φ 0, where Ω be an open connected set in R2,
∈ D ∞ × ≥ R
then multiplying NSE by φu and integrating by parts we obtain the local energy
equation
(2.3)
2ν u2φdxdt= u2(∂ φ+ν∆φ)dxdt+ (u2+2p)u φdxdt
t
|∇⊗ | | | | | ·∇
ZZ ZZ ZZ
4 R.DASCALIUCANDZ.GRUJIC´
where ((0, ) Ω) denotes the space of infinitely differentiable functions with
D ∞ ×
compact support in (0, ) Ω.
∞ ×
We also consider the vorticity form of the 2D NSE by taking the curl of (2.1)
viewed as a 3D equation with the third component zero,
∂
(2.4) ω ν∆ω+(u )ω =0,
∂t − ·∇
where ω = u (with the convention u=(u ,u ,0) and ω =(0,0,ω)).
1 2
∇×
Note that for the full 3D NSE (2.4) would contain the vortex-stretching term
(ω )u.
·∇
Multiplying (2.4) with φω yields the local enstrophy equation,
(2.5) 2ν ω 2φdxdt= ω 2(∂ φ+ν∆φ)dxdt+ ω 2u φdxdt.
t
|∇⊗ | | | | | ·∇
ZZ ZZ ZZ
We will make the following assumptions on the domain Ω and test functions φ.
First, we assume there exists R satisfying
0
(2.6) R >0 such that B(0,3R ) Ω
0 0
⊂
whereB(0,3R )representstheballinR2 centeredattheoriginandwiththeradius
0
3R .
0
Next, let 1/2 δ <1. Choose ψ (B(0,2R )) satisfying
0 0
≤ ∈D
ψ C ∆ψ C
(2.7) 0 ψ 1, ψ =1 on B(0,R ), |∇ 0| 0, | 0| 0 .
≤ 0 ≤ 0 0 ψ0δ ≤ R0 ψ02δ−1 ≤ R02
For a T > 0 (to be chosen later), x B(0,R ) and 0 < R R , define φ =
0 0 0
∈ ≤
φ (t,x) = η(t)ψ(x) to be used in (2.3) and (2.5) where η = η (t) and ψ =
x0,T,R T
ψ (x) are refined cut-off functions satisfying the following conditions,
x0,R
η′ C
0
(2.8) η (0,2T), 0 η 1, η =1 on (T/4,5T/4), | | ;
∈D ≤ ≤ ηδ ≤ T
if B(x ,R) B(0,R ), then ψ (B(x ,2R)) with
0 0 0
⊂ ∈D
(2.9)
ψ C ∆ψ C
0 ψ ψ , ψ =1 on B(x ,R) B(0,R ), |∇ | 0, | | 0 ,
≤ ≤ 0 0 ∩ 0 ψδ ≤ R ψ2δ−1 ≤ R2
and if B(x ,R) B(0,R ), then ψ (B(0,2R )) with ψ = 1 on B(x ,R)
0 0 0 0
6⊂ ∈ D ∩
B(0,R ) satisfying, in addition to (2.9), the following:
0
(2.10)
ψ =ψ on the part of the cone in R2 centered at zero and passing through
0
S(0,R ) B(x ,R) between S(0,R ) and s(0,2R )
0 0 0 0
∩
and
ψ =0 on B(0,R ) B(x ,2R) and outside the part of the cone in R2
0 0
\
(2.11) centered at zero and passing through S(0,R ) B(x ,2R)
0 0
∩
between S(0,R ) and S(0,2R ).
0 0
Figure 1 illustrates the definition of ψ in the case B(x ,R) is not entirely con-
0
tained in B(0,R ).
0
2D TURBULENCE IN NSE 5
ψ=ψ0
ψ=1
regioncontainingtherestofsupp(ψ)
2R R
R0
0
2R0
Figure 1. Regions of supp(ψ) in the case B(x ,R) B(0,R ).
0 0
6⊂
Remark 2.1. The additional conditions on the boundary elements (2.10) and
(2.11) are necessary to obtain the lower bound on the fluxes in terms of the same
versionofthe localizedenstrophyE inTheorems4.1 and6.2 (see Remarks4.3 and
6.3).
3. Averaged enstrophy and energy flux
Let x B(0,R ) and 0 < R R . We define the localized versions of energy,
0 0 0
∈ ≤
e, enstrophy, E, and palinstrophy, P at time t associated to B(x ,R) by
0
1
(3.1) e (t)= u2φ2δ−1dx,
x0,R 2| |
Z
1 1
(3.2) E (t)= ω 2φ2δ−1dx or E′ (t)= ω 2φdx ,
x0,R 2| | x0,R 2| |
Z (cid:18) Z (cid:19)
and
(3.3) P (t)= ω 2φdx.
x0,R |∇⊗ |
Z
In the classical case, the total – kinetic energy plus pressure – flux through the
sphere S(x ,R) is given by
0
1
( u2+p)u nds= ((u ) u+ p) udx
2| | · ·∇ ∇ ·
Z Z
S(x0,R) B(x0,R)
wheren isanoutwardnormaltothe sphereS(x ,R). Similarly,the enstrophyflux
0
is given by
1
ω 2u nds= (u )ω ωdx.
