Attractors of Navier–Stokes Equations

CHAPTER 6

Attractors of Navier-Stokes Equations

A.V. Babin Department of Mathematics, University of Cal~f'ornia at Irvine, Irvine, CA 92697-3875, USA E-mail: ababine @math. uci. edu

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

! 71

!. S o l u t i o n s e m i g r o u p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

174

2. D y n a m i c a l s y s t e m s in f u n c t i o n spaces

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3. H a u s d o r f f and fractal d i m e n s i o n s o f attractors

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

178 188

3.1. Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

188

3.2. Basic t h e o r e m s

19 I

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3. M o r e a s p e c t s r

finite d i m e n s i o n a l i t y

4. A t t r a c t o r s of the 2 D N S s y s t e m 4. I. D i m e n s i o n r

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

attractors for the 2D NS s y s t e m

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

194 196 196

4.2. Periodic b o u n d a r y c o n d i t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

199

4.3. l . o w e r e s t i m a t e s for the d i m e n s i o n r

200

4.4. E q u a t i o n s in an u n b o u n d e d d o m a i n 5. T h e 3 D N a v i e r - S t o k e s e q u a t i o n s 5. I. T h e d i m e n s i o n r

attractor

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

203

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

206

r e g u l a r invariant sets

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2. E q u a t i o n s in a rotating f r a m e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. T h i n d o m a i n s

206 207

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21()

5.4. G e n e r a l i z e d attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

212

Moditications r

N S and related h y d r o d y n a m i c e q u a t i o n s

Acknowledgements References

HANDBOOK

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

OF MATHEMATICAL

FLUID DYNAMICS,

E d i t e d by S.J. F r i e d l a n d e r and D. Serre 9 2003 Elsevier S c i e n c e B.V. All rights r e s e r v e d

169

VOLUME

II

212 2 !4 2 !4

Attractors of Navier-Stokes equations

171

Introduction

Dynamics of fluids is governed by the equations of hydrodynamics. The long-time behavior of fluids can be adequately described in terms of attractors of the equations. Here we consider mostly attractors associated with the Navier-Stokes system of hydrodynamics (for brevity we will refer to it as the NS system). We consider two-dimensional (2D) and three-dimensional (3D) Navier-Stokes systems with either no-slip or periodic boundary conditions. The theory of the NS system has to deal with non-scalar equations and with the divergence-free condition that leads to non-local operators. Therefore, mathematical methods developed for the NS system have to be rather general and usually can easily be extended to more complicated physical systems. The Navier-Stokes system takes the form Otu = F(u), after the pressure is excluded; it describes the evolution of divergence-free vector fields u(t) = u(x, t) in an appropriate function space. In many problems of hydrodynamics, the influence of the initial data has vanished after a long time has elapsed. Therefore permanent regimes are of importance. The simplest permanent regimes are described by time-independent vector fields that are solutions of the equation F(u) = 0, such a solution is an equilibrium of the dynamical system. The local theory of equilibria and their perturbations is very rich and includes their stability, bifurcations, theory of local invariant manifolds through them (see Chossat and looss 1521, Guckenheimer and Holmes 11031, Marsden and McCracken 11551, Sattinger ! 173 I, Stuart I 1811, and Section 2 of this article). Nevertheless, time-independent regimes are very special and it is widely believed that time-dependent permanent regimes are of importance. To describe spatially and temporally chaotic turbulent flows one has to consider non-trivial time-dependent regimes (see Chorin et al. 1511, Ruelle and Takens I1711). Time-dependent regimes may include time-periodic, time-quasiperiodic, and chaotic regimes; their common feature is that they are defined for all times, both positive and negative. A mathematically rigorous description of such regimes and related questions of asymptotic behavior and stability are given by the theory of attractors. There are aspects of the dynamical theory approach that are common to finitedimensional and infinite-dimensional systems. They are related to bifurcations, to the chaotic temporal behavior of solutions, and to the fractal structure of attractors (see Lorenz ! 148 i, Barnsley !35 i, Falkoner i 701, Marsden and McCracken I 1551, Guckenheimer and Holmes !103 I). We consider here questions that are specific for infinite-dimensional dynamical systems, in particular for PDE. The theory of global attractors of partial differential equations (PDE) in general and NS equations in particular is developed in the works of Babin and Vishik I321, Hale 11041, Ladyzhenskaya ! 1341, Temam ! 1851, Vishik 11891. The theory applies the ideas of finite-dimensional dynamical systems to infinite-dimensional dynamical systems described by PDE. Many aspects of the theory are specific to an infinite-dimensional case. In particular, the boundedness of energy does not mean that a blow-up of solutions is impossible and does not imply the global solvability of the 3D Navier-Stokes system. There are completely new phenomena, for example, the dimension of attractors tends to infinity when viscosity tends to zero; such behavior and its asymptotics make sense only in an infinite-dimensional situation. Another phenomenon that has no analogue in the finite-dimensional case is the

172

A. V. Babin

presence of a spatial variable in addition to the time variable. Relations between spatial and time variables manifest themselves in trivialization of dynamics on the attractor of the NS system in unbounded channels near spatial infinity. Another example is higher regularity with respect to spatial variables of functions on the attractor compared to a generic function from the function space. Many aspects of the theory of attractors are important for applications, in particular to geophysics and meteorology (see Lions et al. [ 143-145]). Remarkably, the global attractor is a finite-dimensional set. Dimension may be large, global attractors include high-dimensional invariant sets. Examples from the theory of finite-dimensional dynamical systems (the Lorenz attractor) show that dynamics on such sets may exhibit a chaotic temporal behavior. One of approaches to turbulence is based on this observation (see Ruelle and Takens [ 171 ]). On the other hand, the attractor is embedded into a function space and the points of the attractor are functions that represent spatial patterns. Since the attractor has a large size and dimension, its different points correspond to different spatial patterns. Therefore the dynamics on the attractor generates the dynamics of spatial patterns that may lead to a spatial chaotic behavior. Attractors are very interesting objects from a computational point of view. Since the attractor is a finite-dimensional object in the infinite-dimensional phase space of the system, the dynamics on it is subjected to restrictions that are additional to the original NavierStokes system that describes dynamics in the whole phase space. Understanding these restrictions may lead to better computer simulations of physical flows. Dimension of the attractor may serve as an estimate of number of degrees of freedom required to describe the long-time behavior of the Navier-Stokes system in detail. A global attractor also contains all the information on the instability of the dynamical system (see Babin and Vishik [30,32]). Quantitative expression of the instability is given by Lyapunov exponents, which are closely related to the dimension of the global attractor. The purpose of this paper is to give a sketch of the core of the classical theory of attractors with minimum technicalities and to point out major directions in which the theory develops. When exact formulations of results are too technical, we refer to the literature. The Navier-Stokes equations for viscous incompressible fluids have the form po(Otu + u . V u ) -

vAu = f+

Vp

(1)

with the divergence-flee condition V.u =0.

(2)

Here, u = ( u l , u e , u3) is the velocity field, u = u ( x , t ) = U(Xl,X2, X3, t), v > 0 is the kinematic viscosity, f represent volume forces; the density po of the fluid is constant and we set Po = 1. We use the notations V . u = OlUl -t- 02u2 q- 03U3,

u 9 V/g --/4101/4 q- u 2 0 2 u -t- U 3 0 3 u .

Attractors of Navier-Stokes equations

173

W h e n u, f do not depend on x3 and u3 -- f3 -- 0 we obtain the 2D N a v i e r - S t o k e s system. On the boundary 0 D of the domain D, no-slip boundary conditions are imposed u(x)--O,

x E OD.

(3)

An important case, in which boundary layer effects are absent, is the case of periodic boundary conditions U(Xl + 2zral, x2, x3) -- U(Xl, X2 -I- 27ra2, x3) = U(Xl, x2, x3 + 2sra3) -- U(Xl, X2, X3).

(4)

Usually, we take a l -- a2 -- a3 -- 1. In the case of periodic boundary conditions, we impose the following zero average condition on u and f" ~2sra' f() 2rr''212sr''3 d x ul ( x l. ' x2, 1 x3, 0 t)

dx2 d x 3 - - 0 ,

(5) Condition (5) for u(t) holds for all t if it is satisfied at t = 0. To introduce classes of solutions for the Navier-Stokes system, Sobolev function spaces are c o m m o n l y used. We recall that the norm in the Sobolev space H, = / 4 , ( D ) of a function u(x) defined in a domain D C R" is given by the formula

Ilull,,2.- ~ f~ la~ul2d-',,

(6)

Ic~l<~.~' ) where s is an integer, 0 ~' - 0~'~ 9.. o,, '~u" are partial derivatives of order Iot l - c~ + . . . + ~,,. When s - - 0 , we obtain the L2 norm

Ilu Iio - f~

lul 2 dx.

(7)

We denote by H , , s >~ 1, the space of vector fields with a finite norm (6) which satisfy (2) and either (3) or (4). The space Hi is obtained as the completion in the H i - n o r m of infinitely smooth divergence-free vector fields u - (ui . . . . . u,,) that satisfy the boundary conditions. We denote by H0 the closure in the space H = (H0)" of H i . Clearly H0 C H. The space H _ i is dual to H i . We denote b y / 7 the orthogonal H e l m h o l t z - L e r a y projection 17"H -+ H0.

(8)

Gradient fields are in the null-space of the Leray projection. Applying the projection we can rewrite (1) in the form Otu + B(u, u) + vAu - - f ,

(9)

174

A. V. Babin

where B ( u , v) = H

lgi Oi V

A u -- - / T A u .

,

(10)

i=1

The space H2 coincides with the domain of the Stokes operator A and the norm Ilu li2 is equivalent to [[au[I2; Ilull2 is equivalent to Ila 1/2u112 - ( a u , u)o -- I[Vul[ 2. We consider here the no-slip boundary problems in either bounded domains or unbounded domains with exits to infinity of bounded width. For periodic boundary conditions we impose zero average requirement (5), therefore Poincar6 inequality I[Vull 2 >~ Zi Ilull 2 holds with X1 > 0 (see (19)) and IlVui]2 is a norm. The space H - i C H0 is dual to Hi. The bilinear Euler operator B ( u , v) has the following important skew symmetry property (B(u,v),V)o--O

for allu, v 6 H l .

(11)

For basic properties of the spaces I-Is and the bilinear operator B, see, for example, Babin and Vishik [32], Constantin and Foias [57], Ladyzhenskaya [128], Lions [141], Temam I]831.

1. Solution semigroup The two-dimensional Navier-Stokes equations for viscous incompressible fluids have the form

Solution semigroup.fiw the 2D NS system.

"Otu + u . V u - v A u -

f+

(12)

V p,

together with the divergence-free condition (2). Here, u -- (u i,//2) is the velocity field, u -- u ( x , t) = u ( x l , x2, t), v > 0 is the kinematic viscosity, and . f ( x l , x 2 ) represents a volume force. The Euler nonlinearity and divergence for 2D Navier-Stokes equations are U 9 VU - - ttl01l) + U2a2V,

V'U--OlUl

if- 02U2.

On the boundary 0 D of the domain D no-slip boundary conditions are imposed u(x)-O,

x 6 OD.

(13)

We assume that the boundary is smooth enough (see Ladyzhenskaya [135] for a nonsmooth boundary). A special case of interest is periodic boundary conditions u ( x l + 27ral, x2) - - u ( x l , x 2 + 2 n ' a 2 ) - t t ( x i , x2).

