OPTIMAL BOUNDS ON THE DIMENSION OF THE

ATTRACTOR OF THE NAVIER-STOKES EQUATIONS

Mohammed Ziane

Department of MathematicsIndiana University, Bloomington, Indiana 47405, USA

e-mail: mziane@indiana.edu

November 19, 2003

Abstract. In this article we derive optimal upper bounds on the dimension of the attractorfor the Navier-Stokes equations in two-dimensional domains, these bounds fully agree with the

lower bounds obtained by Babin and Vishik [BV] (see also Ghidaglia and Temam [GT], and Liu

[L2]). As in [BV], we consider here elongated domains and, leaving the density of volume forcesand the viscosity fixed, we let the shape ratio of the domain become large so that the Grashof

number is large. The estimates derived here are based on the general methods for estimating

attractors dimensions as in [CFT2], on a new version of the Lieb-Thirring inequalities forelongated domains and on techniques developed for such domains in [RS] and [TZ].

At the end of the article, we also give some partial results in the 3D case for which we needa physical assumption on the Reynolds number introduced in [GT].

0. INTRODUCTION

In recent years, the concept of finite dimensionality of a turbulent flow has been mademathematically rigorous. The first estimate on the dimension of the attractor for the Navier-Stokes equations was established by C. Foias and R. Temam [FT]; the bound obtained therewas very high and physically irrelevant. Later on, a new approach based on the dynamicalsystem theory led P. Constantin, C. Foias, O. Manley, and R. Temam to estimates on thedimension of attractors [CFMT],[CFT1] which agree with the physical estimates on thenumber of degrees of freedom of turbulent flows in the conventional theory of turbulence[Ko], [Kr].

In this article we mainly consider the two dimensional case. To the best of our knowl-edge, there are no previous estimates giving optimal upper bounds on the dimension of the

1991 Mathematics Subject Classification. 34C35, 35Q30, 76D05.Key words and phrases. Navier Stokes Equations, Lieb-Thirring inequalities, Global attractors, Haus-

dorff and fractal dimensions, Grashof and Reynolds numbers, Elongated domains.

Typeset by AMS-TEX

2 MOHAMMED ZIANE

attractor for the Navier-Stokes equations. This problem was implicitley raised by a numberof authors [BV], [CFT1,2], [GT], [FT], and explicitly raised by V. X. Liu in [L2]: “... itseems that the question whether there are estimates such that the lower bound and upperbound on the dimension of the attractor are of the same order is still an open problem”.Our goal in this work is to give such estimates for two dimensional flows; our result showsthat estimates of the type of those in [CFMT], [CFT1,2], which were known to be physicallyrelevant, are also, in some sense, mathematically optimal.

The lower bound on the dimension of the attractor of the 2D Navier-Stokes equationswas given in a work of Babin and Vishik [BV], where they studied a two-dimensional spaceperiodic flow with period 2πL in direction x2 and 2πL/α in direction x1, 0 < α ≤ 1, αsmall. The approach in [BV] was to leave fixed the viscosity and the density of volumeforces and to let α → 0. The authors choose a specific volume force for which a simplestationary solution can be found. Then, using properties of the Orr-Sommerfeld equations,they were able to estimate the number of unstable modes around the stationary solution .This yields a lower bound on the dimension of the unstable manifold of this solution, andthus a lower bound on the dimension of the attractor of the form:

dim A ≥ cα

;

see [BV] and [GT] for more details.Another approach is to find the lower bound in terms of the Grashof number when the

domain is the unit square; this was undertaken by Liu in [L1]; the lower bound derived in[L1] agrees with the upper bound obtained in [CFT2] up to a logarithmic factor.

In this work we adopt the approach of [BV], and derive an upper bound on the dimensionof the attractor of the 2D Navier-Stokes equations, which agrees fully with the lower boundobtained in [BV]; see also [GT] and [L2]. We obtain for small α the following:

c0α

≤ dim A ≤ c1α,

where c0 and c1 depend on the viscosity and the density of the volume forces but not onthe shape ratio α. In other words, we derive upper bounds which are optimal with respectto the shape factor of the domain; i.e., the dimension of the attractor can be as large as theupper bound. The dependence of the dimension of the attractor with respect to the shapeof the domain is also considered in [DW] for a shear flow in a rectangle and in [MZ] for theBénard problem; see also [Z].

The methods developped in the present article extend to the three dimensional case.We give at the end some partial results on the upper bound on the dimension of regularinvariant sets of the Navier-Stokes equations obtained by using the same methods.

The article is organized as follows: In Section 1, we introduce the problem and give somepreliminary results. Section 2 is devoted to the new estimate on the upper bound in termsof the shape ratio, and in Section 3 we present some partial results for the three dimensionalNavier-Stokes equations.

NAVIER-STOKES EQUATIONS 3

1. Preliminaries

Let Ωα = (0, L/α) × (0, L) be a rectangle elongated in the the direction x1. Here L isa typical macroscopic length and 0 < α ≤ 1 is a small non dimensional parameter. TheNavier-Stokes equations are given by:

∂u

∂t− ν∆u+ (u · ∇)u+ ∇p = f in Ωα × (0,∞),(1.1)

div u = 0 in Ωα × (0,∞),(1.2)u(·, 0) = u0(·) in Ωα,(1.3)

where the unknowns are the velocity u = u(x, t) of the particle at point x at time t and thepressure p = p(x, t) at x at time t. The kinematic viscosity ν, the volume force f and theinitial velocity u0 are given.

