Asymptotic analysis of the navier-stokes equations

Physica 9D (1983) 157-188 North-Holland Publishing Company

ASYMPTOTIC ANALYSIS OF THE NAVIER-STOKES EQUATIONS C. FOIAS Indiana University, Bloomington, IN 47405, USA

O.P. MANLEY U.S. Department of Energy, Washington DC 20545, USA

R. TEMAM Laboratoire d'Analyse Num~rique, C.N.R.S. and Universit~ Paris-Sud, 91405 Orsay, France

Y.M. TREVE La Jolla Institute, La Jolla, CA 92038, USA Received 5 April 1983

New bounds are established on the number of modes which determine the solutions of the Navier-Stokes equations in two dimensions. The best bound available at present is nearly proportional to the generalized Grashof number (defined in the paper), and less than logarithmically dependent on the spatial structure, or the shape of the force driving the flow. To the extent that for the case of 2-dimensional Rayleigh-Brnard convection, the generalized Grashof number may be identified with the usual Grashof number, the resulting bound on the number of modes is found to differ only slightly from a bound obtained earlier on heuristic grounds.

1. Introduction

As shown in [1] a remarkable property of any two-dimensional solution of the viscous incompressible Navier-Stokes equations is that, for large times, it is completely determined by a finite number~ say N, of spatial Fourier modes. In [1] a rigorous upper bound on N was derived which essentially depends on the rate of viscous dissipation of energy associated with the solution considered. Here we present new rigorous estimates of an upper bound on the number N of the required, so-called determining modes; these estimates are of the order of the L 2 norm of the driving force. It must be emphasized at the outset that the number of modes N considered in [1] and in the present paper define truncations of exact Fourier representations of the solution. Thus, in all rigor, the bounds obtained in [1] and in the following do not apply to Galerkin approximations without further justification. However, pending such justification we will assume that N is close to the minimum number of modes needed in order for the Galerkin approximation to describe adequately the exact solution. This assumption is supported by the fact that the Galerkin approximations also enjoy the property of being determined by the same number N of modes which is independent of the order of the approximation [2]. The present study was stimulated by attempts to establish whether turbulence should be modeled along the lines suggested by Landau [3], or those suggested by Ruelle and Takens [4]. These attempts have centered on the so-called Lorenz [5] system which governs the temporal evolution of the simplest nontrivial Galerkin approximation to the two-dimensional Brnard convection problem. It is not our purpose here to elaborate further on the behavior of the Lorenz system as such. There exists a significant literature on this specialized subject [6], which lies outside the scope of this paper. What 0167-2789/83/0000-O000/$03.00 © 1983 North-Holland

C. Foias et al./Asymptotic analysis o f the Navier-Stokes equations

158

is of interest here, however, is the minimal set of modes required in order for the Galerkin approximations to be like the exact solutions of the Navier-Stokes equations. The problem of finding this minimal set has come to the forefront as in recent years attempts have been made to improve on the Lorenz model by adding more modes, e.g. [7]. Numerical experiments with these augmented models have yielded ambiguous results in that for some combinations of modes the strange attractor seemed to vanish, and the flow simulations became periodic or even steady. Thus without further guidelines, one is at a loss as to which set of modes corresponds a more or less realistic model of the flow. This is especially taxing since the customary ad hoc procedure of testing for small differences in the output as the order of the approximation is increased, becomes meaningless if one is not sure about the qualitative validity of the tested approximation. Faced with this uncertainty some of us have inquired whether a combination of physical and mathematical principles might provide such guidelines. It was found that at least in the case of two-dimensional flows the answer is in the affirmative and the results were presented earlier elsewhere [8]. In order to complement and amplify these results, here we provide a rigorous justification for the earlier estimates of the required number of modes. It must be emphasized that the results reported here are in the nature of sufficient conditions, and much work is still needed to establish the corresponding necessary conditions. Fulfilling this last need is of great practical importance because, admittedly, the present estimates suggest that an uncomfortably large number of modes may be required to simulate even relatively simple flows. The paper is organized as follows: in the next section the mathematical problem is stated, the so-called set of determining modes is defined, a heuristic estimate of the number of determining modes for the case of two-dimensional Brnard convection is recalled from [8], and a generalized Grashof number is defined. In section 3 some mathematical preliminaries, including some important inequalities are reviewed. Section 4 presents an improved, more compact proof of the relevant theorem of [1] together with the lemmas needed to obtain that proof. In section 5 it is shown first that a rigorous but naive estimate of the number of determining modes is considerably larger than that obtained on the heuristic physical basis, and then it is argued that the heuristic estimate is in some well-defined sense "almost true". Section 6 bridges the gap between the "almost true" heuristic estimate and a rigorous one; here it is shown that independent of the shape of the force driving the flow, the difference between the two estimates is of less than logarithmic order. In section 7 it is proved that taking into account the structure of the driving force introduces but a slight correction to the heuristic estimate, a correction again of less than logarithmic order. Section 8 is devoted to some results concerning improved convergence, and it is followed by some concluding remarks.