(14)

In this case we impose the zero average condition (5) on u, f" therefore the functions from the space H0 satisfy (2), (14), and (5). (In the periodic 2D case the integral in x3 in (5) is skipped.) Applying the Leray projection/7 we rewrite the NS system (12) in the equivalent form (9).

Attractors of Navier-Stokes equations

175

We consider the initial value problem u l,=0 -- u0 9 H0.

(15)

The existence and uniqueness of solutions of the 2D NS system with periodic or no-slip boundary conditions are well known (see Constantin and Foias [57], Ladyzhenskaya [ 128], Lions [ 141 ], Temam [ 183]). THEOREM 1.1. Let f 9 H - l . For any uo 9 Ho, there exists a unique solution u(t) o f the 2D NS system (12) with initial data (15). This solution belongs to Ho f o r all t ~ O. The solution mapping S T " U It=0 ~

U It=T

(16)

determines a family of operators {ST }, T ~> 0. One also can solve the system with initial data at t = T! and find a solution v(t), t >~ Tj. If v]t=Tj -- u]t=0, the solutions v]t=Tj w+ V]t=T2 and ult=o w+ u]t=(T2-T~) give the same result vlt=T2 -- u]t=(T2-T~). Therefore, the operators ST form a semigroup that acts in the space Ho. Many important properties of the NS equations can be formulated in terms of the solution semigroup {St }. These basic properties are described in the following theorem (see Babin and Vishik [32] for a detailed proof).

THEOREM 1.2. Let f 9 Ho. Then, the semigroul~ {St} that corresponds to tile 2D NS system (12). ( 13 ) in a bounded domain or to the periodic problem (12), (14) has the.fidlou'in~,, properties. The solutions u(t) = Stul) are bounded in Ho un~f'ormly when t >~ 0 and uo is bounded in Ho. The operators St are continuous./hint Ho to Ho f o r t >~ O. Moreover, i f 0 <~ t <~ T and [[u0[[i) <~ R, then Stuo depends on ul) uniformly continuously for.fixed R, T. The following smoothing property hohls: the.f'unctions St u are bounded in H l and H2 uniformly in t and uo when t >~ to > 0 and ]]u0[]0 <~ R. The operators St, t > O, are contpa~'t. Since the statements of the theorem contain basic information on the semigroup and their proofs include important estimates, we sketch the principal points of the proof and give basic estimates on the solutions. We deduce the boundedness statements in this theorem from the fundamental energy estimates which we give below. Multiplying (9) by u and using the skew symmetry (1 1), we obtain the energy equation ib Ilu I1~ + 2~llVu II2 - 2(f, u)o

(17)

and a differential inequality for smooth solutions 3,]lull 2 + 2vllVull 2 ~< 211fll01lull0.

(18)

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Attractors of Navier-Stokes equations

177

After multiplication by w and application of ( 1 l ) one obtains 1 -~atllw]l 2 + vllVwll 2 - - - ( B ( w ,

(25)

u2), W)o.

Using Sobolev embedding theorems, we estimate (B(u2, w), w)0 in (25) and get in two space dimensions: OtllWl[ 2 ~ C(t),~.l)litoll21iVu2112.

(26)

Therefore 2

[lu, (t) - u2(t) IIo

~

u2(O)ll~exp c(v,)~,)

] Vuz(r)ll~dr .

(27)

Since (23) gives an estimate of the exponential term, this inequality implies the Lipschitz dependence on the initial data and the Lipschitz continuity of operators St. Thus, Theorem 2. I is proven. Lor 3D semignmp. In three space dimensions, the Navier-Stokes system has global weak solutions (see Constantin and Foias [57], Lions [141 ], Temam [183]). There exist weak solutions (may be non-unique) that satisfy the energy estimates we derived above for the 2D NS system. The formal derivation is the same.

THEOREM 1.3. Let ./ E H - i , u(0) e HI). Then, there exists a weak solution u(t), 0 <~ t < oo, of the 3D NS system (9) which sati,sfies, f o r ahnost every Tl, T2 sur that 0 ~ T! < T2, the energy estimate

II,,(r)llg- I1,,
IlVullodt <~ 2

(f,u)odt.

(28)

One can prove the local existence and uniqueness of strong solutions if the initial data and the forcing are smooth enough. THEOREM 1.4. Let s > I/2 and f ~ H,.-l. For every uo e H,. there exists Tl and a unique stnmg solution u(t) o f the 3D NS system that is defined f o r t < T! and belongs to H~. f o r 0 <~ t < Ti. The time Ti depends on [[./'[[.~-l, [[u()lls, v. For the 3D NS system, the local operators St uo, 0 <~ t < Tl, are well-d 1/2. These norms cannot be estimated by (28). Avail-

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Attractors of Navier-Stokes equations

179

Absorption and attraction.

Let E be a complete metric space with distance de(u, v). In many examples E is a Banach space and

de(u, v) = Ilu - vile. Let a semigroup of operators {St, t/> 0} act in E

St : E--+ E,

t >~O.

T h r o u g h o u t we assume that operators Stu are continuous with respect to u in the metric E. We introduce a (nonsymmetric) distance 3 e ( B l , B2) from a set Bl to a set B2,

SE(BI, B2) = sup

inf dE(xl,x2).

(33)

xlEBl X2EB2

A set B is invariant if St B C B for all t ~> 0. A set B is strictly invariant if St B = B for a l l t >~0. A set B0 is called an absorbing set if for every bounded in E set B there exists T such that StB C Bo for all t ~> T. When {St} has a bounded absorbing set B0 and the operators St are uniformly bounded on bounded sets and continuous, the set

Bo'r -- closuret.: ( U st Bo ) t >>.'/" is also absorbing and is invariant. Therefore the existence of a bounded absorbing set is equivalent to the existence of a closed bounded invariant absorbing set. A set B0 is called an attracting set (in a space E) if for every bounded in E set B lim 3 e ( S t ( B ) , B0) = 0 .

(34)

t -----~cx3

DEFINITION 2.1. A set .,4 is called the global attractor of {St } in E if (i) r is compact; (ii) .,4 is strictly invariant: St r = A; (iii) .,4 is an attracting set for {St} in E, that is 6 E ( S t ( B ) , A ) --+ 0 as t --+ ~ bounded in E set B.

for every

The following properties characterize global attractors. A global attractor is a minimal set a m o n g all compact sets that attract all bounded sets. A global attractor is a maximal set a m o n g all bounded strictly invariant sets (see Babin and Vishik [32], Hale [104], Lad y z h e n s k a y a [134], T e m a m [185]); it has a maximal domain of attraction. Sometimes, global attractors are called maximal attractors or minimal attractors. Here we often call a global attractor simply an attractor. Note that if a global attractor exists, it is unique.

A.V. Babin

180

REMARK. The above constructions and theorems in the next subsection are applicable to the case where E is a metric space. In addition to a n o r m e d topology of a Banach space, a metrizable weak topology of a linear separable Hilbert space or a metrizable topology of convergence on b o u n d e d sets can be used. Another important example is the case where a semigroup is defined on a bounded invariant subset of a linear Banach space. See Babin and Vishik [32], Hale [ 104], Temam [ 185] for details. REMARK. An absorbing set for the 2D NS system was constructed by Foias and Prodi [75]. The attractor of the 2D N a v i e r - S t o k e s system was constructed and many of its important properties were established by L a d y z h e n s k a y a [ 129,130].

Upper semicontinuity. If Equation (29) depends on a parameter 0, Otu = F(u, 0), we have a semigroup that depends on the parameter and the global attractor of this semigroup also depends on 0, ,,4 = A(O). Under very mild assumptions A(O) depends on 0 upper semicontinuously. This means that aE(,A(0), A(Oo)) --+ 0 as 0 --+ 00 (since the distance aE from one set to another is not symmetric, this does not mean that aE(A(Oo), A(O)) --+ 0!). For example, it is proven by Babin and Vishik [28] that the attractor of 2D NS system in a b o u n d e d domain depends upper semicontinuously on the domain, see Babin and Vishik [28,32] for details. Strict invariance. A c u r v e u ( t ) , - o o < t < oo, iscalledatrajectorvof {St} if Sr, u(t2) = u(tl + t2) for all - 0 0 < t2 < 00, 0 < tl < cxa. The following important property of a global attractor is equivalent to its strict invariance. For every point el c A there exists a bounded in E trajectory u(t) of {St} defined for all - o o < t < + o o such that u(0) = a. Strict invariance is a strong property. We show it on an important example that we will use later. If u(t) is a bounded trajectory of the 2D NS system, inequality (21) gives

IIu(T' )l[~ ~< e-V~"lT'-T~

u(TO)

for any 7b, Tl >~ 7b. We assume 7b --+ -0r

+ (1 - e -''z''T' -To,) V2)v~

II.fll 2,

(35)

since Ilu(~))ll 2 is bounded we get for all Ti

I[u( T,)I]~ ~ Ilftl~

V2~"2 .

(36)

This inequality and ( 2 3 ) i m p l y

V frW IIVu II2 dt T - Tl

I+T-TI

[[u (T1)][2 [1u (TI)l[ o2e -X,~(T-T,)+ v (T T, ) ilfll2o" T - TI Xl (37)

Attractors of Navier-Stokes equations

Letting

TI --~

lim

181

--oo we obtain

sup

2

fTT IlVull2dt ~< ~.llfll~ ~.lv

v

TI-+--oo T -

TI

(38)

!

The right-hand sides of (36) and (38) do not depend on T and u. Therefore, if an attractor ,,4 of the NS system exists (we prove the existence of the attractor of 2D NS system in the next section), then every trajectory u(t) = Stuo on the attractor satisfies (36) and the estimate

1 LT IlVull2dt~

.oeAsuplim T---+ooSUp-~

Ilfll~ ~'lV 2 .

(39)

Here, f is the body force, v is the viscosity, and X! is the first eigenvalue of the Stokes operator; X i depends only on the spatial domain.