2| | · ·∇ ·
Z Z
S(x0,R) B(x0,R)
6 R.DASCALIUCANDZ.GRUJIC´
Considering the NSE localized to B(x ,R) leads to the localized versions of the
0
aforementioned fluxes,
1
(3.4) Φ (t)= ( u2+p)u φdx
x0,R 2| | ·∇
Z
and
1
(3.5) Ψ (t)= ω 2u φdx,
x0,R 2| | ·∇
Z
where φ = ηψ with η and ψ as in (2.8-2.9). Since ψ can be constructed such that
φ=η ψ isorientedalongthe radialdirectionsofB(x ,R)towardsthe center of
0
∇ ∇
theballx ,Φ andΨ canbe viewedasthe fluxesintoB(x ,R)throughthe
0 x0,R x0,R 0
layerbetween the spheres S(x ,2R)and S(x ,R) (in the case ofthe boundary ele-
0 0
mentssatisfyingtheadditionalhypotheses(2.10)and(2.11),ψ isalmostradialand
the gradient still points inward). In addition, (2.3) and (2.5) imply that positivity
of these fluxes contributes to the increase of e and E , respectively.
x0,R x0,R
NotethatthetotalenergyfluxΦ consistsofboththekineticandthepressure
x0,R
parts. Without imposing any specific boundary conditions on Ω it is possible that
the increase of the kinetic energy around x is due solely to the pressure part,
0
withoutanytransferofthe kineticenergyfromlargerscalesintoB(x ,R)(see[8]).
0
Aswementionedintheintroduction,underphysicalboundaryconditions,like(2.2),
the increase of the kinetic energy in B(x ,R) (and consequently, the positivity of
0
Φ )implieslocaltransferofthe kineticenergyfromlargerscalessimplybecause
x0,R
thelocalkineticenergyisincreasingwhiletheglobalkineticenergyisnon-increasing
resultingindecreaseofthekineticenergyinthecomplement. Thisisalsoconsistent
with the fact that in the aforementioned scenarios one can project the NSE to
the subspace of divergence-free functions effectively eliminating the pressure and
revealing that the local flux Φ is indeed driven by transport/inertial effects
x0,R
rather than the change in the pressure.
Henceforth,followingthediscussionintheprecedingparagraph,inthesettingof
decayingturbulence(zerodrivingforce,non-increasingglobalenergy),thepositivity
and the negativity of Φ and Ψ will be interpreted as transfer of (kinetic)
x0,R x0,R
energy and enstrophy around the point x at scale R toward smaller scales and
0
transfer of (kinetic) energy around the point x at scale R toward larger scales,
0
respectively.
For a quantity Θ = Θ (t), t [0,2T] and a covering B(x ,R) of
x,R i i=1,n
∈ { }
B(0,R ) define a time-space ensemble average
0
n
1 1 1
(3.6) Θ = Θ (t)dt.
h iR T n R2 xi,R
Z i=1
X
Denote by
(3.7) e = e (t) ,
R x,R R
h i
′ ′
(3.8) E = E (t) or E = E (t) ,
R h x,R iR R h x,R iR
(cid:0) (cid:1)
(3.9) P = P (t) ,
R x,R R
h i
(3.10) Φ = Φ (t) ,
R x,R R
h i
2D TURBULENCE IN NSE 7
and
(3.11) Ψ = Ψ (t) ,
R x,R R
h i
the averagedlocalized energy, enstrophy, palinstrophy, and inward-directed energy
and enstrophy fluxes over balls of radius R covering B(0,R ).
0
Also, introduce the time-space average of the localized energy, enstrophy and
palinstrophy on B(0,R ),
0
1 1 1 1 1
(3.12) e = e (t)dt= u2φ2δ−1dxdt,
0 T R2 0,R0 T R2 2| | 0
Z 0 0 ZZ
1 1 1 1 1
E = E (t)dt= ω 2φ2δ−1dxdt
0 T R2 0,R0 T R2 2| | 0
(3.13) Z 0 0 ZZ
1 1 1 1 1
or E′ = E (t)dt= ω 2φ dxdt ,
0 T R2 0,R0 T R2 2| | 0
(cid:18) Z 0 0 ZZ (cid:19)
and
1 1 1 1
(3.14) P = E (t)dt= ω 2φ dxdt
0 T R2 0,R0 T R3 |∇⊗ | 0
Z 0 0 ZZ
where
(3.15) φ (t,x)=η(t)ψ (x)
0 0
with ψ defined in (2.7).