2. Determining modes and heuristic estimate of their number 2.1. The initial value problem We consider the Navier-Stokes equations for a viscous incompressible fluid in a bounded domain f2 of R 2. Denoting b y f = f ( x , t) the volume forces, by u = u(x, t) the velocity vector, and by p = p(x, t) the pressure, we have &u Ot + (u " V)u = vAu + lip + f , V'u=0,

in t2 x (0, oo),

inf2 x ( 0 , oo),

where v > 0 is the kinematic viscosity.

(2.1) (2.2)

C. Foias et al./Asymptotic analysis of the Navier-Stokes equations

159

In order to define a well-posed initial boundary value problem, we have to supplement eqs. (2) with an initial condition on u (which we do not write here) and a boundary condition on u. Two types of boundary conditions are of interest for us; one o f them is the non-slip condition on 012: u =0,

on 012.

(2,3)

The second interesting possibility arises when we assume that 12 is a square (0, L ) x (0, L ) and we look for space-periodic solutions, i.e. for u's satisfying the conditions

u(O, x2, t ) - - u ( L , x2, t ) ,

O
u(xl, O,t)=U(Xl, L , t ) ,

0
(2.4)

in which case we assume that the integrals of u and f on 12 vanish at all time. Note that we have assumed 12 to be a square solely for the sake of simplicity. All the arguments presented below can be easily modified for the case where f2 is a rectangle, say: (0, L) × (0, H). 2.2. Determining modes Let u be a solution of (2.1), (2.2), (2.3) (or (2.4)) and let

u(x, t) = qbl(t)wl(x) + c~2(t)w2(x) + " "

(2.5)

be a Fourier type expansion of u(x, t) with respect to an appropriate infinite (and complete) sequence of divergence free vector functions w~(x), w2(x). . . . . In the sequel, the modes

dpl(t)w,,. . ~, Ct~m(I)wm

(2.6)

will be called determining for this solution if they determine completely its behavior in the limit t ~ oo. More precisely let us assume that another function v satisfies the same equation,

OV+(v'V)v Ot

vAv+Vq+g,

in12 x ( O , ~ ) ,

(2.1)'

with perhaps a different force g, and (2.2), (2.3) (or (2.4)), that

v(x, t) = ~k,(t)w,(x) + ~2(t)wz(x) + " "

(2.5)'

and

f

lf(x, t ) - g ( x ,

t)12dx--,0,

as t ~ .

(2.7)

t2

Then the modes q~,(t)Wl. . . . . d~m(t)w,, will be called determining if the condition Iqq(t) - 4,~(t)12 + . . . + I~,.,(t) - 4~,.(t)12-,0,

as t ~ o o .

(2.8)

is necessary and sufficient for

f

[v(x,t)-u(x,t)12dx~O,

ast~oo.