Equilibria and local invariant manff'olds. DEFINITION. A point z is an equilibrium point of {St } if 5'i z = z for all t. Since :: does not depend on t, h)r semigroups defined by (29), ::. satisfies the equation F(z) = 0 . DEFINITI()N. An unstable manifohl M+(z) through an equilibrium point : of St is the set of all points v E E such that St v is defined for all t ~< 0 and St v ---> z. in E as t --> -cx3. If {St } has a global attractor ,,4 and z: is an equilibrium point of {St }, then M+(z.) C ,A. DFFINITION. A stable manifold M_(z.) through an equilibrium point z. of St is the set of all points v E E such that St v is defined for all t ~> 0 and St v --> 7. in E as t ~ +cx3. The behavior of a dynamical system near an equilibrium is described by the theorem on stable and unstable manifolds of semigroups in Banach spaces; this theorem is fairly similar to the finite-dimensional theorem. We formulate the theorem skipping technical details, in particular the differentiability conditions (see Babin and Vishik [32], Henry [106] for details; see also Bates et al. [36], Chen et al. [44] for details, generalizations, and more references). Let St be a nonlinear differentiable (of class C a, c~ ~> 1) semigroup in a Banach space E. Let ::. be an equilibrium of St, that is St z - - z for all t >~ 0. The differentials S;(z) form a semigroup of linear bounded operators in E. The properties of this semigroup play important role; the behavior of St near z is in many respects similar to that of S~(z). The most important assumption is the existence of a circular gap in the spectrum of S~(z). Namely, we assume that the spectrum of S~(z) does not contain a ring I ~ ' l - pt in the complex plane. Therefore, a ring (p - s) t ~< I~l ~ (p + s) t is not in the spectrum if s is small. We conclude that the spectrum is divided by the ring into two parts: external or+

182

A. V. Babin

and internal o_. Therefore, the Banach space E splits into two complementary invariant subspaces E+(p) and E_(p), S ; ( z ) ( E _ ( p ) ) C E _ ( p ) , S ; ( z ) ( E + ( p ) ) - E+(p) for all t >~ 0. We assume that E+ (,o) is finite-dimensional. Under these conditions, the nonlinear semigroups have local invariant manifolds M+(z., p) and M_(z,, p) through z. in a neighborhood of z. (the local manifolds may be non-unique). A set M is called local invariant (in a neighborhood of z ) when St u ~ M if u ~ M and Sru stays in the neighborhood of z. for 0 ~< ~- ~< t. When p = 1, M+(z, p) is called a local unstable manifold of St, M_(z, p) is called a local stable manifold of St. When I~'l = 1 is in the spectrum and p < 1, M+(z, p) is called a center-unstable man~#>ld of St. THEOREM 2.1. There exists a man~fold M+(z., p) that is in the neighborhoodq#z a graph ql~'afunction )'class C~ from E _ ( p ) to E+(p), M_(z., p) is tangent to E _ ( p ) at z. The intersection M+(z., p) r-) M_ (z, p) = z.. In the neighborhood
0 ~ r ~ t

(40)

holds when Sru is in the neighborhood ~ I and

-II,...

o

<4,>

, . , . , --->

then u(t) e M_(z, p). When p >~ 1, St is ~:~temled inside M+ (z, p) to negati~,e t aml distE(Stu, z) <~~'"(p+e) t,

t <~().

(42)

If there are many circular gaps 14"1-- P[ with points of the spectrum between them, one can construct many different local invariant manifolds of St near :.. Intersections of these manifolds are also smooth local invariant manifolds. Therefore, the nonlinear dynamics near z is in many respects similar to the linear dynamics of S~(z).

The Navier-Stokes system near equilibrium. The 3D and 2D Navier-Stokes systems in a bounded domain with a time-independent force always have time-independent solutions (equilibria). Namely, if D is a bounded domain, for any f c H0, there exists a timeindependent solution ::.(x) E H2 (an equilibrium) of the 3D or the 2D NS system ( 1) z-Vz. = v A z - . /

+ Vp,

V . z = O, zlat) = 0,

see, e.g., Lions [ 141 ], Temam [ 183 ]. We consider here the 3D case since the 2D case is simpler. The simplest example is f = 0 and then z = 0 is the only equilibrium. The long-time dynamics described by the NS system is the simplest in this case, every solution tends to

A ttractotw of Na vier-Stokes equations

183

zero as t ~ oo. According to Leray theorem, every weak solution of the 3D NS system u(t) becomes regular for large t and u(t) -+ 0 as t --+ cx:). Therefore when f = 0 the global attractor of 2D and 3D Navier-Stokes system consist of one point z = 0. We consider a more general case f E H0 and consider regular solutions that have a limit,

u(t)--+z

ast~cx~inH,,,,

s>

1/2.

The linear semigroup S~(z)vo is generated by the variation equation (57); this equation takes the form

Otv+B(v,z)+B(z,v)+vAv=O,

(43)

v(0) = v0.

The operator Sf (z) is compact when t > 0. Its spectrum is discrete and has only one accumulation point, namely, zero. Therefore almost any circle ]~']= r lies in the resolvent set of St= I (z) and one can choose a sequence of circles [r - rj in the resolvent set of S ! (z) with rj ---> 0 as j --> oo. Let a sequence of positive numbers rj, j - 1 determine the circles 1 r rt. in the !

!

.

.

. ,

j

!

resolvent set of S t (-), rj --> 0 as j --> oo. According to T h e o r e m 2.1, we have invariant manifolds M + ( r j ) and M_ (r j) through z that are tangent to the corresponding invariant subspaces of Sf(z)" M + ( r j ) are finite-dimensional. As we discussed before (see Theorein 2.1 ), the behavior of St near an equilibrium is similar to that of the linearization ,St (-). In particular, the following theorem (see Babin and Vishik [32]) describes the tracking property. -,t

THEOREM 2.2. Let tt(t) be a solution o f the 3D NS system su~'h that o

~

(44)

where e, > (). Thell, f o r e~'erv .j, there exists a solution h.j (t) qf'the 3D NS svstent that lies ill the.finite-dimeltsional lo~'al in~,ariant man~f'ohl M+ ( z, r j ) f o r all t >~ 0 such that (45) Here, we can approximate an infinite-dimensional solution u(t) of the NS system by a solution {ti(t) that lies in a finite dimensional manifold, the dynamics of t~(t) is described by a system of ODE. REMARK. In the case of potential fl)rces when .1" = 0 and z = 0, results of Foias and Saut [82,831 give more detailed information on the dynamics. A normal form of the NS system is given and the structure of the normalization map is described. The normalization map reduces the NS system near zero in the non-resonant case to the linear problem and in a resonant case to a more complicated normal form.

184

A. V Babin

Exponential attractors. Another important notion of the theory of dynamical systems is an exponential attractor, also called an inertial set (see Eden et al. [66,68] for a construction). DEFINITION. A set C C E is called exponential attractor of the semigtopup {St} in the Banach space E if (i) ~' is compact and has a finite fractal dimension; (ii) C is (not strictly) invariant StE C C for all t ~> O; (iii) there exist positive constants c and c' such that for all t ~> 0 and for every bounded setBc E distE (St B, ,5') ~< c' e x p [ - c t ] . Note that points (i) and (iii) of this definition are more restrictive than the corresponding points of Definition 2.1 and point (ii) is less restrictive. For the definition of a fractal dimension see Section 3. If an exponential attractor exists, it always contains the global attractor, A c E. An exponential attractor is non-unique. Existence of an exponential attractor for 2D NS system is proven by Eden et al. [66]; see also Eden et al. [67,68]. Trajectory attractors. When the fl)rcing f depends on t, . / = ./(t), Equation (29) takes the form 0tu = Fb(u) + f ( t ) . it becomes non-autonomous and operators St do not form a semigroup anymore. The function u(t + s), t ~> 0, is a solution of a shifted equation atu = Fo(u) + f ( t + s ) . Following Sell 1174,175], Dafermos 161 ], Chepyzhov and Vishik [49] we define an operator S., "(u(t), f ( t ) ) v--, (u(t + s), f ( t + s)). We can consider functions (u(t), f ( t ) ) , t ) 0, as elements of a topological space ('-)+ of time-dependent functions. The topology in this space is defined by the convergence in an appropriate norm on bounded intervals [ti, t2], for example, (.jOlt2 ]lu(r)]]o d r ) I/2 for all positive t I, t2. This convergence can be described by a metric, so results of Section 1 are applicable. The closure of shifts f ( t + s) is called the hull of f . Under appropriate conditions, one can prove the existence of the global attractor of the semigroup {ST }. The attractor of this semigroup is called trajectoO, attractor (see Chepyzhov and Vishik [48 ]). This attractor consists of the solutions u (t) defined for -~:~ < t < m and corresponding to an element f ( t ) of the hull that is defined for - m < t < oct. The construction is directly applicable to the 2D NS equations with time-dependent forcing term f ( t ) and can be easily generalized to more general non-autonomous equations. This construction is also useful for treating equations without uniqueness, in particular the 3D Navier-Stokes equations. For details of the theory of trajectory attractors see Sell [ 176] and Chepyzhov and Vishik [48-50] where existence of trajectory attractors for non-autonomous 2D NS system and non-autonomous 3D NS system is proven. For a related method of "short trajectories" see Mtilek and Ne(zas [150], Mfilek et al. [152]. A toy model.

To illustrate the above concepts, we consider a linear equation

3tu -- - A o u ,

(46)

Attractors of Navier-Stokes equations

185

where Ao is a self-adjoint operator in a Hilbert space H with an orthonormal eigenbasis gj Aogj -- ~ j g j ,

j -- 1 . . . . .

One may take Ao -- - A 4- c I , where A is the Laplace operator with appropriate b o u n d a r y conditions and c is a constant. The numeration o f e i g e n v a l u e s is in increasing order, ~.j+l ~> ,kj. Let No be the n u m b e r of non-positive eigenvalues )~1 . . . . .

/~'No ~ 0,

~'No+! > 0.

Equation (46) can be written as an infinite system of uncoupled equations 0t u j -- - X.j u j,

j -- 1. . . .

for the coefficients u.j(t) of the expansion of u(t) in the basis gi. The general solution of (46) is given by or

u ( t ) -- Z

e - ~-it tI.j g.j. I

it can be split into two parts

tt(l) - - t t + ( l ) 4- tt_(t), N() U+ (t) -- ~ e - z it u .j g.j,

OX3 u_(t)

-

Z

e - )~it uj g.j.

.j=No+!

j= I

We have

II"-(')lIH

II"(O)IIH

')

Obviously, this estimate is uniform when the initial data u(0) belongs to a b o u n d e d set B in H. The linear subspace EN() with basis gl . . . . . gUo E N() -- E()+ -- span {g I . . . . . gnu) } is strictly invariant with respect to the linear semigroup St -- e -A(~t St E()+ -- E()+

(47)

and is finite-dimensional. It satisfies the attraction property

(~H(St(B), Eo+) ~< C ( B ) e -~N~

(48)

A. V. Babin

186

for every bounded set B. Therefore, we may consider E0+ an unbounded global attractor of the linear semigroup e - A ~ The only condition in the definition of a global attractor which is not satisfied is the compactness (so an unbounded global attractor is not a global attractor in the usual sense). The set E0+ is strictly invariant and attracting; it is not compact, but it is locally compact, that is every bounded part of it is compact and it is finite-dimensional. Nontrivial examples of unbounded attractors of nonlinear PDE are given by Chepyzhov and Goritskii [47]. When 1.N0 > 0, the subspace E0+ is an unbounded exponential attractor with the rate of attraction

distE(St B, ENo) ~ c'e -)~N~ If one takes a larger subspace with the basis g~ . . . . . rate of attraction

gNi

with N~ > N0, one gets a higher

distF~(St B, ENi ) ~ c'e -XNl+'t 9

(49)

More interestingly, one may increase E No just a little, taking NI

S

E N,, +

NI

E .j=N()+ I

I,, jl 2 ;

/.2

.j=NI~+ I

with arbitrary small r, and still (49) remains true (with a different t" ), so it is an unbounded exponential attractor, too. The set L" is invariant, $I,5' C s but it is not strictly inwlriant, St S --/: s This toy example shows that by expanding the global attractor a little and increasing its dimension, one may drastically increase the rate of convergence to it and obtain an exponential attractor. A nontrivial generalization of this idea leads to the proof of existence of an exponential attractor of a dynamical system, see Eden et al. 166-68 ]. The set E0+ can serve as an illustration to the notion of an unstable manifold. Clearly, zero is an equilibrium of the linear semigroup. If 2.N0 < 0, the subspace EN0 -- E0+ is the unstable manifold through zero of the linear semigroup S~ generated by (46). When )'.No ---= 0 the subspace E No is a center-unstable manifold of S~. The center-unstable manifold of this semigroup is non-zero if I.! ~< 0, it has the basis gl . . . . . gNo with )~! . . . . . 2.N0 ~< 0, )~No+ I > O. A stable manifold of the linear semigroup through zero is a linear subspace with the basis gNo+ ! . . . . . This toy model illustrates the basic properties of attractors: the set E0+ is an attracting set, it is locally compact; moreover, it is finite-dimensional. It is invariant, that is on E0+ every trajectory u(t) can be extended to - o c < t < + e c . When A0 = --A + e l , where A is a negative operator, then for c - - 0 the attractor consists of only one point" zero. The nontrivial attractor E0+ is a result of the perturbation e l , with sufficiently large c > 0 that creates instability. Of course, the Navier-Stokes system is much more complicated than this toy model. Still in many cases one may consider the global attractor of the N a v i e r Stokes system as a result of perturbation of the linear Stokes system by the nonlinear term