0
Finally, define Taylor and Kraichnan length scales associated with B(0,R ) by
0
1/2
e
0
(3.16) τ =
0 E′
(cid:18) 0(cid:19)
and
1/2
E
0
(3.17) σ = .
0
P
(cid:18) 0(cid:19)
To obtain optimal estimates on the aforementioned fluxes we will work with
averagescorresponding to optimal coverings of B(0,R ).
0
Let K ,K > 1 be absolute constants (independent of R,R , and any of the
1 2 0
parameters of the NSE).
Definition 3.1. We say that a covering of B(0,R ) by n balls of radius R is
0
optimal if
2 2
R R
0 0
(3.18) n K ;
1
R ≤ ≤ R
(cid:18) (cid:19) (cid:18) (cid:19)
(3.19) any x B(0,R ) is covered by at most K balls B(x ,2R).
0 2 i
∈
Note that optimal coverings exist for any 0 < R R provided K and K are
0 1 2
≤
large enough. In fact, the choice of K and K depends only on the dimension of
1 2
R2, e.g, we can choose K =K =8.
1 2
Henceforth, we assume that the averages are taken with respect to optimal
R
h·i
coverings.
The key observation about these optimal coveringsis contained in the following
lemma.
8 R.DASCALIUCANDZ.GRUJIC´
Lemma 3.1. If the covering B(x ,R) of B(0,R ) is optimal then the aver-
i i=1,n 0
{ }
ages e , E , and P satisfy
R R R
1
e e K e ,
0 R 2 0
K ≤ ≤
1
1 1
′ ′ ′
(3.20) E E K E E E K E ,
K 0 ≤ R ≤ 2 0 K 0 ≤ R ≤ 2 0
1 (cid:18) 1 (cid:19)
1
P P K P .
0 R 2 0
K ≤ ≤
1
Proof. Note that since the integrand is non-negative, using (3.19) and the lower
bound in (3.18) we obtain
1 1 1 n u2 1 1 1 u2
e = | | φ2δ−1dxdt K | | φ2δ−1dxdt
R T R2n 2 i ≤ T R2n 2 2 0
i=1ZZ ZZ
X
1 1 R 2 u2
K | | φ2δ−1dxdt=K e .
≤ 2T R2 R 2 0 2 0
(cid:18) 0(cid:19) ZZ
Next, we use the upper bound in (3.18) and the non-negativity of the integrand to
bound e from below,
R
1 1 1 n u2 1 1 1 u2
e = | | φ2δ−1dxdt | | φ2δ−1dxdt
R T R3n 2 i ≥ T R3n 2 0
i=1ZZ ZZ
X
1 1 1 R 2 u2 1
| | φ2δ−1dxdt= e ,
≥ T R3K R 2 i K 0
1 (cid:18) 0(cid:19) ZZ 2
arriving at the first relation of (3.20). The other two relations are proved in a
similar manner.
(cid:3)
Note that the lemma above shows that for the the non-negative quantities, like
energy,enstrophy,andpalinstrophy,the ensemble averagesoverthe ballsofsize R,
e , E , and P are comparable to the total space-time average. This is not so for
R R R
the quantities that change signs, like the energy and enstrophy fluxes. In fact Φ
R
andΨ provideameaningfulinformationastoenergyandenstrophytransfersinto
R
balls of size R. Positivity of Ψ , for example, implies that there are at least some
R
regions of size R for which the enstrophy flows inwards.
Moreover,note that the space-time ensemble averagesof energy, enstrophy, and
palinstrophythatcorrespondtotheseoptimalcoverings(overfinitenumberofballs)
areequivalenttotheuniformspace-timeaverage. Wedefinetheuniformspace-time
averageof Θ=Θ (t) as
x,R
2T
1 1 1
(3.21) Θu = Θ (t)dxdt;
R T R2 R2 x,R
0 Z Z
B(0,R0) 0
thuswehavethefollowinguniformaveragesofenergy,enstrophy,palinstrophyand
fluxes in regions of size R: eu, Eu (E′u), Pu, Φu and Ψu.
R R R R R R
2D TURBULENCE IN NSE 9
Lemma 3.2. The following estimates hold
1
e eu 42e ,
22 0 ≤ R ≤ 0
1 1
(3.22) E Eu 42E E′ E′u 42E′ ,
22 0 ≤ R ≤ 0 22 0 ≤ R ≤ 0
(cid:18) (cid:19)
1
P Pu 42P .
22 0 ≤ R ≤ 0
Proof. We will prove the first relation in (3.22), the others follow in a similar way.
Note that the definition of uniform average applied to the energy e (t) yeilds
x,R
1 1 1 u2
eu = | | φ dxdt dy.