Equations (1.1)-(1.3) are supplemented with the space periodic boundary conditions:

(1.4) u(x1 +L

α, x2, t) = u(x1, x2 + L, t) = u(x1, x2, t),

(and the same for p). We also assume that

(1.5)

∫

Ωα

u0(x)dx = 0 and

∫

Ωα

f(x)dx = 0.

For the functional setting of the problem (1.1)-(1.5), we consider the space

(1.6) Ḣm(Ωα) =

{

u ∈ Hm(Ωα);∫

Ωα

udx = 0

}

,

and the spaces Hmper(Ωα), which are defined with the help of Fourier series; we write

(1.7) u(x) =∑

k∈Z2

ukexp(2ik ·x

L),

with ūk = u−k (so that u is real valued). Then, u is said to be in L2(Ωα) if and only if

∑

k∈Z2

|uk|2

4 MOHAMMED ZIANE

Now, we introduce

H =

{

u ∈ L2(Ωα); div u = 0;∫

Ω

udx = 0, uj |Γj = uj |Γj+2; j = 1, 2}

,

V = {u ∈ Ḣ1per(Ωα); div u = 0},

These spaces are endowed with the usual norms and scalar products which we denote by| · | and (·, ·) for H and by || · || and ((·, ·)) for V. The Stokes operator A is defined as anisomorphism from V onto the dual V ′ of V, by

∀v ∈ V ; < Au, v >V ′,V = ((u, v)).

The operator A can be considered also as a linear unbounded operator in H with domainD(A) = H ∩ Ḣ2(Ωα).

Let b(., ., .) be the continuous trilinear form on V defined by:

b(u, v, w) =

2∑

i,j=1

∫

Ωα

ui∂vj∂xi

wjdx, u, v, w ∈ H1(Ωα).

We denote by B the bilinear form defined for (u, v) ∈ V × V by

< B(u, v), w >V ′,V = b(u, v, w), ∀w ∈ V, and B(u) = B(u, u).

We assume in this work that the data ν, u0 and f are fixed and satisfy

ν > 0, u0 ∈ H (or V ), f ∈ H.

The system of equations (1.1)–(1.4) can be written as a differential equation in V ′ :

(1.8)

{

u′ + νAu+B(u) = f,

u(0) = u0,

where u′ denotes the derivative (in the distribution sense) of the function u with respect totime.

1.1. The lower bound on the dimension of the attractor.

The lower bound on the dimension of the attractor was derived in [BV] for special forcesf of the form f(x1, x2) = (g(x2), 0), with

(1.9)

∫ L

0

g(x2) dx2 = 0.

NAVIER-STOKES EQUATIONS 5

It is then readily seen that (1.1)-(1.3) possess a stationary solution us of the form (U(x2), 0),and p = 0. The derivation of the lower bound is based on a linearization of (1.1)-(1.4) aroundus, which reduces to the Orr-Sommerfeld equation

(1.10) (D4 − 2α2D2 + α4)Ψ = iαν

[(U − iσ)(D2 − α2) − U ′′]Ψ, D = ddx2

,

where Ψ is given in terms of the stream function Φ by

(1.11) Φ(x1, x2, t) = Ψ(x2) exp(α

Lσt+ iα

x1L

).

Then, if θ is the 2π-periodic function with 0 average stisfying θ′ = U , it was shown in [BV]that

(1.12) For ν satisfying ν2 <1

L

∫ L

0

[θ(y)]2dy,

there exists δ0 that depends onUν

(or g/ν), such that, if 0 < α ≤ δ0, there exists a familyof solutions {Ψα, σ(α)} of (1.10) satisfying

(1.13)

∫ 2πL

0

[Ψα(y)]2dy = 1, Re σ(α) > 0, and

d

dαReσ(α) > 0.

Hence for fixed α, α ≤ δ0, and if k is the largest integer satisfying αk ≤ δ0, we can associatewith each l = 1, . . . k a pair of solutions of (1.10), Ψlα, σ(lα), that yields an unstablesolution to the equations (1.1)-(1.3) linearized around us. Now according to (1.13) theσ(lα), l = 1, . . . k are different and thus Ψlα, l = 1, . . . , k are linearly independent. Thisshows that the dimension of the unstable manifold of us has dimension at least equal tok ≥ δ0

α− 1. Therefore, (see [T]) the lower bound on the dimension of the attractor, when

the volume forces are of the form (g(x2), 0) with zero average, is given by

(1.14) dim Aα ≥δ0α

− 1.

In the three-dimensional case, using Squire’s transformation, it was shown in [GT] (seealso [L2]), that for the special class of forces of the form (g(x3), 0, 0) with zero average,there exists d0 = d0(

gν) and a0 = a0(

gν) such that for α2 + β2 ≤ a20, we have

(1.15) dim Aα,β ≥d0αβ

− 1.

Remark 1.1. As far as the dependence on the shape ratio is concerned, the lower boundsobtained in [L2] are of the same order as the ones given by (1.14) and (1.15).

6 MOHAMMED ZIANE

2. The upper bound on the dimension of the attractor in the 2D case

The purpose of this section is to obtain an optimal upper bound on the dimension ofthe attractor in terms of the shape ratio. We do this by working in an elongated domain

Ωα = (0, L/α) × (0, L) and finding the upper bound dim A ≤c1α

, for the class of volume

forces f = f(x1, x2) satisfying

∫ L

0

f(x1, x2)dx2 = 0.