Attractors of Navier-Stokesequations

187

and the external forcing. The attractor is no longer a linear subspace. Finite-dimensional nonlinear models (for example, the Lorenz system) show that the structure of such set may be very complicated and this set does not look like a smooth manifold neither locally nor globally. Nevertheless, many observations are still true. The global attractor is a compact, finite-dimensional (in the sense that will be discussed below) set. This set is invariant, that is the dynamics on the attractor is invertible. Remarkably, it is a compact bounded set, that is, in a sense it is smaller than a linear subspace (unbounded). And this set uniformly attracts all the solutions of the dynamical problem, in particular of the NS system. This toy model illustrates another general property of attractors: the upper semicontinuous dependence on parameters. Let the parameter be 0 = k No+! > 0. When 0 --+ 0 the attractor changes continuously (in fact, it does not change at all when 0 > 0), but at the limit value 0 = 0 the attractor E No -- E uo+J changes its dimension: it becomes larger. At the same time distE(Eu0, Euo+i) = 0 . Existence o f attractors. Here we give basic existence theorems from the theory of attractors. More details are given in Babin and Vishik [32], Ladyzhenskaya [ 134], Temam [185]. A very detailed treatment of general aspects of the theory of existence of attractors of operator semigroups is given by Hale [104]; see also Ball [34]. We consider an operator semigroup {St} in a complete metric space E. The operators St are assumed everywhere to be continuous bounded (nonlinear) operators in E. A semigroup {St} is called asymptotically compact if, for any u0 6 E and a sequence tj --+ cx~, the sequence Sq u~) has a convergent subsequence.

THEOREM 2.3. Let a semigroup {St} have a b o u n d e d absorbing ball B R, Ut>~(){StB R } be bounded, and {St} be asymptotically compact. Then {St} has a global attractor A. The attractor contains an equilibrium uo, Slit()~ lg() for all t. If St is r in t, the attractor .A is a connected set. The attractor .,4 is defined as an omega-limit set by the formula

.A- N T>()

B(T),

where B ' ( T ) - closureL, ( U

St

BR).

(50)

t/>7"

All the properties of ,,4 can be derived from the definition, see Babin and Vishik [32], Ball [34 ], Hale [ 104 ], Ladyzhenskaya [ 134 ], Temam [ 185 ]. COROLLARY 2.1. Let {St } have a bounded absorbing ball. Let the operators St, f o r t >~ O, be continuous and un~fbrmly bounded and compact f o r every t > O. Then {St} has a global attractor.

The existence of the attractor for the 2D NS system is based on fundamental properties of its semigroup {ST} which are stated in Theorem 1.2. Using Theorem 1.2, we obtain the tbllowing theorem.

188

A. V. Babin

THEOREM 2.4. Let f ~ Ho. Let {ST} be the semigroup in the space Ho that is generated by the 2D Navier-Stokes system in a bounded domain. Then, {St} has a global attractor .f4. The attractor contains an equilibrium uo, is a connected set, and is bounded in Hi. PROOF. Inequality (21) readily implies the existence of an absorbing ball Boo--

u" lulo ~< Ro, R 2 -

2 ~--~[Ifl[o2

(5~)

Note that Boo is bounded in Ho, but is unbounded in Hi. We set Bo~ -- Si Boo,

(52)

Si -- St=j.

When ST B C Boo, we have ST+IB C Bol. Therefore, the set Bol is absorbing. The smoothing property from Theorem 1.2 implies that the functions St u are bounded in Hi, if t > 0, and Ilullo ~< R0; this is true in particular for t = 1. Therefore, the set Bol is absorbing and is bounded in HI. The compacteness of the embedding Hi C Ho implies that Bol is compact. The smoothing property and the continuity of St in H! also implies that operators St are compact. Hence the global attractor exists by Corollary 2.1. 1--7 A stronger property of attraction ((Hr H2)-attraction)holds (see Babin and Vishik 127, 32]):

THEOREM 2.5. Let .[" ~ Ho. The attrar /'or any bounded in Ho set B.

is r

in H2 alul ~ H ~ ( S t ( B ) , A )

--+ ()

REMARK. Usually, the smoothness of functions on the attractor is determined by the smoothness of the forcing term and by the boundary. Interestingly, the attraction can be in a stronger norm than boundedness of solutions (see Babin and Vishik 132 ]) since difference of two functions from Hi may belong to H2. The regularity of functions on attractors and attraction in stronger norms are studied in Babin and Vishik [27,32], Ghidaglia and Temam [97 ], Ternam 1185 ]. In particular, when f and i~D are infinitely smooth, the attractor ,14 consists of infinitely smooth functions (see Temam 11851, Foias and Temam 1891).

3. Hausdorff and fractal dimensions of attractors

3.1. Basic definitions A fundamental characteristic of an attractor of a dynamical system is its dimension. The physical meaning of the dimension of an attractor is roughly speaking the number of degrees of freedom required to describe the large-time dynamics of the dynamical system. The attractor may be a very complicated set, so a definition of dimension has to be applicable to general sets. First, we give the definition of Hausdorffdimension of a set in a Banach space.

Attractors of Navier-Stokes equations

189

If K is a compact set, we consider finite coverings C K of K by balls Br~ (Xi) of radius ri centered at xi, Bri(Xi) = {u: ]u - XiJE < ri}. We denote by ICKI the maximum of r for the covering C K. Let inf Z r ~ " ICKI<~~

(53)

# u ( K ) = lim #o,.e(K).

(54)

lzot.~(K)

-

-

i

Let ~--~0

The Hausdorff measure #c~ (K) equals oo for small c~ and equals 0 for large c~. The Hausdorff dimension is defined as

(55)

dimH(K) --inf{ot" /zot(K) --0}.

Note that since E is infinite-dimensional, dimH (K) may be equal to infinity. Now, we define thefractal dimension of a compact set K. We consider finite coverings of K by balls Br(xi) of radius r centered at xi with a fixed radius r. We denote by n(r, K) the minimum number of balls in such a covering. The box-counting dimension (fractal dimension) of K is log n (r, K) dimv( K ) -- lim sup . ,--,o log( 1/ r)

(56)

We always have dimH(K) ~< dimF(K). Note that if K is a smooth compact d-dimensional manifold (or a piecewise smooth manifold, or a manifold with a boundary) that lies in E, then d = dimH(K) = dimF(K). An important property of the fractal dimension is the existence of Mafi6's projection (see Marl6 [154]). Namely, if a set K has dimension dimF(K), there exists a projection onto a linear subspace with dimension d < 2 dimF(K) + 1 which is one-to-one on K. This follows from the following Mafi6's theorem (see Eden et al. [68], Foias and Olson [81 ]). THEOREM 3.1. Let H be a separable Hilbert space with Hausdorff dimension dimH X ~ k and dimH(X projection of rank equal to k' + 1. Then f o r every 6 projection P = P(3) o f the same rank in H such that X is one-to-one.

and X be a compact subspace of H x X) <~ k'. Let Po be an orthogonal E (0, 1) there exists an orthonormal II P o - P(~)II < ~ and P restricted to

Note that the fractal dimension has the following natural property: if dimF X <~ k" then dimF(X x X) <~ 2k" (the Hausdorff dimension does not have this property). Since dimH(X x X)<~ dimF(X x X), the conditions of Mafi6's theorem are fulfilled if dimF(K) is finite.

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192

A. V. B a b i n

The following theorem that allows to get quantitative estimates is proven by Douady and Oesterle [64]. THEOREM 3.2. Let S X = X, and let S be uniformly quasidifferentiable on X, with uniformly bounded on X quasidifferentials S' (u). Let d be such that f o r some k < 1 cbd ~< k < 1.

(61)

Then, the Hausdorff dimension of X is finite and is not greater than d. The following theorem on the fractal dimension is proven by Constantin et al. [58], see also Constantin and Foias [56]. THEOREM 3.3. Let -

-(d-j)/(n+l)

cojco,,+i

(62)

< 1 f o r j = 1. . . . . n.

Then, the fractal dimension of X is finite and not greater than d. Chepyzhov and Ilyin [46] estimate the fractal dimension under conditions similar to that of Theorem 3.2: THEOREM 3.3' Under the h37~otheses of Theorem 3.2 suppose that the quasidifferentials S'(u ) continuously depend on u E X in the operator norm. Then the fractal dimension of X is not greater than d. The statements of Theorems 3.2 and 3.3 can be illustrated as follows. Inequality (61) means that the mapping S strictly decreases the d-dimensional Hausdorff measure with a coefficient k < I. Since SX = X, this is possible only when the Hausdorff measure of X is zero. This sketch can be made rigorous (see Babin and Vishik [32], Temam [185]). Here, we follow mainly Temam [185 ], skipping some technicalities to show the most important ideas. If S~ (u0) is the solution operator for the variation equation (57), then using a Liouville type formula for solutions of linear equations of the form (57), we get the estimate

Cod(S;(uo)) <~supexp Ed

If'

tr(F'(u(r))17E,,(~))dr

1 ,

(63)

where /TE,/(r) is the orthoprojection in H onto E d ( r ) = S'r(uo)Ed, the supremum being taken over all d-dimensional subspaces and tr(F'(u(r))17E,l(r)) being the trace of the finite-dimensional operator F'(u(r))17Ea(r). From (61) and (63), we obtain that dimH X <~ d if, for some t > 0 and k < 1, for all trajectories u (t) on X sup Ed

Ifo'

tr(F'(u(r))FIE,,(r))dr

]

<~ Ink < 0.

(64)

=.

,H

9

9

.~.

<

~

,.

= = "~-

=~

b9

~'r~

~

~

~ .

..

~,.

~ ~

.~, ~

"~

~.~:~

~.~

~

II

*~

~,~-~ ~ ~.~-

~.~ ~

~"

.

" 0

~ - r

-

~',-

-

--.

=

:z

:

--]

~'~

.--

--

.~

~,.

I

/A

~=

.~

~_,

~

~
a~

i-, l

=

~

~

i.- t

~

o.. ~ .

~ ~

~

B

~ =

,_.

=

a~

>-

~

~,

~

-.

~

,_

~

~

=

F

<

A

~

,..,

~

~ _.

~"

~

a~

~

~"

~

-

~

=r

9

r

-

~.=

,..--

:"

r

=

a~

-~=

~,

~"

= ~-

,~

m:~

=

~"

_..~

:I:

~.~" -- ~-

=

=2

~

-

.~

"

~.

;~"

~.~

._.

::1

-.

II

-~

~

a~

-

~

II

a~

:=r'

~

~

-"

,-,

.

~"

-

~ '

I-I

i...., o

,--0

i.mo

-t~,~

I~

i.... l

~'o

.~.