(2.9)

f/

Clearly, (2.9) implies (2.8) and when the modes are determining (2.8) implies (2.9) too. Actually, as we shall

C. Foias et al./Asymptotic analysis of the Navier-Stokes equations

160

show later (see section 8), in many cases, (2.9) implies that sup I v ( x , t ) - u ( x , t ) l ~ O ,

ast~,

(2.10)

x~2

where ].[ is the Euclidean norm and that other norms of v(., t) - u(., t) converge to zero as well. Hence, in a less formal way, we can say that for t ~ ~ the asymptotic behavior of the exact solutions is totally determined by that of a set of determining modes. 2.3. Number of determining modes A first estimate of the number of determining modes was given in [1], under the assumption that 812 is regular (i.e. of class C2), that the boundary conditions are (2.3) and that the set wl, w2. . . . . is formed by the eigenfunctions of the Stokes operator, i.e.

-- Awj q- Vqj : J.jwj, wj= O ,

V " wj : O, i n f 2 ,

(2.11)

on&O,

where the 2/s are the eigenvalues corresponding to the w/s. In the case of the space-periodic boundary condition (2.4), the eigenvalues are of the form 4~zZ(h2/H 2 + 12/L2), h, l = 1, 2, 3 .... They are labelled with the single subscript m = 1, 2, 3 . . . . . corresponding to the pairs (h, l), and ordered in an increasing sequence of s i z e s 0 < 2 1 ~ 4 2 ~ < . . . . The estimate [1] is as follows (see also [9] or section 4 below): the modes (2.6) are determining if

2m+'>21imSUPv '~ f IV x u(t"x)12dx"

(2.12)

(2

An important application of inequality (2.12) is the estimate of the number of modes needed to represent adequately the Rayleigh-B6nard convection. As is well known, this is the convective motion induced in a plane-parallel layer of fluid heated from below. This motion is described in the Boussinesq approximation

x 2

~ T , : T o - AT

H

Y

/

/

/

/

L

2L

T= To Fig. 1.

x~

C. Foias et al./Asymptotic analysis of the Navier-Stokes equations

161

by eqs. (2) with f = g [ 1 + e ( T o - - T)],

(2.13)

where g = (0, 0, - g = gravitation constant), T (resp. To) is the temperature within the fluid (resp. at the bottom of the fluid) e is the compressibility, and OT 0---[ + v • V T = x A T .

(2.14)

The appropriate boundary conditions may be different from (2.3) but we conjecture that in this case the estimate (2.12) is at least approximately correct. Here of course we restrict our attention to the case of two-dimensional convection, i.e. the fluid motion is in the form of long rotating rolls. Then (following an idea of [8]), one can estimate, by a purely physical argument the right-hand side of inequality (2.12). The next section is devoted to this estimate. 2.4. A h e u r i s t i c e s t i m a t e Our immediate aim is to estimate the integral

fi

r x u[~dx.

To this end we consider the motion of a single roll of the fluid in a cell of length L and thickness H. A fluid element near the vertical edges of the cell moves essentially in the vertical direction upward on one side and downward on the other side, with at most a negligible velocity component in the horizontal direction. We argue from symmetry that the fluid achieves its maximum speed when it is half way between the top and the bottom of the cell. Thus for some ¢ ~ ( - L , L) we have (see fig. 1) t3u2

_ 2Vmax

Oxl x = ~,.n/2)

L

and we assume that this value is a good approximation of the average

IVxul 2 Therefore,

f lv x u[ 2 dx ~ 4 H L - ' v 2 m a x .