2.1. Preliminaries.

We define the average operator in the direction x2 “the thin direction” as

(2.1) Mu(x1, t) =1

L

∫ L

0

u(x1, x2, t)dx2,

and set Nu = u−Mu. The following statements are straightforward (see [TZ] for details):

M and N are orthogonal projectors in L2(Ωα) and H1p(Ωα); More precisely

(2.2) M2 = M, N2 = N, MN = 0, M∂

∂xi=

∂

∂xiM, i = 1, 2,

(2.3)

∫

Ωα

∇Nu · ∇Mvdx = 0, for all u, v ∈ H1p(Ωα),

(2.4) |u|2 = |Mu|2 + |Nu|2 and ||u||2 = ||Mu||2 + ||Nu||2, ∀ u ∈ H1p(Ωα).

Note that Nu satisfies the following Poincaré inequality with constant independent of α(see [TZ]):

(2.5) |Nu|2 ≤ L2||Nu||2.

We will need the following version of the Lieb-Thirring inequalities established in [Z].First we introduce some notations. Let ϕ1, . . . , ϕm be a finite family of functions in H

1(Ωα)which are sub-orthonormal in L2(Ωα), in the sense that

(2.6)m

∑

i,j=1

ξiξj

∫

Ωα

ϕiϕjdx ≤m

∑

k=1

ξ2k, ∀ ξ ∈ Rm.

NAVIER-STOKES EQUATIONS 7

We set for almost every x ∈ Ωα,

ρ(x) = ρϕ(x) =

m∑

j=1

|ϕj(x)|2,(2.7)

ρM (x) = ρMϕ(x) =

m∑

j=1

|Mϕj(x)|2, and(2.8)

ρN (x) = ρNϕ(x) =

m∑

j=1

|Nϕj(x)|2,(2.9)

where |.| denotes the absolute value. Note here that with (2.4)

(2.10)

∫

Ωα

ρ2ϕ(x)dx =

∫

Ωα

ρ2N (x)dx+

∫

Ωα

ρ2M (x)dx.

Proposition 2.1. Let T be a linear bounded operator in L2(Ωα) with norm ≤ 1. Letϕ1, . . . , ϕm be a finite family of functions which are sub-orthonormal in L

2(Ωα), in thesense of (2.6) Then Tϕ1, . . . , Tϕm are sub-orthonormal in L

2(Ωα)

Proof. The proof is based on the following simple calculation:

m∑

i,j=1

ξiξj

∫

Ωα

TϕiTϕjdx =

∫

Ωα

( m∑

k=1

ξkTϕk(x)

)2

dx =

∫

Ωα

∣

∣

∣

∣

T(

m∑

k=1

ξkϕk(x))

∣

∣

∣

∣

2

dx

≤∫

Ωα

( m∑

k=1

ξkϕk(x)

)2

dx

≤m

∑

k=1

ξ2k, ∀ ξ ∈ Rm.

Now, we state an anisotropic Lieb-Thirring inequality proven in [Z] (see (2.11)) and someconsequences:

Lemma 2.1. There exists an absolute constant c0 (independent of L and α) such that, for

every finite family{

ϕj}m

j=1in H1(Ωα) which is sub-orthonormal in L

2(Ωα), we have

(2.11)∫

Ωα

ρ2(x)dx ≤ c0( m

∑

j=1

∣

∣

∂ϕj∂x1

∣

∣

2

L2(Ωα)+α2

L2|ϕj |2L2(Ωα)

)1

2

( m∑

j=1

∣

∣

∂ϕj∂x2

∣

∣

2

L2(Ωα)+

1

L2|ϕj |2L2(Ωα)

)1

2

.

Moreover, we have

(2.12)

∫

Ωα

ρ2N (x)dx ≤ c0m

∑

j=1

∣

∣

∣

∣∇Nϕj∣

∣

∣

∣

2,

8 MOHAMMED ZIANE

and

(2.13)

∫

Ωα

ρ2M (x)dx ≤c0L

( m∑

j=1

∣

∣Mϕj∣

∣

2

L2(Ωα)

)1

2

( m∑

j=1

∣

∣

∣

∣Mϕj∣

∣

∣

∣

2

L2(Ωα)

)1

2

.

Proof. The proof of (2.11) is given in [Z, Lemma 4.1]. For the proof of (2.12) and (2.13), wenote that, thanks to Proposition 2.1, if ϕ1, . . . , ϕm is a finite family of functions in H

1(Ωα)which are sub-orthonormal in L2(Ωα), then the families Mϕ1, . . . ,Mϕm and Nϕ1, . . . , Nϕmare both sub-orthonormal in L2(Ωα). Now (2.12) and (2.13) follow from (2.11).

Finally, we derive an a priori estimate on the solutions of the Navier-Stokes equations(1.1)-(1.3), which shows the boundness of the Grashof number when α goes to zero. Wewrite, (with Mf = 0),

(2.15)1

2

d

dt|u|2 + ν||u||2 = (f, u) = (Mf,Mu) + (Nf,Nu) = (Nf,Nu).

Thanks to the Poincaré inequality (2.5), we have

(2.16)d

dt|u|2 + ν||u||2 ≤ L

2|f |2ν2

.

Hence,

(2.17) lim supt→∞

1

t

∫ t

0

||u(τ)||2dτ ≤ L2|f |2ν2

.

2.2. The upper bound on the dimension of the attractor in the two-dimensional

torus.

We will use the general theory developed by P. Constantin, C. Foias and R. Temam[CFT2], and follow the presentation given in [T], to which we refer the reader for moredetails. Let u0 ∈ H and u(t) = S(t)u0, t ≥ 0. The linearized flow around u is given by

U ′ = F ′(u)U,

whereF (u) = f −Au−B(u, u),

andF ′(u)U = −νAU −B(u, U)−B(U, u),

i.e.,

(2.18)U ′ + νAU +B(u, U) +B(U, u) = 0 in V ′,

U(0) = ξ, ξ ∈ H given.