I

~,,

i--o

--.

< ~

("1:1

~

?

-..,. ,...,.

194

A. V. Babin

REMARK. Estimates of dimension of attractors can be made in terms of the Lyapunov exponents of the operators S~(uo)(t). We define, for j ~> 2, a global j t h Lyapunov exponent by

/zj = In(A j),

Aj_

lim [ &J(t) ] l/t t--,~ &j-I (t)

(see Temam [185] for more detail). One can derive from (63)

#l +""

.'

+/z,,+l ~< sup lira sup 2- supexp u(O)cX

t---,oQ

E,I

If,,'

tr(F'(u(r))/TE~

i

(70)

Condition (61) is replaced by the following condition: /~l + " - +/.t,,+l < O.

(71)

THEOREM 3.5. Let St X = X, St be quasid~ff'erentiable on X, let (71 ) hold and, for every t, the quasidiff'erentials S;(uo)(t ) be un~lbrmly bounded on X. Then, the Hausdorffdimension ql'X is not greater than dt~ = n +

max(gt + . . . + t~,,. 0)

(72)

lu,,+i I

(the izumber dr. is called the Lyapunov dimension qf" X). The fractal dimelzsioll q['X is not greater than (n + 1) max 1-+I<~i~<,,

rnax(/~ I + ... + i~i, 0) I/~i + " " + l~,,+ll

1

.

(73)

3.3. More a,v~ects q/ilinite dimensionalitv

Parametrization ql'attractors. The first statements related to finite-dimensional parametrization of attractors were given by Foias and Prodi [75] and Ladyzhenskaya [ 129,130]. Foias and Prodi [75] proved that the asymptotic behavior as t -+ cx~ of the solutions of 2D NS system in many cases is determined by the asymptotic behavior of their finitedimensional projections. Ladyzhenskaya [129,130] constructed the attractor of the 2D NS system and proved that a trajectory on the attractor is completely determined by its orthogonal projection onto the space spanned by the first n eigenfunctions of the Stokes operator. There are several ways to parametrize attractors. The first one is to project the attractor onto a finite-dimensional linear subspace, and if the projection is injective, the subspace gives a parametrization of the attractor. By Mafi6's Theorem such a projection always exists. A related question, which arises in connection with spectral numerical methods is the following: Do lower Fourier modes give the parametrization and how many modes does

Attractors <~'Navier-Stokes equations

195

one have to take? A similar question, which arises in connection with finite-difference numerical methods, is the following: Can one parametrize functions on the attractor by their values at given points (determining nodes) and how many nodes does one need? More general question is: When do values of a finite number of functionals determine the longtime dynamics? Such problems, important for computational applications, are addressed in papers by Chueshov [53,54], Cockburn et al. [55], Constantin et al. [59], Foias et al. [79], Foias and Temam [87], Jones and Titi [121,122], Ladyzhenskaya [133], Shao and Titi [179]).

Approximation
196

A. V. Babin

4. Attractors of the 2D NS system 4.1. Dimension of attractors for the 2D NS system If the phase space of a mechanical system is a manifold, the dimension of the manifold is the number of degrees of freedom of the system. The attractor represents the phase space for permanent regimes. Though it is not a manifold, one can determine its Hausdorff or fractal dimension (or at least obtain an estimate of this dimension). Of particular interest is to estimate the dimension in terms of physical parameters of the problem. Methods of deriving such estimates that are applicable to the 2D NS system were developed by Ilyashenko [113,114] (for the Galerkin approximations), Babin and Vishik [23,24,26], Ladyzhenskaya [ 13 l, 133 ], Foias and Temam [86], Constantin et al. [58], Constantin and Foias [56] on the basis of a theorem of Douady-Oesterle. Constantin et al. [60] developed the method of upper estimates of the Hausdorff and fractal dimensions of PDE in its modern form. This method allows to obtain estimates which are physically consistent and are shown to be precise for important examples (see Section 4.3 for details). In particular, estimates of the dimension for 2D NS can be given in terms of the Grashof number G -

II./Iio

(74)

v2ZI '

where X i is the first eigenvalue of the Stokes operator A (or the bottom of the spectrum when the spectrum of A includes a continuous component as in the case of an unbounded domain). Note that X]-l/2 can be taken as a typical length and I]f [l~)/2 has the dimension of the velocity, so the Grashof number is dimensionless. As shown in the preceding section, to get an upper bound for the dimension, it is sufficient to estimate the trace in (68). Here we give a sketch of an estimate of the trace in (68) with X -- ,i4 (see Babin and Vishik [32], Constantin et al. [60], Temam [185] for details). We take an orthonormal in H basis (pj (r) = ~0j (x, r), j = 1. . . . . d that spans E,l(r). We have d

tr(F'(u(r))FIE,,(r)) -

Z

(F'(u(r))qgj(r), qg.j(r))o.

j=! After a straightforward computation, we get d j=l

"

j=l

I1' j--I

II 0

I

/A

>-.o

p~

,'"-t

" ~'~

~

A

~

'-.,,I-

~

Or)

~"a

=

II

%,.

,1, ~

9

00

.~.~

9

!

~,..< ,,,,...

0,o

I

4-

/A

,"',1

,--,i

>...,

I

p-.-

,,,,_,,.

"1"1 ,,_..

<1

"q I - -

mo

/A

I

..,...

,"-,I

mo

.,-,,,

I

<1

/A

,,,.-i

I,,,4

>.-,

...,.

,,,,'-,I

,,,-,1

,,,-,1

,,q

9

~

>-,

~" "" XV

"6, "

~"

<1 ""

.._

~'1

II

,"-t

<1

4-

<1

"6,

<~

/A I bal,c

-.-."" ,"-I

,'.1

<1

<1

+

,".t

/A

,_.]

e~

9

9

,....

O

/A

i.o o

ixa

..,,,

,"-t

r.a

<1

,,,.,q

II

ol.o

i,,,,-, o

~.,,,. o

=

9

"-',

~,.~o "-1

,.-]

I

t'imo

O

i,,-, o

o', ",,,.,4

9

~,,,.,, o

>. 9 "-1 ~,.,,. o

~,,.. o

..,,,,,a

,.,,,-,.

F..

?

198

A. V Babin

El/2 d > c ~ . v3/2~,1

(76)

Therefore, (69) holds when an integer d satisfies d > c'G. Here, c', c are dimensionless constants which depend only on the shape of the domain. Therefore, we can use Theorem 3.4. So we obtain the following result of Constantin et al. [60], see also Constantin and Foias [56], Temam [184], Chepyzhov and Ilyin [45]. THEOREM 4.1. Let d satisfy either (76) or d > c'G. Then d-dimensional volumes on A decay exponentially as t --> oc. Hausdo.rff dimension dimH A o f the attractor A satisfies the estimate

gl/2 dimHA ~< c

v3/2~ i

~ c'G.

The,fractal dimension dimF A o f the attractor ,A satisfies the same estimate 81/2

dimFA ~< c

V3/2~. i

~< {"G.

-I/2 REMARK. Introducing the macroscopical length L ( ) - X i and the dissipation length L,; = s l / a / v 3/4, one can rewrite condition (76) in terms of the ratio of different spatial scales, following Temam [I 85 ]:

d > c

(,0),2 ~--~a

Note that, since II.fllfl)/2 has the dimension of a velocity, a generalized Reynolds number can be introduced

Re

llfllo

~/2

vXl/2 9 i

This number is related to the Grasshof number: Re = G I/2 REMARK. The definitions of the Hausdorff and ffactal dimensions of attractors include the metric of the space, so dimensions computed using the Hi-norm instead of the H0norm could be different. But this does not happen. Clearly, balls Ilvll~ ~< r are smaller than Ilvll0 ~< r. Therefore the number of Hi-balls needed to cover A is larger than the number of H0-balls needed to cover A and therefore Hi-computed dimension may be only larger. When f 6 H, the solution operator St=l is a smoothing operator, in particular for positive t it is Lipschitz from H into Hi with a Lipschitz constant L0. Since S! A = A, the covering by balls {llv - vjll0 ~< r} induces covering by {llu - S, vjll, <~ Lor}. Therefore, the dimensions computed using the metrics in H and H1 are the same.

,_.,.

m .

,...

9' ~

~-

'~

.-.

-,

~

m

+

'~

\v

-~%.~

~

~

~.

~

.,

~"

~

-~

t'~

~~ -~ -~

--

-.

=

-~.

--C]

C~+\V ~

~

P-/A~

--~

--.

-

~

o

.~

~-,~

.

~

,..~

--.

~

~

~.~_

~',.<

,,.4

~

m Z [

~

~

=

=.-. "

~v ~ ~ ~ _~ ~..~

~

~

~

-

~ - - -

~'=_ C) ~

B

/A

--

-

~"u

="~

~."

/A

~

'..~

',.~

"~.

,....

=

..

t"3

~..

,

,....

_~.

...,.

~

M

= 1- - "- - ~

"~

---

~

+

,,,.,.

.__._

.-,

I

m m

...

~~

-..1

;~

b~

II

+

<]

"H

~

/A

<

-.

~

t~

_~

,~

~

Zm-~"

m.

..~

t.m

-

,"--

-

~

_<.

9

--

,.~

Om

~

"a

N..

~

9

_

W<

_

=

_

~

m

b

],

.~

200

A. V. Babin

technical calculation, we obtain (68) in the form lim sup

sup

l[fo

tr (F' (, (r))/7E,, (r))] dr

T-+e~ uoc.A ~

Cl Vkl d2 -Jr-c2v~.l 2

X

2/3

)2/3

v3k~

(1 + log ( V 3 k ~ ) )

(81)

where sup sup 1 f 0 r IIAu< >ll 2 dr ~<

X--vkllim

T ~ ~ u o e A -~

k i l l f 112 1)

3 2 = v k~G .

The right-hand side of (81) is negative when d ~> c3

v3k~

1 + log

v3k~

(82)

with an absolute constant c3. Therefore, if d ~> "~1G2/3( 1 + l~

2/3,

(83)

then condition (69) holds and Theorem 4.2 follows from Theorems 3.3 and 3.4. A similar result can be formulated in terms of X. THEOREM 4.3. /f (82) holds', then d-dimensional volumes on ,A exponentially decay as t ~ cx~. The Hausdolff dimension of the attractor A satisfies dimH A <~ d. The fractal dimension of the attractor A satisfies dimF ,,4 ~< d. REMARK. The ratio (X/(V3k~)) I/3 in (82) can be interpreted (Temam [185]) as a square and a of the ratio of length scales ( L o / L x ) 2 with a macroscopical length L0 -- k -I/2 t ,

microscopical length L x -- ( v3/X )l/6.