(2.15)

12

Next we estimate vmax. First we note that near the vertical edge the fluid element moving in the upward direction starts at the bottom with u2 = 0, ending with u2 = 0 at the top of the heated layer. A similar situation holds for the descending element at the opposite vertical cell edge. This implies that the acceleration is in the upward direction in the bottom half of the fluid layer and in the downward direction in the upper half. Therefore the absolute maximum speed which can be achived by a fluid particle is that

C. Foias et al./Asymptotic analysis of the Navier-Stokes equations

162

due to a uniform acceleration acting up (resp. down) to halfway up (resp. down) the slab for a particle moving upward (resp. downward). The acceleration in question is in fact that due to the buoyancy force. Thus H T, \1/2 Vmax~ 2 ~ - ¢ g ( o - T O ) ,

(2.161

where Tl is the temperature at the top of the cell. By virtue of (2.15) and (2.16), inequality (2.12) becomes H 2 Eg(ro -- T,)

2,,+1 > 8 L

v2

(2.17)

Introducing the Grashof number [3] Gr* = eg(To - T O H 3 v2 ,

(2.18)

inequality (2.17) can be written in the form )I'm + 1

---~-8 Gr* = coGr*

(2.19)

2---~-> HL21

where Co= 4 H / n L is a non-dimensional constant. Here and in the sequel c0, q, c2. . . . will denote dimensionless constants. They are sometimes difficult to evaluate precisely, but generally they are estimated to be of the order of unity. In two dimensions, with the ordering of the eigenvalues as in section 2.3, the number of eigenvalues up to the mth value is asymptotically proportional to m, i.e., 2m+1/2~ ~ m + 1. Proof of a similar asymptotic relation in the case of (2.3) is not trivial [26]. Thus inequality (2.19) has the striking consequence that for the flow considered above the number m of determining modes is at most of the same order of magnitude as the Grashof number Gr*. Although the preceding argument was related to a very special driving force f, namely (2.13), it can be easily reformulated for more general f o r c e s f For this purpose we observe that in the special case considered in sections 2.3 and 2.4, it is likely that

(f

;

[T(t,x)_ T012dx

1 lim sup ~-~

~ Tl_

To.

Q

This assumption is physically plausible and mathematically consistent with a maximum principle for eq. (2.14). This suggests that 1 li m sup v22,

t,

x)l 2 dx

~ e, Gr* ,

(2.20)

f2

where el as Co in (2.19) depends only on the aspect ratio. Accordingly we shall define a generalized Grashof number Gr for any driving force f (see also [10]) to be

Gr = ~

,

Y ~'1

lim sup t~

(f O

If(t, x)[ 2

(2.21)

163

C. Foias et al./Asymptotic analysis o f the Navier-Stokes equations

In the sequel our estimates will always refer to this generalized Grashof number, and therefore we shall omit the label "generalized". In the remaining part of the paper we show that slightly modified forms of the inequality

2,.+1 > co Gr ;q

(2.22)

yield upper estimates for the number of the determining modes, even for general driving forces.

3. Mathematical preliminaries

3.1. Functional setting of the Navier-Stokes equations Let f2 be an open bounded region in R 2, or in the square (0, L) 2. In the first case we set

= {u

2, v - u = 0},

(3.1)

where C~(f2) is the class of C ~ real functions with compact support in f2 while in the second case = {u = R2-values trigonometric polynomial with period L, V'u=0

and ~ u dx = 0 } .

(3.2)

e/

In both cases we set H = the closure of U in L2(g2) 2 , V = the closure of "U in Hi(f2) 2 ,

(3.3)

where L2(f2) = H°(f2) and H~(f2) (l = 0, 1, 2 , . . . ) denote the usual LZ-Sobolev spaces [11, 12]. Let us recall that Ht(f2) is the space of functions u, such that all the weak derivatives D'u up to the order 1~1_<_l belong to L2(12). The norm on HI(f2) is

lul ,,o,= E lal +'llD'ulll=,o, .*

I=l_-
and I[v [12= f [Vv :Vvl 2 dx

o

t/

=

dx =

IV x v(x)l 2dx

(3.4)

j,k= 1 ~2

~2

* Ifl] is the measure of f2. We modified the usual definition of the norm o f Ht(fl) by introducing the weight factors IDIL~I-tso that all the terms o f the sum have the same dimension.