NAVIER-STOKES EQUATIONS 9

One can show that

Given ξ ∈ H, there exists a unique U ∈ L∞(0, T ;H) ∩ L2(0, T ;V ), ∀ T > 0,satisfying (2.18). Moreover, U ′ ∈ L2(0, T ;V ′) and U ∈ C([0, T ];H), ∀ T > 0.

Now we define the linear map L(t; u0) : H 7→ H by L(t; u0)ξ = U(t). One can show thatL(t; u0) is bounded and that

{

S(t)}

t≥0is uniformly differentiable on the attractor A. Con-

sider m solutions U1, . . . , Um corresponding to the initial data ξ1, . . . , ξm, with ξi ∈ H, 1 ≤i ≤ m. With u = u(τ) = S(τ)u0, u0 ∈ H, we denote by Q̃m(τ) = Q̃m(τ, u0; ξ1, . . . ξm) theorthogonal projector in H onto the space spanned by U1(τ), . . . , Um(τ). For each time τ ,

let ϕj(τ), j = 1, . . . , m be an orthonormal basis of Q̃m(τ)H. We have

(2.19)

Tr F ′(

u(τ))

◦ Q̃m(τ) = −νm

∑

j=1

∣

∣

∣

∣ϕj(τ)∣

∣

∣

∣

2+

m∑

j=1

b(ϕj , ϕj, u)

= −νm

∑

j=1

∣

∣

∣

∣ϕj(τ)∣

∣

∣

∣

2+

m∑

j=1

b(ϕj , u, ϕj)

= −νm

∑

j=1

∣

∣

∣

∣Mϕj(τ)∣

∣

∣

∣

2 − νm

∑

j=1

∣

∣

∣

∣Nϕj(τ)∣

∣

∣

∣

2

+

m∑

j=1

b(Mϕj +Nϕj ,Mu+Nu,Mϕj +Nϕj).

Now note that div ϕj = 0 and∫

Ωαϕjdx = 0 imply that the first component Mϕ

1j of Mϕj

is zero, which yields

(2.20) b(Mϕj ,Mϕj,Mu+Nu) = 0.

Moreover, thanks to (1.2), we have

(2.21) b(Mϕj,Mu,Nϕj) = 0 and b(Nϕj ,Mu,Mϕj) = 0.

Hence,

(2.22)b(Mϕj +Nϕj ,Mu+Nu,Mϕj +Nϕj) = b(Nϕj , u, Nϕj) + b(Mϕj, Nu,Nϕj)

+ b(Nϕj , Nu,Mϕj).

Cauchy-Schwarz inequality yields

(2.23)

m∑

j=1

b(ϕj , u, ϕj) ≤ ||u|||ρN |L2 + 2||Nu|||ρM |1

2

L2 |ρN |1

2

L2 ,

and, by Lemma 2.1, we have

(2.24)m

∑

j=1

b(ϕj, u, ϕj) ≤ν

4

m∑

j=1

||ϕj ||2 +c04ν

||u||2 + c02ν

||Nu||2 + ν2c0

|ρM |L2 |ρN |L2 .

10 MOHAMMED ZIANE

Since Mϕ1, . . . ,Mϕm is suborthonormal in L2,

(2.25)

ν

2c0|ρM |L2 |ρN |L2 ≤

ν

2L1

2

m1

4

( m∑

j=1

||Mϕj||2)

1

4

( m∑

j=1

||Nϕj||2)

1

2

≤ ν4

m∑

j=1

||Nϕj||2 +νm

1

2

4L

( m∑

j=1

||Mϕj||2)

1

2

≤ ν4

m∑

j=1

||Nϕj||2 +ν

2

m∑

j=1

||Mϕj||2 +νm

2L2.

Thus

(2.26)

m∑

j=1

b(ϕj , u, ϕj) ≤ν

2

m∑

j=1

||Nϕj||2 +ν

2

m∑

j=1

||Mϕj||2 +3c04

||u||2 + νm2L2

,

and

(2.27) Tr F ′(

u(τ))

◦ Q̃m(τ) ≤ −ν

2

m∑

j=1

||Nϕj||2 −ν

2

m∑

j=1

||Mϕj||2 +3c04

||u||2 + νm2L2

.

Now note that

(2.28) m =

∫

Ωα

ρ(x)dx ≤ Lα

1

2

(∫

Ωα

ρ2(x)dx

)1

2

=L

α1

2

[∫

Ωα

ρ2M (x)dx+

∫

Ωα

ρ2N (x)dx

]1

2

,

so that

(2.29)

αm2

L2≤ |ρ2M |L2 + |ρ2N |L2≤ (with Lemma 2.1)

≤ c0m

∑

j=1

||Nϕj||2 +c0L

( m∑

j=1

|Mϕj|2)

1

2

( m∑

j=1

||Mϕj||2)

1

2

and, since∑mj=1 |Mϕj |2L2 ≤ m, we can write

(2.30)

αm2

L2≤ c0

m∑

j=1

||Nϕj ||2 +c0m

1

2

L

( m∑

j=1

||Mϕj||2)

1

2

≤ c0m

∑

j=1

||Nϕj ||2 + c0m

∑

j=1

||Mϕj||2 +c0m

4L2.

NAVIER-STOKES EQUATIONS 11

Therefore

(2.31) Tr F ′(

u(τ))

◦ Q̃m(τ) ≤ −ναm2

2c0L2+

3c04ν

+3νm

4L2,

and since

(2.32)3νm

4L2=

3√

2νc0L√α

· m√να

L√

2c0≤ ναm

2

4c0L2+

9c0ν

16αL2,

we find

(2.33) Tr F ′(

u(τ))

◦ Q̃m(τ) ≤ −ναm2

4c0L2+

3c04ν

||u||2 + 9c0ν16αL2

.