R R2 T R2 2 y,R
0 Z (cid:18) ZZ (cid:19)
Bx0
Denote
1 1 u2
F(y)= T R2 | 2| φy,Rdydt.
ZZ
Observethat since the solutionu is continuous, F :B(0,R ) R is continuous as
0
→
well.
Cover B(0,R ) in n cubic cells, C of linear size R/2. Note that
0 i
{ }
4 n 8
≤ ≤
and the area of a cell C is
i
R2
A(C )= .
i
4
If a cell intersects the sphere S(0,R ), we extend F to the whole cell by setting
0
F(y) = 0 on C B(0,R ). Naturally, this extension makes F is measurable (but
i 0
\
not necessarily continuous) on C .
i
∪
Let ǫ>0. Since F is bounded, there exist y¯ ,y C such that
i i ∈ i
ǫ ǫ
F(y¯ ) supF and F(y ) infF + .
i ≥ Ci − 2i i ≤ Ci 2i
Consequently,
n
1 1 1 ǫ
F(y)dy = F(y)dy F(y¯ )+ A(C )
R2 R2 ≤ R2 i 2i i
0B(0Z,R0) 0∪ZCi 0 Xi=1(cid:16) (cid:17)
2 n 2
1 R 1 R
F(y¯ )+ ǫ.
i
≤ 4 R 4 R
(cid:18) 0(cid:19) i=1 (cid:18) 0(cid:19)
X
Note that since F 0 and F = 0 outside B(0,R ), without loss of generality
0
≥
we may assume y¯ B(0,R ). Moreover, the balls B(y¯ ,R) form an optimal
i 0 i
∈ { }
covering of B(0,R ) in the sense of Definition 3.1 with K =82. Thus,
0 2
n n 1 u2 1 u2
R2F(y¯ )= | | φ dxdt K | | φ dxdt=K R2e ,
i T 2 y¯i,R ≤ 2T 2 0 2 0 0
i=1 i=1 ZZ ZZ
X X
and so
1 2 K 1 R 2
eu = F(y)dy 2e + ǫ,
R R ≤ 4 0 4 R
0 Z (cid:18) 0(cid:19)
B(0,R0)
for any ǫ>0, which implies the upper bound in the first relation in (3.22).
10 R.DASCALIUCANDZ.GRUJIC´
To obtain the lower bound, proceed similarly,
n
1 1 1 ǫ
F(y)dy = F(y)dy F(y ) A(C )
R2 R2 ≥ R2 i − 2i i
0B(0Z,R0) 0∪ZCi 0 Xi=1(cid:16) (cid:17)
2 n 2
1 R 1 R
F(y ) ǫ.
≥ 4 R i − 4 R
(cid:18) 0(cid:19) i=1 (cid:18) 0(cid:19)
X
Note that even if y B(0,R ), we still can choose ψ satisfying (2.9)-(2.11)
i 6∈ 0 yi,R
and so the supports of ψ will still cover B(0,R ) and
y ,R 0
i
n n 1 u2 1 u2
R2F(yi)= T | 2| φyi,Rdxdt≥ T | 2| φ0dxdt=R02e0 .
i=1 i=1 ZZ ZZ
X X
Consequently,
2
1 1 1 R
eu = F(y)dy e ǫ,
R R2 ≥ 4 0− 4 R
0 Z (cid:18) 0(cid:19)
B(0,R0)
and, since ǫ > 0 is arbitrary, we obtain the lower bound in the first relation of
(3.22). (cid:3)
Thelemmaaboveallowsustotonotethattheestimatesfortheoptimalensemble
averages, = 1 n thatwillfollowwillalsobevalidfortheuniformaverages,
h·iR n i=1·
= 1 dx.
h·iU R20 B(0,R0)·P
R
4. Enstrophy cascade
Let B(x ,R) be an optimal covering of B(0,R ).
i i=1,n 0
{ }
Note that the local enstrophy equation (2.5) and the definitions of P and Ψ
R R
(see (3.9) and (3.11)) imply
n
1 1 1 1
(4.1) Ψ =νP ω 2(∂ φ +ν∆φ )dxdt
R R− n T R3 2| | t i i
i=1 ZZ
X
where φ = ηψ and ψ = ψ is the spatial cut-off on B(x ,2R) satisfying (2.8-
i i i xi,R i
2.11).
If
R2
(4.2) T 0,
≥ ν
then for any 0<R R ,
0
≤
1 C
(φ ) = η ψ C ηδψ ν 0φ2δ−1,
| i t| | t i|≤ 0T i ≤ R2 i
(4.3)
ν C
ν ∆φ =ν η∆ψ C ηψ2δ−1 ν 0φ2δ−1;
| i| | i|≤ 0R2 i ≤ R2 i
hence,
C
0
Ψ νP ν E .
R ≥ R− R2 R