C. Foias et al./Asyrnptotic analysis of the Navier-Stokes equations

164

be the norms in L2(I2) 2 (and thus also in H) and V, respectively. The scalar product in H associated with the norm Il will be denoted by (., .). Let P denote the orthogonal projection in L2(I2) 2 onto H and let us define the H-valued linear, resp. bilinear, operators

Au= -PAu,

resp. B(v,w)=P((v'V)w),

foru, v, w e ~ A =

V('IH2(O)

2.

(3.5)

Then the Navier-Stokes equations (2.1), supplemented in the case (3.1) with the boundary condition (2.3) and in the case (3.2) with the condition (2.4) is equivalent to the differential equation in H: du

dt +vAu+B(u'u)=f'

fort>0,

(3.6)

where from now on f is the projection of the initial f onto H. We assume that f as an H-valued function is (a measurable) bounded function, i.e. sup If(t) I < O
oo.

(3.7)

From now on, by a solution of the Navier-Stokes defined on [0, oo) satisfying (3.6) and such that

equations we shall mean an H-valued continuous function

T

f llu(t)l[2dt
for a l l 0 < T < o o .

(3.8)

0

It is known that for any u0 ~ H there exists a unique solution is continuous from (0, oo) to V,

u(t)

satisfying u(0) = u0. Moreover this solution

I1

f lAurdt
for a l l 0 < t 0 < t i < o o ,

(3.9)

to

and lim sup

]lu(t)ll <

oo

(3.10)

t~oc

([13]; see also [14, 15, 10]). In the proof of these facts, as well as in the sequel, a crucial role is played by some inequalities involving the operators A, B and the norms I'l, II"II"We shall mention here only the classical Ladyzhenskaya's inequality [16]

v), w) l

c21ul'
and the main inequality of [2] (see also [14])

,[C3]U[1/2"U "I/2 "I2 [III/,2'A1)'l/2 , IB(u, v)l < tc41ul,/2 )Aul,/2 iiv

(3.12)

where u, v, w ~ A . In case (3.1) the constant c2 = x/~ (see lemma 1, chap. I in [16]), while in case (3.2) c2 = (2 + w/2/n + 1/4n2)~/2 (see the appendix). Some other inequalities of this type will be introduced subsequently as needed.

c. Foias et al./Asymptotic analysis of the Navier-Stokes equations

165

Finally, we mention very useful orthogonality properties of B: for u, v ~ @A (B(u, v), v) = 0,

in both cases (3.1) and (3.2),

(B(u, u), Au) = 0,

in case (3.2) only.

(3.13) (3.14)

3.2. Eigenfunctions Let us recall that there exists an orthonormal basis w~, w2. . . . in H (which is orthogonal in V as well) formed by the eigenvectors of A, i.e. Aw,, = 2rowm

(3.15)

(m = 1,2 . . . . ). The eigenvalues are positive and we write them in increasing order (0 < ) 2, < 2 2 < . . -

(3.16)

Eq. (3.15) is the functional form of eqs. (2.11) in case (2.3), and of

{

-Awj+Vqj=2jwj, w(x + Le;) = w ( x ) ,

V-wj=0, in[2, V x R 2 , i = 1, 2,

(3.17)

in case (2.4) where (el, e2) is the canonical basis of R 2. Therefore, if P,, denotes the orthogonal projection of H onto the linear space spanned by w~, w2. . . . , Win, the fact that the modes (2.6) are determining can be restated in the following way: the set of modes (2.6) of the solution u(.) of the Navier-Stokes equations, i.e. (u(t), w,)w,, (u(t), w2)w2. . . . .