We set

(2.34)

qm(t) = supu0∈A

supξi∈Hα|ξi|≤1i=1,...,m

(

1

t

∫ t

0

Tr F ′(

u(τ))

◦ Q̃m(τ)dτ)

,

qm = lim supt→∞

qm(t),

Inequality (2.33) yields

(2.35) qm ≤ −ναm2

4c0L2+

3c04ν

γ +9c0ν

16αL2,

where

(2.36)

γ ≤ lim supt→∞

supu0∈A

1

t

∫ t

0

||u(τ)||2dτ

≤ (with (2.17))

≤ L2|Nf |2ν2

≤ L2|f |2ν2

.

Hence,

(2.37)

qm ≤ −ναm2

4c0L2+

3c04ν

L2|f |2ν2

+9c0ν

16αL2

≤ − να4c0L2

[

m2 − 3c20G2

α− 9

4c0

1

α2

]

,

where G = L2|f |ν2 is a nondimensional Grashof number.

Finally, note that if m ≥ c1(

1α +

G

α1

2

)

, where c1 = 2(c0 +√c0) is an absolute constant

(independent of L and α), then qm < 0. Therefore, the procedure developed in [T] yieldsthe following upper bound on the Hausdorff and fractal dimensions of the attractor:

(2.38) dim Aα ≤ c1( 1

α+

G

α1

2

)

.

12 MOHAMMED ZIANE

Theorem 2.1. The global attractor Aα that describes the long time behavior of (1.1)-(1.3)in Ωα =

(

0,L

α

)

× (0, L) together with the space periodic boundary condition, and with

volume force f = (f1(x1, x2), f2(x1, x2)) satisfying

∫ L

0

f(x1, x2)dx2 = 0 has a Hausdorff

and a fractal dimensions bounded by

c( 1

α+Ḡ

α

)

,

where c1 is non dimensional absolute constant, and Ḡ = α1

2G is independent of α.

Note that the condition Mf = 0 was used only to find an appropriate estimate for

lim supt→∞

1

t

∫ t

0

||u(τ)||2dτ.

In the case where Mf 6= 0, the procedure given above yields the following general result:Theorem 2.2. For f ∈ H and Mf 6= o, the global attractor Aα that describes the long timebehavior of (1.1)-(1.3) in Ωα =

(

0,L

α

)

× (0, L) together with the space periodic boundarycondition has a Hausdorff and a fractal dimensions bounded by

(2.39) c1

[

L2|Nf |L2ν2α

1

2

+1

α+L2|Mf |L2ν2α

3

2

]

,

where c1 is non dimensional absolute constant.

3. Partial Results in the three-dimensional case

In this section we apply the methods above in space dimension three and obtain partialresults: we derive an upper bound for regular functional invariant sets which, due to [GT]appears to be optimal with respect to the shape factor. Let Ω = (0, L/α)×(0, L/β)×(0, L),where 0 < β ≤ α ≤ 1. Since the existence and uniqueness of solutions to the 3D Navier-Stokes equations is still an open problem, we will make the following assumptions, whichare shown to be physically reasonable in [GT]:

(i) The H1-norm of the solution u(t) remains bounded for all time on the attractor orthe regular invariant set Aαβ under consideration, i.e.,(3.1) sup

t∈R||u(t)||

NAVIER-STOKES EQUATIONS 13

Preliminaries.

For notational sake, we write L1 = L/α, L2 =, L/β and L3 = L. Let M3 and M1 be theaverage operators in the directions x3 and x1, i.e.

(3.3) M3u =1

L

∫ L

0

u(x1, x2, x3)dx3, M1u =α

L

∫ Lα

0

u(x1, x2, x3)dx1.

Set

(3.4) N3u = u−M3u, Pu = M1(M3u) and Qu = M3u−M1(M3u).

Note that M3, N3, P and Q are all orthogonal projectors in H and V.

Let{

ϕj}j=m

j=1be a finite family of functions inH1(Ω) which are sub-orthonormal in L2(Ω).

Then, thanks to Proposition 2.1,{

M3ϕj}j=m

j=1,{

N3ϕj}j=m

j=1,{

Pϕj}j=m

j=1and

{

Qϕj}j=m

j=1are

all sub-orthonormal in L2(Ω).Now we recall the anisotropic version of the Lieb-Thirring inequality of [Z] in dimension

three (see also [TZ] for a number of anisotropic functional inequalities):

Lemma 3.1. Let Ω = (0, L1) × (0, L2) × (0, L3). There exists an absolute constant c0(independent of L1, L2 and L3) such that, for every finite family

{

ϕj}m

j=1in H1(Ω) which

is sub-orthonormal in L2(Ω), we have

(3.5)

∫

Q

ρ5

3 (x)dx ≤ c03

∏

i=1

( m∑

j=1

∣

∣

∂ϕj∂xi

∣

∣

2

L2(Ω)+

1

L2i|ϕj |2L2(Ω)

)1

3

,

where

(3.6) ρ(x) = ρϕ(x) =

m∑

j=1

|ϕj(x)|2.

We introduce the following:

(3.7)

ρN (x) = ρNϕ(x) =m

∑

j=1

|Nϕj(x)|2,

ρP (x) = ρPϕ(x) =

m∑

j=1

|Pϕj(x)|2,

ρQ(x) = ρQϕ(x) =m

∑

j=1

|Qϕj(x)|2.