4.3. Lower estimates for the dimension of attractor The method of the previous subsection gives upper estimates on the dimension of the attractor, which allow the dimension to grow according to the given formulas as the parameters vary. Though the estimates are physically reasonable, one may ask the question: Does the dimension really grow, and do the methods yield precise estimates? A positive answer is based on the following observation of Babin and Vishik [23]. Their idea (see also Babin and Vishik [32]) is to estimate the dimension of the attractor from below by using the inclusion M+(z0) C .,4, where M+(zo) is the unstable manifold of the semigroup St through an equilibrium point (time-independent solution) z0. The unstable

Attractors ~?f"Na vier-Stokes equations

201

manifold M+(z0) contains a local invariant manifold M+(z0, r), r > 1 (see Section 2). The local invariant manifold M + ( z o , r) is a smooth manifold through z0. The tangent linear subspace for this manifold at the point z0 is the invariant subspace of the linear operator t (z0) corresponding to eigenvalues g" with ]9] ~> r > 1 (unstable invariant subspace). St=l Note that S~(zo)vo is generated by the variation equation (60) which takes the form (43). To prove that the dimension of the attractor is large it is sufficient to find a steady-state solution z0 with a high dimension of the unstable invariant subspace. The unstable invariant subspace coincides with the invariant subspace of the operator L v = B ( v , z.o) + B(z.o, v) + vAv corresponding to eigenvalues Z with Re Z > 0, g" = e x. We consider the 2D NS system with periodic boundary conditions u ( x l + 27r/or0, X2) -- U(Xl, X2),

U(Xl, X2 -~- 2 7 r ) = U(Xl, x2),

(84)

with or0 ~< 1, that is, in an elongated periodic box. We take a small c~0 > 0. Let the forcing term have the form

f=

(-•

f_

~x2), 0),

rc gl (x2) dx2 -- O, 7(

with a 27r-periodic gl (x2). The steady-state 2D NS system v A z + B(z, z) = f

has a solution

:o(x) - (•

u(x2), 0),

where a periodic function U (x2) is found from the equation

f

a~U(x2) - ,~l (x2),

'r U ( x 2 ) dx2

-

-

0.

715

We take the parameter V of the form ~, = iv 2. A fixed number / is chosen so that

12fs }O(x2)[" --

Yr

-dx'~-

_

1 >

O,

7r

where 0(x2) is a periodic function that satisfies 030(x2) - gi (x2). It is proven in Babin and Vishik [23,32] that the unstable manifold M + ( z o ) C ,,4 of the semigroup St through this point has dimension not less than c/c~0 with a positive constant c. The proof uses the analysis of the Orr-Sommerfeld equation made by Yudovich [191]. This implies the following theorem of Babin and Vishik [23].

202

A. V. Babin

THEOREM 4.4. The dimension ffractal and Hausdorff) of the global attractor of the 2D NS system in the periodic box [0, 27r/cg0] x [0, 2n'] with the special force f is estimated from below as follows: dim(A) ~> c/oto,

c > O.

(85)

Now, we deduce a corollary for a square (or almost square) periodic box. When m is an integer, solutions u which satisfy (84) can be periodically extended; they form an invariant subspace in the space of solutions that are 27r m-periodic with respect to x2 and 2n'/c~0 with respect to x~. We consider the 2D NS system in 2rr/c~0 • 2zr[l/c~0]-box, where [l/c~0] = m is the integer part of 1/or0 (for simplicity one can take l/c~0 integer, so that [l/c~0] = 1/c~0). The solutions which form M+(zo) and satisfy (84) lie in the attractor of the system in the square 2n/ot0 • 2n'[ 1/ot0]. Therefore (85) is true for this system too. Since f = lvZg where g = g(x2) is 2zr-periodic, we have the norm in L2([0, 2n'/c~o] x [0, 2rr[ l/c~0]]) Ilfll 2L2(lO,2~/o~olxlO,2rcl l /uoll)

-- - 12v4(l/c~o)ll/c~oJllgll 2L-,(10,27r l) - - C 124 ( l / o , o ) [ I / ~ ( ) ] , and the first Stokes eigenvalue is X l - c~o . Therefore, for this problem the Grasshof number is equal to

G

m

Ilfllo

c'

U2,~I

~()3 '

and we obtain from (85) the estimate dim.,4 >~c" G !/3.

(86)

REMARK. The original lower estimate (85) was given by Babin and Vishik [23] for an elongated periodic box [0, L/c~o] • [0, LI in terms of c~0- I . in terms of the Grasshof number this estimate implies that dim.A ~> ~,tG2/5. The above elementary argument based on repeating the space period [1/c~0] times yields (86) from the original estimate (85). Estimates in terms of G of the form dim.4 ~> ~'G 2/3 w e r e obtained by Liu [ 146,147] by a direct treatment of 2zr • 27r periodic box. REMARK. The upper estimate of Theorem 4.2 differs from the lower estimate in Liu [ 146, 147] by a logarithmic factor (1 + log(G)) 2/3. One would like to obtain lower and upper estimates of exactly the same order by improving either the lower or the upper estimate. Ziane [192] considers the case of an elongated periodic box [0, L/oto] x [0, L] for which the lower estimate (85) was given. He obtains the following upper estimate of the dimension of the attractor which is sharp.

Attractors ~'Navier-Stokes equations

203

THEOREM 4.5. The following estimate holds f o r the fractal and Hausdorff dimensions of the attractor o f the 2D NS system with periodic boundary conditions (84) in an elongated box: dim(A) ~< c ' ( l / a o + G / a o ) , "~

1/2

(87)

c ' > O,

2

where G - o t 0 L [[fIIL2(IO,L/~oIxlO, LI)/V 2 and c ~ is an absolute constant; f has zero average with respect to x2. This estimate is exactly of the same order with respect to or0 as the estimate (85) of Babin and Vishik [23]. REMARK. Doering and Wang [63] consider 2D NS system in an elongated box with periodic horizontal and no-slip boundary conditions on top and bottom boundaries. They have obtained upper estimate of dimension of the attractor dim(A) ~< cRe3/2/~o with the same kind of dependence on the aspect ratio c~0 as in (87).

4.4. Equations in an unbounded domain Flow of a fluid in a pipe or in a channel can be described by the NS system in an unbounded domain. When the domain D is unbounded, the operators St are not compact anymore in the usual Sobolev metric. One can prove the existence of attractors of such semigroups in a weak topology, but generally speaking the attractors can be infinite-dimensional (see Babin and Vishik [33 ]). This can also be seen from the lower estimates of the dimension of the attractor of 2D NS system in the previous section that show that the dimension of the attractor tends to infinity when the size of the domain tends to infinity. Nevertheless, when no-slip boundary conditions are imposed, and the domain is not too wide at infinity (like a channel or a pipe) and the forcing term is spatially localized, one can prove the existence of a finite-dimensional attractor. We consider here for simplicity a rectilinear channel D = {(xl,x2): 0 < ,,cl < b, - o o < x2 < oo} along xz-axis. This case was studied by Abergel [1] (in zero flux case) and by Babin [5-7]. Similar results on the existence of the attractor in a curvilinear channel are proven by Babin [5,6]. We consider a strongly perturbed Poiseuille flow, that is, we assume that the velocity field has Poiseuille component

(o, w(x,

(o, v,,x,(h-x,)).

This component gives the leading contribution as x2 --~ ~ since it does not tend to zero; the flux through the cross-section generated by the Poiseuille flow is non-zero when V0 r 0. Clearly, V (x) is a steady-state solution of the 2D NS system V. VV - vAV = Vp,

VaD = 0 ,

V. V =0.

0

\V

II

<1

c~

I

~

~

'--h

~

_.~"

~-~

+

~:

'~

_l ~ . ~~

~~

o""

_

9

II

9

'~

+

......

.-~, p__

0

~,"

rh

~

V

r~

~

~

~

~= - .

El.

=-~

~

~

0

-.~

.

~:.

--

~. ~

~

._.

~

~-

~-.

~

.--o

E

.~

9- - ~

-.

~

~.~

_.

~

T~

%

ro P_...

to

~

~

,~

\V

"~

<

~

~

--

o

--

~

..-.

P--"

0

~"

~:~

~.o

~

~-~

~-h ~"h

-.

~

~

~

"

~.~

,".I

+

,....

0

~=-

f~q

~.

o

Attractors ~" Navier-Stokes equations

205

THEOREM 4.6 (Babin [7]). Let v ~ ' - v - k ~ > 0 and .[" E HI,• with y > O. Then the 2D NS system (88) with no-slip boundary conditions (3) determines a semigroup St o f b o u n d e d continuous operators in H. This semigroup has a global attractor A. The attractor is bounded in the spaces H2,• and HI,• and has finite H a u s d o r f f and fractal dimensions which satisfy the estimate d i m A ~< Cl Ilfll2k]-2v ''-4.

(90)

The condition v" = v - k(~ > 0 imposes a restriction on the magnitude of the flux through the channel. Without this kind of restriction the basic Poiseuille flow with a very large flux is unstable everywhere in the infinite channel and one cannot expect existence of a finitedimensional or compact attractor. REMARK. When the flux through a cross-section is zero (V = 0), existence and finite dimensionality of the attractor for the case of rectilinear channel is proven by Abergel [ 1]. For a curvilinear channel, existence and finite dimensionality is proven by Babin [5 ]. When the forcing is not assumed to belong to a weighted space, the existence of the attractor and estimate of type (90) for f E H-i,() is proven by Rosa [170]. Ju [123] extended the compactness and convergence results of Rosa [ 170] to the Hi norm for more regular f E Ho,o. These results are proven in the zero-flux case. Moise et al. [I 61] considered a flow in a rectilinear channel past an obstacle. They consider flows that are perturbations of a constant flow U ~ at the spatial infinity. Compared with the Poiseuille flow such a flow has a simpler behavior at infinity; namely, because of the absence of shearing for the constant flow, it is stable for any value of U ~ . Existence of a global attractor is proven in Moise [161] for perturbations that are not assumed to lie in weighted spaces and for arbitrary large U ~ . Linear stability analysis shows that small disturbances of the Poiseuille ttow do not propagate along the channel if the flux is not too large. The next theorem proved by Babin [7] shows that arbitrary large disturbances do not propagate along the channel, too: at the spatial infinity any time dependence dies out. So the turbulence is spatially localized. THEOREM 4.7. Let /" E Ho.~t,~ with ~o> 0, and v - X~,~ > 0. Then, there exists a timeindel~endent solution ' z ( x ) _ _ _ . , • o f too). Any solution h(x, t),-o<~ < t < + ~ , that lies on the attractor A, admits an asymptotic expansion as Ix l ~ ~ : cy~

u(x, t) "~ Z z.j(x), j =I N

It(X, t) -- Z

z.i(x) E H2..i~,, (9i)

7..j(X) E HI,(N+I)y,

j=l

where ttle -.i(x) are explicitly written (see Babin [7]). Moreover, u ( x , t ) is ~:~:ponentiall3' close to x(x ) at infinity, namely, there exists a constant VI > 0 such that f o r any solution u(x,t)

206

A. V. Babin

and

~T+,fDeY' 1"21(I A (z'(x) - u(x,t))l 2

+

[a,u(x, t)l 2) dx

~ C

f o r all T.

REMARK. If the original channel is rectilinear only at infinity and has a local narrowing, or an obstacle, then a change of coordinates that makes the channel rectilinear everywhere results in a localized force as considered above (it also results in a spatially localized linear term which can be treated similarly). One can consider in a similar way the 3D NS system in a cylindrical domain (in a pipe). In this case, existence of attractors and non-trivial solutions defined for all t, -cx~ < t < +cx~, is not proven. But one can prove that if such solutions exist and if the flux through cross-section of the pipe is not too large, then time-independent asymptotic expansion of form (92) holds, see Babin [7]. So, in the 3D case turbulence does not propagate to infinity if the flux is not too large. REMARK. Navier-Stokes equations in domains with a finite area are studied by llyin [1 19], he proves the existence of the attractor and its finite dimensionality.