(u(t), w,,)w,,,

is determining if for any other solution of the Navier-Stokes equations dv dt + vAv + B(v' v) = g '

fort>0,

(3.18)

where g enjoys the same properties as f we have lim Iv(t) - u(t)l = 0,

(3.19)

t~oO

whenever lim

Ig(l)

- f ( t ) I --- 0

(3.20)

and lira IP,,v(t) - P,,u(t)l = 0. t~oo

(Compare (2.7), (2.8) and (2.9) with (3.20), (3.21) and (3.19).)

(3.21)

C. Foias et a l . / A s y m p t o t i c analysis o f the N a v i e r - S t o k e s equations

166

4. Mathematical lemmas and first estimates

4.1.

A lemma

We start with a useful elementary fact. Namely, let c~ be a locally integrable real valued function on (0, oo), satisfying for some 0 < T < oo the following conditions: t+T

lim,~ooinf f c~ dr = ?, > 0,

(4.1)

t

t+T

lim,~.sup f e - d~ = F < oo,

(4.2)

l

where ~ - = max{ - e, 0}. Moreover let B be a measurable real valued function defined on (0, oo) such that fl(t)~0,

Lemma d

as t--*~ .

(4.3)

4.1. Let ~ be an absolutely continuous non-negative functon on (0, oo) such that ~+~
(4.4)

a.e. o n ( 0 , ~ ) .

Then, under the assumptions (4.1)-(4.3), ¢(t)~O,

Proof

(4.5)

ast~.

Using the Gronwall lemma, we infer from (4.4) that for 0 <

0 < ~ ( t ) < ~= ( t 0 ) e x p ( - =

•(a)da

+

to

<~(to) e x p ( -

ada

"r)exp-

•(a)da

to

dz

r

+(,oZ~_-<,sup ~

to

to < t:

exp to

ada

(4.6)

d~.

r

By virtue of the conditions (4.1), (4.2), (4.3), we can choose to > 0 such that for all s > to we have s+T

s+T

f ~-da < F + l and f ~ da >7/2. s

s

Therefore, if to < r < t and the integer k is chosen such that r + exp(-f~da)=exp(-f r

otda)
~da).exp(r

kT <_t <_r + (k +

z+kT

1)T, we have

~-dtr) r+kt

r+(k+l)T

=
f r+kT

°t-d~7)
,

C. Foias et al./Asymptotic analysis of the Navier-Stokes equations where r ' =

167

e r+t+~/z. Thus from (4.6) it follows that

o _<_¢(t) =<~(to)r' e - (~/~'~('-'o>+ ,o_SUp_,l/~(~)Ii t ' e-~'/2'~('-" d~ to

2F'T < ~(to)r'e -(~/2rg'-'°) + - sup [fl(z)[,

y to6~
2F'T lim supl~(t)l < sup Ifl(T)[, t~oo )' to<~
qm = (1 -- ?m)(V -- u) . The following proposition will play a basic role in the sequel:

Lemma 4.2. If for some m = 1, 2 , . . . and some 0 < T < oo we have t+T

~(B(qm'u), q,) dz = 8 > 0

limt_.oo inflT f v Ilqm[Iz +[q,i

(4.8)

t

and

limlem(v(t)

-

-

u(t)) I = O,

(4.9)

then we have also limlv (t) -

u(t) I---,o .

(4.1o)

Proof. We subtract (3.18) from (3.6) and take the scalar product in H of the resulting equality with q = qm. Then from (3.12), (3.13), (3.15) and (3.16) we infer that ld 2 dt [q12+ v IIq II2 + (B(q, u), q) = (g - f , q ) - (BfPm(v - u), u), q) - (B(v, Pm(v

-

-

U), q)

<= [g -- f[ [q[ + c4[Pm(v -- u)[~/2[APm(v - u)[m[lu [[ [q[ + c3[VI~/2Hv [['alIP,~(v - u)l[I/2[Ap,~(v _ u)ll/21q I

< [g --f[ [q[ + c42~2[pm(v -- u)[ [[u [[ [q[+ c3[v[l/E[[v [[1/22~4[P,,(v -- u)[ [q[ --<--[g--fl26 "~ c2~ra[]u ]12..~c2,~ 3m/2[v[[[z)I1 IPra(V __ u)[2 + 43~lql 2 •

(4.11)

C. Foias et al./Asymptotic analysis o f the Navier-Stokes equations

168

We set

¢ =[q[ 2, vllql[2 +(B(q,u),q) Iq[2

~=2

3 2 6,

3 -- ;2 ]g - f [ 2 + g2 (c~ + c])2m(llu tlz +

2~zlvl IIv II)lPm(v -u)12.