With these notations, we state and prove

14 MOHAMMED ZIANE

Lemma 3.2. There exists an absolute constant c2 such that, for every finite family{

ϕj}m

j=1

in H1(Ω) which is sub-orthonormal in L2(Ω), we have

|ρN |L5/3 ≤ c2( m

∑

j=1

||Nϕj||2)3/5

,(3.8)

|ρP |L5/3 ≤ c2m2/5

L4/5α2/5

( m∑

j=1

||Pϕj||2)1/5

,(3.9)

|ρQ|L5/3 ≤ c2m1/5

L2/5

( m∑

j=1

||Qϕj ||2)2/5

.(3.10)

Proof. (i) Note that Nϕj , j = 1, . . . , m satisfies the Poincaré inequality

(3.11) |Nϕj |L2 ≤ L||Nϕj||.

Hence, since 0 < α ≤ 1, 0 < β ≤ 1, inequality (3.8) follows from Lemma 3.1.(ii) The functions Pϕj , j = 1, . . . , m depend only on x2 and, since

∫

Ω

ϕj(x)dx = 0, we have

(3.12)

∫ L/β

0

Pϕj(x)dx2 = 0.

Hence, Poincaré’s inequality yields

(3.13) |Pϕj |L2 ≤L

β||Pϕj||

and Lemma 3.1 implies that

(3.14) |ρP |L5/3 ≤ c3/50

α2/5

L4/5

( m∑

j=1

|Pϕj |2)2/5( m

∑

j=1

||Pϕj||2)1/5

.

Since

m∑

j=1

|Pϕj |2 ≤m

∑

j=1

|ϕj |2 = m, we obtain (3.9).

(iii) Note that the Qϕj , j = 1, . . . , m do not depend on x3 and Poincaré’s inequality yields

(3.15) |Qϕj |L2 ≤L

α||Qϕj||.

Thanks to Lemma 3.2, we have

(3.16) |ρQ|L5/3 ≤ c3/50

m1/5

L2/5

( m∑

j=1

∣

∣

∂Qϕj∂x1

∣

∣

2+α2

L2|Qϕj |2

)1/3( m∑

j=1

∣

∣

∂Qϕj∂x2

∣

∣

2+β2

L2|Qϕj |2

)1/3

,

and since β ≤ α, we infer (3.10).

NAVIER-STOKES EQUATIONS 15

Lemma 3.3. There exists an absolute constant c3 such that, for every finite family{

ϕj}m

j=1

in H1(Ω) which is sub-orthonormal in L2(Ω), we have

|ρϕ|L5/3 ≤ c3m

∑

j=1

||ϕj||2 +αm

L2+m

L2,(3.17)

m∑

j=1

||ϕj ||2 ≥ c3m5/3(αβ)2/3

L2− αm

L2− mL2.(3.18)

Proof. (i) Inequality (3.17) is a consequence of Lemma 3.2 and of the Young inequalities

(3.19)

m2

3α2

3

L4

3

(

m∑

j=1

||Pϕj ||2)

1

3 ≤m

∑

j=1

||Pϕj ||2 +αm

L2,

m1

3

L2

3

(

m∑

j=1

||Qϕj ||2)

2

3 ≤m

∑

j=1

||Qϕj ||2 +m

L2.

(ii) For (3.18) we first recall that if a, b, c are positive numbers,

(a+ b+ c)5

3 ≤ 2(a 53 + b 53 + c 53 ).Now we write

(3.20)

m =

∫

Ω

ρϕ(x)dx ≤(

∫

Ω

dx

)2

5

|ρϕ|L

5

3 (Ω)

≤ L6

5

(αβ)2

5

[|ρN |L

5

3 (Ω)+ |ρP |

L5

3 (Ω)+ |ρQ|

L5

3 (Ω)].

Hence

(3.21)

m5

3 (αβ)2

3

L2≤ 2[|ρN |

5

3

L5

3 (Ω)+ |ρP |

5

3

L5

3 (Ω)+ |ρQ|

5

3

L5

3 (Ω)]5

3

≤ c3[ m

∑

j=1

||Nϕj ||2 +m

2

3α2

3

L4

3

(

m∑

j=1

||Pϕj ||2)

1

3 +m

1

3

L2

3

(

m∑

j=1

||Qϕj||2)

2

3

]

,

and (3.18) follows, thanks to (3.19). �

We are ready to derive the upper bound on the Hausdorff dimension of any regularinvariant set of the 3D Navier-Stokes equations. Keeping the same notations as in Section2, we have

(3.22)

Tr F ′(

u(τ))

◦ Q̃m(τ) = −νm

∑

j=1

∣

∣

∣

∣ϕj(τ)∣

∣

∣

∣

2 −m

∑

j=1

b(ϕj , ϕj, u)

≤ −νm

∑

j=1

∣

∣

∣

∣ϕj(τ)∣

∣

∣

∣

2+m

1

2 |u(τ)|L∞(Ω)(

m∑

j=1

∣

∣

∣

∣ϕj(τ)∣

∣

∣

∣

2) 12

≤ −ν2

m∑

j=1

∣

∣

∣

∣ϕj(τ)∣

∣

∣

∣

2+m

ν|u(τ)|2L∞(Ω).