5. The 3D Navier-Stokes equations 5.1. The dimension q['regular inwiriant sets

It is not known if a solution of the 3D Navier-Stokes system with regular initial data stays regular for all times. One can construct weak solutions defined for all times, but it is not known if such solutions are unique. This situation makes it impossible to apply the standard theory of global attractors which is applicable in the 2D case. An approach to a general 3D NS system is to consider regular invariant sets not requiring attraction to them. Though dynamics is not defined for arbitrary initial data, one may assume (cases when this assumption is proven to be true are discussed below) the existence of bounded strictly invariant sets. One may try to estimate the dimension of such sets. A regular invariant set 2 of the 3D NS consists of values of regular solutions u(t) that are defined for all t, -cx~ < t < cx~, and are bounded in H0 (92) It follows from (36) that such solutions are uniformly bounded in H0 9

2-C

v:

sup,,Ilvll~3~

,-)

v2~. ~

9

For the 2D NS system with f E H0, such solutions are also bounded in Hi and in H2. But for the 3D NS system uniform boundedness in a norm better than H0 is not known, even for regular solutions, and only the energy estimate is available to estimate time averaged

207

Attractors of Navier-Stokes equations

Ilu(t)ll if, see (38). This is not sufficient to get estimates of the dimension in the 3D case by known methods. Therefore we consider regular invariant sets of 3D NS system of the form -- { v" v -- u(t), -cxz < t < oo} A {additional boundedness conditions}.

The simplest boundedness condition is boundedness in Hi. One can estimate the Hausdorff and fractal dimensions of such sets. Such sets always exist, for example, when the force f 6 H0 does not depend on time, there always exists a regular time independent solution z E H2, see Lions [ 141 ], Temam [ 183]. Such a solution gives a one-point regular invariant set. Estimates of the dimension of regular invariant sets bounded in Hi can be given in the following form (see Constantin et al. [58 ], Temam [185 ]). THEOREM 5.1. Let Z C HI be a bounded in H~ invariant set o f the f o r m (92). Let the quantity ~ be defined by 4/5

-- 'im'up'up I ' f'f,'

V u ( x , t)[ 5/2 dx dt

where u(t) is a regular sollttion that lies in Z and domaill D. Let m be an integer that sati.sfies

ll(O)

- - ll().

,

(93)

Here, [D[ is the ~'olume o[the

m >~ c lDI

(94)

Their, ...ttle set Z has Jinite Haltsdor[.]" and fractal dimensions that sati,sfv the estimates dimH(.T) ~ m, diml:(/) <~ m.

The proof of this theorem is similar to that of Theorem 4.2. The question of the existence of a non-trivial (that is, more than one-point), locally attracting regular invariant set of the 3D NS system in the general case is ()pen. The existence of nontrivial regular invariant sets of the 3D NS system, in fact of global attractors of the 3D NS system, is proven in two cases: by Raugei and Sell for domains that are thin in one direction and by Babin, Mahalov, and Nicolaenko for 3D NS system with a large Coriolis lorce. We discuss these important cases below in more detail.

5.2. Equations in a rotating franle

The 3D Navier-Stokes equations in a rotating frame are: OtU -- - U . V U + Y2e3 • U + v A U + f + V p,

V . U --O,

(95)

208

A. V. Babin

where U = (U l , U 2, U 3) is the velocity field, S2e3 • U is the Coriolis force term. The literature on rotating fluids is large, see Greenspan [ 101 ], Pedlosky [ 165]. The axis of rotation is along the x3-axis. We impose periodic boundary conditions (4) and zero average condition (5) over the periodic box D = [0, 27ral] x [0, 2yra2] x [0, 27ra3]. We take a l = 1. The case of a small Rossby number corresponds to large f2. Applying Leray projection/7 onto solenoidal vector fields, we rewrite (95) in the form (96)

OtU = B(U, U) + I-2SU + v A U + .f,

(

")

whereS-/TJ/7, J0]~ ~0 0 being the rotation matrix. The linearized version of (95) was studied by Sobolev [ 180], who continued the analysis of Poincar6 [166] (cf. Arnold and Khesin [2]). The extension of this analysis to the nonlinear equations (95) was done by Babin et al. in [10,11,13]. First results on regularity of Euler and Navier-Stokes systems in rapidly rotating frame were obtained by Babin et al. [ 11,13 ]. First results on the regularity in the context of three-dimensional geophysical flows were obtained by Babin et al. [12,15 ]. Mathematical papers on this and related equations include works by Grenier [ 102 ], Embid and Majda [69 I, Gallagher [92,93 ]. First, we give the results of Babin et al. [ 161 on the global regularity of solutions of the 3D NS system with a large Coriolis force in a simplified form that is sufficient in view of the application to attractors. The proofs of these results use techniques based on a detailed study of resonant three-wave nonlinear interactions of dispersive Poincar6 inertial waves. Note that the skew-symmetric Coriolis force term does not affect energy estimates at all. The regularization cffcct of the Coriolis force is purely nonlincar, since it does not affect energy of the Fourier modes of the linear Stokes problem at all. THEOREM 5.2. Let a, > I/2, v > 0. Let M,~, M~F be arbitrary large.fixed nttmbers. Let IIU (0)I1,~ <~ M~, let ll./'11,~-! ~< MuF,

(97)

and let s >~ F2()(MuF, Mu, v). Then, there exists a regular solution U(t), 0 <~ t < oo, o f (95) such that

where C'u depends only on Ma, M~ F, v, and on aspect ratios a2, a3 q/'the period box.

THEOREM 5.3. Let (97) hold with o~ = I. Let Ro be arbitrary large. Let IIU(0)llo ~ Ro and T - - Ro/v. Then,for everyfixed F2 >~ ~ ' , where f2' is a numberwhich depends only on M~F, v, a2, a3 (and does not depend on U(O)), and.[or any weak solution U(t) o f the 3D rotating Navier-Stokes equati~ms (95) that is defined on [0, T] and sati,sfies the classical energy inequality (28) on [0, T], the following proposition is true: U (t) can be extended to 0 < t < + o o a n d i s regular tbr T ~< t < +oo; itbelongsto Hi and [IU(t)l]l <~ Ci(MuF, v) ./b r ever), t >~ T . A

A

Attractors of Navier-Stokes equations

209

Now, we define the semigroup and the attractor. Let B0 = {U: IlUl[, ~ CI(M~F, v)}. According to Theorem 5.3, B0 is an absorbing set for all weak solutions that satisfy the energy inequality. We take S-2~ large enough and according to Theorem 5.2 for S2 ~> I-2~ solutions with initial data in B0 are regular and bounded in H! for all t. Since such solutions are unique, we have operators StU for U e B o . The set X -- I,_Jr>0 s r B0 is invariant, St X C X , and the semigroup St is generated by the 3D NS system on the invariant set X. Note that St on X is a semigroup of continuous operators which are compact for t > 0. Formula (50) defines, according to Theorem 2.1, the global attractor of the semigroup; it is bounded in HI according to (98) with ot = I. So, we obtain the following theorem. THEOREM 5.4. l.f f e Ho and S2 >~ s2'(llfll0, v, a2, as), there exists a global attractor of the 3D Navier-Stokes equations, which is bounded in HI and H2. Ever)' weak solution that satiafies the classical energy inequality is attracted to the global attractor as t --+ +oc. Quantity (93) where ]D] -- (2rr)Sala2as, is finite. The attractor has Hausdorff dimension dimH(.A) <. m and fractal dimension dimF(.A) <~ m, where m is a minimal integer that sati,~fies

m > ~"lOl

8. )3/4

(99)

PR()()F. The boundedness ot,- fl)liows from the boundedness of the attractor in HI and related boundedness of

,f,,f

-T

IAt,(.,- ,) l- d.,- dr.

To get estimates of the dimension we use Theorem 5.1. The conditions assumed in Theorem 5.1 (boundedness of the invariant set in H i ) and boundedness of s are satisfied for the 3D NS system in rapidly rotating frame according to Theorem 5.3. The estimates used in the proof of Theorem 5. ! (see the proof of Theorem 4.1 or Section VII.2.3 of the book Temam 1185] for more details) are directly applicable since the Coriolis term does not affect energy estimates at all and the trace of the skew-symmetric operator a"2SU is equal to zero. Therefore, all the computations can be done without any change and we get estimates of the dimension. [-1 REMARK. In the inertial coordinate system, alter the change of variables

U'(v) . = e S ~ ' l ' / 2 U ( e O J ' / 2 y ) - -S-2 ~e3 • y, (95) turns into the Navier-Stokes system (1) without Coriolis term but with a modified initial data

ibU' = - U ' . V U ' + v A U ' + . f ' + V p ' , t'2 curl(U)'(0) -- curl U(0) + -:-es. 2

V.U'=0,

II

CD

X

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= <~m &

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2+.,

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= r

~=

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+._. +

+"

~ "+"+ o ~. =~..+ =+ ._, ..+

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~= ,+

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K+

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o ~ + 5 ~

~

~ = +-' ~ ~ 8

8

~

+:>.,+ o

O

~;

-,

,.-, Pu ~

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g=

""

+ =

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_ = m~ + - ', ~ +

.

,.+-,

..

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~

t,--+

V/

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~

t"-J

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t:::::::

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+,...,,

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=

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+-,

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=

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t,'-'

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+''

212

A. V. Babin

REMARK. The 2D NS system in a thin domain with combined periodic no-slip boundary conditions is considered by Iftimie and Raugel [ 112], Moise et al. [ 162], Temam and Ziane [187]. REMARK. No-slip boundary conditions in a thin domain for the 3D NS system with initial data bounded in an appropriate norm stabilize the dynamics. The dynamics on bounded sets is defined for all times and the attractor consists of one point, see Avrin [3] and Montgomery-Smith [163] for details.

5.4. Generalized attractors Multivalued semigroups. One of the ways to overcome the difficulty of non-uniqueness of weak solutions is to use theory of semigroups of multivalued operators. The theory of attractors of multivalued semigroups was started by Babin and Vishik [127]. Further works in this direction are by Babin [9], Ball [34], Melnik [156], Melnik and Valero [157]. In these works, the existence of different types of generalized global attractors for 3D equations is proved. These attractors do not have as good properties as the attractors of the 2D NS system. It is not known if they are finite-dimensional in general. The global attractor of the 3D NS system with a large Coriolis force in Theorem 5.4 attracts solutions that may be multivalued for t < T. Therefore it can be considered as an example of a finite-dimensional attractor of a multivalued semigroup. Trajectoo' attractors. The construction of trajectory attractors (see Section 2) can be applied to treat equations without uniqueness, in particular the 3D Navier-Stokes equations. For details of the theory of trajectory attractors, see Sell [! 76], Chepyzhov and Vishik [48, 49], Kapustyan and Melnik [125 ], Feireisl [71]. Measure attractors. Kuksin and Shirikyan in [ 127] prove the existence of a unique invariant measure for the stochastic nonautonomous 2D NS system. The support of the measure is called a measure attractor. Schmalfuss [178] studies relations between measure attractors and random attractors. Flandoli and Schmalfuss [72,73] prove existence of a generalized stochastic attractor for a stochastic 3D NS system. Non-standard analysis. Capinski and Cutland [43] considered attractors of the 3D NS system in the framework of non-standard analysis.

Modifications of NS and related hydrodynamic equations The methods originally developed for the NS system are applied successfully to many different problems. Here, we refer to some of related papers" the list of papers presented here is inevitably not complete. We did not intend to give a complete bibliography but rather

Attractors ~g'Navier-Stokes equations

213

to show the vast scope and some of directions of continuing mathematical research on attractors of problems related to the Navier-Stokes system. First we briefly mention papers which treat the 3D equations of hydrodynamics which include additional physical effects.