16 MOHAMMED ZIANE

Thanks to Lemma 3.3, we have

(3.23) Tr F ′(

u(τ))

◦ Q̃m(τ) ≤ −c0m

5/3(αβ)2/3

2νL2+c4 αm

2νL2+

c0m

2νL2+m

ν|u(τ)|2L∞(Ω).

where c0, c4 are constants independent of α and β. We then set

(3.24)

qm(t) = supu0∈Aαβ

supξi∈H|ξi|≤1i=1,...,m

(

1

t

∫ t

0

Tr F ′(

u(τ))

◦ Q̃m(τ)dτ)

,

qm = lim supt→∞

qm(t),

Inequality (3.23) implies that

(3.25)qm ≤ −

c6 ν m5/3(αβ)2/3

2L2+c6ν m

L2+ν m

L2Re2

≤ −c6 νmL2

[

(mαβ)2

3 − 1 − Re2],and an upper bound on the dimension of the attractor or the regular invariant set Aαβ isgiven by

(3.26) dim Aαβ ≤ c71 + Re3

αβ,

where c7 is an absolute constant.

Theorem 3.1. Assume that the density of volume force has zero average in the thin direc-tion x3 and that the assumptions (i) and (ii) hold. Let Aαβ be a functional invariant setbounded in V . Then, there exists an absolute constant c7 such that (compare to (1.15)):

(3.27) dim Aαβ ≤ c71

αβ.

Appendix. The Lieb-Thirring inequality in rectangular domains

We establish a version of Lieb-Thirring inequalities in the rectangular domain Q =(0, L1)×(0, L2).The inequality that we obtain yields the explicit dependence of the constantson the shape of the domain (i.e. the ratio L1/L2). The proof given here is borrowed from[Z]

Let ϕ1, . . . , ϕN be a finite family of functions in H1(Q) which are sub-orthonormal in

L2(Q) in the sense that

(A.1)

N∑

i,j=1

ξiξj

∫

Q

ϕiϕjdx ≤N

∑

k=1

ξ2k, ∀ ξ ∈ RN .

We set for almost every x ∈ Q,

(A.2) ρ(x) = ρϕ(x) =

N∑

j=1

|ϕj(x)|2,

where |.| denotes the absolute value. With these notations, we state

NAVIER-STOKES EQUATIONS 17

Proposition A.1. Let Q0 = (0, 1)2, and let

{

ϕj}N

j=1be a finite family of functions in

H1(Q0) which is sub-orthonormal in L2(Q0), i.e.

(A.3)

N∑

i,j=1

ξiξj

∫

Q0

ϕiϕjdx ≤N

∑

k=1

ξ2k, ∀ ξ ∈ RN .

Then, there exists an absolute constant c2 such that

(A.4)

∫

Q0

ρ2(x)dx ≤ c22

∏

i=1

( N∑

j=1

∣

∣

∂ϕj∂xi

∣

∣

2

L2(Q0)+ |ϕj|2L2(Q0)

)1

2

,

where

ρϕ(x) =N

∑

j=1

|ϕj(x)|2, a.e. x ∈ Q0.

Proof of Proposition A.1. The proof is done in several steps:

Step1. Let{

ϕj}N

j=1be a finite family of functions in H10 (R

2) which is sub-orthonormal in

L2(R2), and for k1 > 0 and k2 > 0, define the family{

ψj}N

j=1as

(A.5) ψj(y1, y2) =1√k1k2

ϕj(y1k1,y2k2

).

It is clear that ψj ∈ H10 (R2), j = 1, . . . , N, and that the family{

ψj}N

j=1is sub-orthonormal

in L2(R2). Therefore, Corollary 2.4 of [GMT] implies the existence of an absolute constantc1 such that

(A.6)

∫

Q

ρ2ψ(x)dx ≤ c12

∑

i=1

N∑

j=1

∣

∣

∂ψj∂yi

∣

∣

2

L2(R2).

Now we note that

(A.7)

∫

R2

ρ2ψ(x)dx =

∫

R2

1

(k1k2)2(

N∑

j=1

ϕ2j (x))2k1k2dx =

1

k1k2

∫

R2

ρ2ϕ(x)dx,

and

(A.8)

∫

R2

∣

∣

∂ψj∂yi

∣

∣

2dy =

1

k2i

∫

R2

∣

∣

∂ϕj∂xi

∣

∣

2dx.

Therefore, (A.6) can be rewritten as

(A.9)

∫

R2

ρ2ϕ(x)dx ≤ c1(k1k2)2

∑

i=1

1

k2i

( N∑

j=1

∫

R2

∣

∣

∂ϕj∂xi

∣

∣

2dx

)

.

18 MOHAMMED ZIANE

The left hand side of (A.9) is independent of k1 and k2, while the right hand side has aminimum (as a function of k1 and k2) which is obtained when

(A.10) k2i =N

∑

j=1

∫

R2

∣

∣

∂ϕj∂xi

∣

∣

2dx.

Therefore,

(A.11)

∫

R2

ρ2ϕ(x)dx ≤ 2c1( N

∑

j=1

∫

R2

∣

∣

∂ϕj∂x1

∣

∣

2dx

)1

2

( N∑

j=1

∫

R2

∣

∣

∂ϕj∂x2

∣

∣

2dx

)1

2

.

Step2. The localization argument:We extend each function ϕj , j = 1, . . . , N to a larger cube Q1 = (−12 , 32)2 by the Babitch

extension operators

(A.12) E1kϕj(x) =

ϕj(x) for x1 ∈ [0, 1],2

∑

r=1

(−r)kαrϕ(−rx1, x2) for x1 ∈ [−1

2, 0),

2∑

r=1

(−r)kαrϕ(1 − r(x1 − 1), x2) for x1 ∈ (1,3

2),

where (α1, α2) is the unique solution to the system

2∑

r=1

(−r)kαr = 1, k = 0, 1.

We have, see [TZ] for details,

(A.13) ∀ ϕ ∈ H1(Q0),∣

∣

∂kEϕ

∂xki

∣

∣

2

L2(Q1)≤ c23

∣

∣

∂kϕ

∂xki

∣

∣

2

L2(Q0), k = 0, 1; i = 1, 2,

where c3 is an absolute constant. Now we show that{

1c3Eϕj

}N

j=1is sub-orthonormal in

L2(Q1). Indeed

(A.14)

1

c23

N∑

i,j=1

ξiξj

∫

Q1

EϕiEϕjdx =1

c23

∫

Q1

∣

∣

∣

∣

E(

N∑

k=1

ξkϕk)

∣

∣

∣

∣

2

dx

≤ (with (A.13))

≤∫

Q0

∣

∣

∣

∣

N∑

k=1

ξkϕk

∣

∣

∣

∣

2

dx

=N

∑

i,j=1

ξiξj

∫

Q0

ϕiϕjdx ≤N

∑

k=1

ξ2k.

NAVIER-STOKES EQUATIONS 19

Now let ϕ ∈ C∞0 (R) be such that

(A.15) support (ϕ) ⊂(

−12,3

2

)

, 0 ≤ ϕ ≤ 1 and ϕ = 1 on [0, 1].

Set

Φ(x1, x2) = ϕ(x1)ϕ(x2),

and let

(A.16) ψj = Φ(x1, x2)E(ϕj)

c3, j = 1, . . . , N.

It is clear that ψj ∈ H10 (R2). Moreover,

(A.17)N

∑

i,j=1

ξiξj

∫

R2

Φ2

c23E(ϕi)E(ϕj)dx ≤

N∑

i,j=1

ξiξj

∫

R2

E(ϕi)E(ϕj)

c21dx ≤

N∑

k=1

ξ2k.

Therefore, the family{

ψj}N

j=1is sub-orthonormal in L2(R2), and (A.17) yields

(A.18)

∫

R2

ρ2ψ(x)dx ≤ 2c12

∏

i=1

( N∑

j=1

∫

R2

(∣

∣

∂ψj∂xi

∣

∣

2)dx

)1

2

≤ 2c1c232

∏

i=1

( N∑

j=1

∫

R2

(∣

∣

∂ϕj∂xi

∣

∣

2+ |ϕj|2

)

dx

)1

2

.

Moreover,

(A.19)

∫

R2

ρ2ψ(x)dx =

∫

Q1

( N∑

k=1

∣

∣

ΦE(ϕj)

c3(x)

∣

∣

2)2

dx

=1

c43

∫

Q1

Φ4( N

∑

k=1

∣

∣E(ϕj)(x)∣

∣

2)2

dx ≥∫

Q0

( N∑

k=1

∣

∣ϕj(x)∣

∣

2)2

dx.

Hence,

(A.20)

∫

Q0

ρ2ϕ(x)dx ≤ c22

∏

i=1

( N∑

j=1

∣

∣

∂ϕj∂xi

∣

∣

2

L2(Q0)+ |ϕj|2L2(Q0)

)1

2

.

The proof of Proposition A.1 is complete. �

20 MOHAMMED ZIANE

Lemma A.1. Let Q = (0, L1)× (0, L2). There exists an absolute constant c0 (independentof L1 and L2) such that, for every finite family

{

ϕj}N

j=1in H1(Q) which is sub-orthonormal

in L2(Q), we have

(A.21)

∫

Q

ρ2(x)dx ≤ c02

∏

i=1

( N∑

j=1

∣

∣

∂ϕj∂xi

∣

∣

2

L2(Q)+

1

L2i|ϕj |2L2(Q)

)1

2

.

Proof. Let{

ψj}N

j=1defined in H1(Q0) as

(A.22) ψj(x1, x2)√

L1L2ϕj(L1x1, L2x2)

where ϕj ∈ H1(Q) and{

ϕj}N

j=1is sub-orthonormal in L2(Q). It is clear that

{

ψj}N

j=1is

sub-orthonormal in L2(Q0). Hence, Proposition A.1 implies that

(A.23)

∫

Q0

ρ2ψ(y)dy ≤ c22

∏

i=1

( N∑

j=1

∣

∣

∂ψj∂yi

∣

∣

2

L2(Q0)+ |ψj |2L2(Q0)

)1

2

,

where c2 is an absolute constant independent of L1 and L2.

Finally, note that

∫

Q0

ρ2ψ(y)dy = L1L2

∫

Q

ρ2ϕ(x)dx,∣

∣

∂ψj∂yi

∣

∣

2

L2(Q0)= L2i

∣

∣

∂ϕj∂xi

∣

∣

2

L2(Q),

and |ψj |2L2(Q0) = |ϕj|2L2(Q). Thus, inequality (A.21) follows promptly. �

Repeating the proof of Lemma A.1, we obtain the 3D version of the Lieb-Thirring in-equalities. We have (see the details in [MZ])

Lemma A.2. Let Q = (0, L1) × (0, L2) × (0, L3). There exists an absolute constant c0(independent of L1, L2 and L3) such that, for every finite family

{

ϕj}N

j=1in H1(Q) which

is sub-orthonormal in L2(Q), we have

(A.24)

∫

Q

ρ5

3 (x)dx ≤ c03

∏

i=1

( N∑

j=1

∣

∣

∂ϕj∂xi

∣

∣

2

L2(Q)+

1

L2i|ϕj |2L2(Q)

)1

3

,

where

ρ(x) = ρϕ(x) =

N∑

j=1

|ϕj(x)|2.

NAVIER-STOKES EQUATIONS 21

Acknowledgment

This work was partially supported by the National Science Foundation under Grant NSF-DMS-9400615, by the Office of Naval Research under grant NAVY-N00014-96-1-0425, andby the Research Fund of Indiana University.

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