3D problems. Lions et al. [142-144] study the primitive equations of geophysics. The equations in many respects are similar to the 3D NS system but are more complex since they take into account more physical effects, in particular rotation and stratification, and contain more unknown functions. The authors have built a general theory of such equations and have analyzed their properties; in particular, they have estimated the dimension of regular invariant sets (see also Lions et al. [145]). The existence of global strong solutions of the primitive equations is not known in general case and the existence of the global attractor in the classical sense is not known, too. Babin et al. [ 12,16] consider the primitive equations (Boussinesq system) under periodic boundary conditions. They prove that when stratification or rotation is strong enough, the equations have global regular solutions; the dynamics in corresponding function spaces is well-defined and the global attractor exists, consists of regular solutions and, therefore, has a finite fractal and Hausdorff dimension. (These results are similar to Theorems 5.2-5.4 of this paper.) In a number of papers, modifications of the classical 3D NS system are considered. Modified equations often have better regularity properties than the original 3D NS equalions and in many cases existence o l a tinite-dimensional global attractor can be proven. Ladyzhenskaya in 1136,1371 considers modified Navier-Stokes equations which admit global regular solutions. She proves the existence of the global attractor, studies its properties and, in particular, gives estimates of its dimension. Mfilek and Ne(.as [ 15()1 consider a system of Navier-Stokes type which has unique weak solutions in appropriate spaces and prove the existence of a finite-dimensional global attractor. Mfilek et al. [I 52] study Boussinesq approximation in three dimensions with a modified stress tensor. They prove existence ot'a finite-dimensional global attractor. Mfilek and Pra~fik [ 151 ] consider non-Newtonian fluids and prove existence of the global attractor with a finite fractal dimension. Bellout et al. [371 consider non-linear bipolar viscous fluids. Upper bounds are obtained for the Hausdorff and fractal dimensions of the global attractor. Foias et al. [741 consider three-dimensional viscous Camassa-Holm equations (also called Navier-Stokes alpha-equations). They prove existence ot'a linite-dimensional global attractor and give an estimate of its dimension in terms of the physical parameters of the equations. 2D problems. The theory of attractors for the 2D NS system can be extended to flows on 2D compact manifolds. Such questions were considered by Cao et al. [411 and by llyin [115-118]. They proved the existence of the attractors and obtained upper estimates of their dimension. The estimates of llyin [ 118] have the same form as the sharp estimates for the 2D NS system with periodic boundary conditions (see Theorem 4.2). Cao et al. [41 ] estimate the number of determining functionals (such as determining modes and determining nodes) for the 2D NS system on a rotating sphere.

214

A. V. Babin

Foias et al. [77] consider the B6nard problem. In the 2D case they prove existence of a global attractor and give an estimate of its dimension. In the 3D case they estimate dimension of regular invariant sets. Ghidaglia and Temam [98] consider the equations of slightly compressible fluids proposed by Chorin and Temam. They prove the existence of a global attractor and estimate its fractal dimension. Hoff and Ziane [109,110] prove the existence of a compact global attractor for the Navier-Stokes equations of compressible flow in one space dimension. This equation does not fit into standard framework due to a lack of compactness. Since properties of the semigroup do not allow to prove finiteness of the fractal dimension, they describe the properties of the attractor in terms of determining nodes. Ilyin [ 120] considers a non-autonomous 2D NS system with a rapidly oscillating almost periodic forcing. He proves that its attractor tends to that of the averaged equation when the frequency tends to infinity. Ladyzhenskaya and Seregin [ 138] study the 2D equations of the dynamics of generalized Newtonian liquids. They prove existence of attractors and estimate their dimension. Miranville and Wang [ 158] consider the 2D NS system with a tangential boundary condition (see also Brown et al. [40]). They give an estimate of dimension of the global attractor. They also estimate dimension of regular invariant sets for 3D channel and Couette-Taylor flows. Miranville and Ziane [ 160] estimate the dimension of the attractor of the B6nard problem with free surfaces in an elongated rectangle. Njamkepo [164] studies a thermohydraulic problem for the slightly compressible 2D Navier-Stokes equations. Bounds on the fractal dimension of the global attractor are given in terms of the physical data of the problem. Ziane [I 93] considers 2D NS system with free boundary condition in elongated rectangular domains and gives an upper bound on the dimension of the attractor.

Acknowledgements The effort of the author was supported by AFOSR grants F49620-99-1-0203 and F4962001-1-0567. The author would like to thank Professor E. Titi and Dr. A.A. Ilyin for very useful discussions and valuable remarks.

References [I] F. Abergei, Attra~'tor.fiw a Navier-Stokes .flow in an unhounded ~hmmin, RAIRO Mod61. Math. Anal. Num6r. 23 (3) (1989), 359-370. [2] V.I. Arnold and B.A. Khesin, T~q~ological Methods in Hydrodynamics, Appl. Math. Sci., Vol. 125, Springer-Verlag, Berlin (1997). [3] J.D. Avrin, A one-point attractor theory for the Navier-Stokes equation on thin domains with m,-slip boundao' conditions, Proc. Amer. Math. Soc. 127 (3)(1999), 725-735. [4] J. Avrin, A. Babin, A. Mahalov and B. Nicolaenko, On regularity of solutions of3D Navier-Stokes equations, Appl. Anal. 7 (1999), 197-214.

Attractors of Na vier-Stokes equations

215

15] A.V. Babin, The asymptotic" behavior as x ~ cx~ offunctions lying on the attractor of the two-dimensional Navier-Stokes system in an unbounded plane domain, Mat. Sb. 182 (12) ( 1991 ), 1683-1709; Translations in Math. USSR-Sb. 74 (2) (1993), 427-453. [6] A.V. Babin, The attractor of a Navier-Stokes system in an unbounded channel-like domain, J. Dynamics Differential Equations 4 (4) (1992), 555-584. [7] A.V. Babin, Asymptotic expansion at il![inity of a strongly perturbed Poiseuille flow, Properties of Global Attractors of Partial Differential Equations, Adv. in Soviet Math., Vol. I(), Amer. Math. Soc., Providence, RI (1992), 1-85. [8] A.V. Babin, Asymptotic behavior as x ~ cx~ of steady.flows in a po~e, Mat. Zametki 53 (3) (1993), 3-14; Translation in: Math. Notes 53 (3-4) (1993), 245-253. [91 A.V. Babin, Attractor of the generalized semigroup generated bv an ello~tic equation in a ~'vlindrical ~h)main, Izv. Ross. Akad. Nauk Ser. Mat. 58 (2) (1994), 3-18. ll01 A. Babin, A. Mahalov and B. Nicolaenko, Long-time averaged Euler and Navier-Stokes equations,fi~r rotating.fluids, Structure and Dynamics of Nonlinear Waves in Fluids, K. Kirchg/.issner and A. Mielke, eds, World Scientific, Singapore (1995), 145-157. 11] A. Babin, A. Mahalov and B. Nicolaenko, Global splitting, integrability and regularity of 3D Euler and NavierStokes equations for unff'orntly rotating.fluids, European J. Mech. B/Fluids 15 (1996), 291-300. 121 A. Babin, A. Mahalov and B. Nicolaenko, Resonances and regularity for Boussinesq equations, Russian J. Math. Phys. 4 (4) (I 996), 417-428. 13] A. Babin, A. Mahalov and B. Nicolaenko, Global regularity am/inte~rability o f 3 D Euler and N a v i e r Stokes equations for lllti['ormly tvmtting.fltdds, Asymptotic Anal. 15 (1997), 1()3-15(). 14] A. Babin, A. Mahalov and B. Nicolaenko, Global .wlitting and reguhtrity of'rotating shalhm'-water equations, European J. Mech. B/Fluids 16 (5)(1997), 725-754. 151 A. Babin, A. Mahalov and B. Nicolaenko, Regtdarity am/integrahility q/rotating .vhalh)w-water equations, C. R. Acad. Sci. Paris Scr. I 324 (1997), 593-598. 16] A. Babin, A. Mahalov and B. Nicolacnko, On the regularity of three-dimensional rotating l:'uh'rBottssine.sq equations, Math. Modcls Methods Appl. Sci. 9 (7) ( ! 999), I ()89- I 12 I. 17] A. Babin, A. Mahalov and B. Nicolacnko, (,;lol~al regularity o131) rotating Na~'ier-Stokes eqtmtions.for resonant domains, Indiana Univ. Math. J. 48 (3) (1999), ! 133-I 176. 181 A. Babin, A. Mahahw and B. Nicolacnko, Fast .singttlar os~'illating limits and global regldarity.for the 31) primitive equations ofgeol~hysi~'s, Mod61. Math. Anal. Num6r. 34 (2)(2()()()), 201-222. 191 A. Babin, A. Mahalov and B. Nicolacnko, 31)Na~'ier-Stokes aml Etder equations with initial ~htta ~'hara~'terized by uniformly large ~'orti~'itv, Indiana Univ. Math. J. 5ti (Special Issue) (2()() I ), 1-35. 1201 A. Babin, A. Mahalov, B. Nicolaenko and Y. Zhou, On the a.symptoti~" reginu, s and the strongly stratified limit o./'rotating Boussinesq equations, Thcorct. Comput. Fluid l)ynamics 9 (I 997), 223-251. 1211 A. Babin and B. Nicolaenko, Evponential attra~'tors aml im'rtiallv stabh" algorithms for Navier-Stokes equations, Progress in Partial Differential Equations: The Mctz Surveys, Vol. 3, Pitman Rcs. Notes Math. Scr., Vol. 314, l~ongman, Harlow (1994), 185-198. 1221 A.V. Babin and S.Yu. Pilyugin, Contintums dependence q/'attractors on the shal~e q/'domain, Zap. Nauchn. Scm. S.-Petcrburg. Otdel. Mat. Inst. Stcklov. (POMI) 221 (1995), Kracv. Zadachi Mat. Fiz. i Smczh. Voprosy Tcor. Funktsii. 26, 58-66, 254: Translation in: J. Math. Sci. 87 (2)(1997), 3304-33 i(). [231 A.V. Babin and M.I. Vishik, Atlra~'lors q/l~arlial di['['erential equations and estimates of their dimension, Uspekhi Mat. Nauk 38 (4) (1982), 133-187; English transl.: Russian Math. Surveys 38 (4) (1983), 151213. 1241 A.V. Babin and M.I. Vishik, A ttra~'tors' o/'the Navier-Stokes svstent and parabolic equations and estintates of their dimension, in Botouhu'v value problems o./ mathemati~'ai physics and related questions in the theory q//im~'tions, 14, Zap. Nauchn. Sere. l~eningrad. Otdcl. Mat. Inst. Steklov. (LOMI) 115 (1982), 3-15, 3O5. [25] A.V. Babin and M.I. Vishik, 77re dimension q/'attractotw of the Navier-Stokes system and other evolution equations, Dokl. Akad. Nauk SSSR 271 (6) (1983), 1289-1293. [26 ] A.V. Babin and M.I. Vishik, Estimates ./)'ore above and./)'om below of dimension of attra~'tors" of evolution partial differential equations, Sibirsk. Mat. Zh. 24 (5) (1983), 15-30; English transl.: Siberian Math. J. 24 (1983).

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