Obviously, (4.11) is then the same as (4.4) and by virtue of (4.8) ~ satisfies the condition (4.1), while fl satisfies the condition (4.3) by virtue of (3.20), (4.9) and the condition (3.10) for u and v. Moreover, because of (3.11),

~ > 2 v llq ll2 - c2lql2 llq ll llU

236 = 2v{llq lIV\lq-~)- 2c21]u l[ I~q¢135 ~ >-- c211u

3

and hence

2v

=

Therefore, with (3.10), we conclude that e satisfies the condition (4.2). Then we can apply lemma 4.1 to = [q[2 and we obtain

Iq(t)[2--O,

ast--~.

(4.12)

We conclude the proof by observing that limlv(t) --

u(t)p= limlPm(v(t)- u(t))12 + limlq(t)l~

and this vanishes because of (4.9) and (4.12). We shall give now some simple consequences of lemma 4.1 which are of intrinsic interest. (See also corollary 5.1.)

Corollary

4.1. If for some T > 0 we have t+T

C~limsu p 1 .~-5

t~

"T

f 1Oul/=d~<,t~+, t

then the m modes (2.6) are determining.

(4.13)

C. Foiaset al./Asymptoticanalysisof the Navier-Stokesequations Proof. Indeed,

169

by (3.11),

t+T

t+T

1 f v llqm]12+(B(q~,u),qm) iq [ 2 dt

>--~v f

t

t+T

[Iq,~]12d~_ l f [q,~[2 c2 -~

t

t+T

t

t+T

t+T "~112i 1

=>VT f ~

dz - (C~T f ]'u ]12dz ) ~T f Ilqm]12 2 dz J '

l

1

t

so that if I+T

,/2 _

lim sup

IIu II2 d~

> 0

t

(i.e. if (4.13) holds) then t+T

lim infl~.o~T

f

HqmH2-i-(B(qm'U)'qr~)[q,.[2 \T

dz >_-~/ lim infv(lt~oo

t

t+T

f -~][qml[2dz~l/2j> rl vr~ rn + I >

Ij.

t

We can now conclude the proof by referring to lemma 4.2.

Remark 4.1.

For the boundary condition (2.3), we have Cz = V/2 and the condition (4.13) in corollary 4.1 is stronger than (2.12), since it involves a time average of Ilu II2 rather than supllul] 2. Later we shall also need the following

Corollary 4.2.

If for some T > 0 we have t+T

21imsupl

v t~

T

f

sup

d~<2,~+1,

~OuJ

j,k = 1,2

v.. x

(4.14)

L~(19)

t

then the modes (2.6) are determining.

Proof. The

argument, based on the obvious inequality

[(g(qm, u), qm)[ <~--- 2 j,ksup

= 1,2

duj ~X k

[qml2

(4.15)

L~(~)

(which replaces (3.11)., is otherwise similar to that of corollary 4.1. 5. An almost rigorous proof of the heuristic estimate 5.1.

A first (rigorous) estimate

Here, based on corollary 4.1, we express in terms of the Grashof number.a simple rigorous upper estimate of the number of determining modes of all the solutions of eq. (3.6) corresponding to an arbitrary

170

C. Foias et al./.4symptotic analysis o f the Navier-Stokes equations

initial condition and a fixed driving force. We recall that the Grashof number was defined in (2.21) and is finite because of assumption (3.7). Now we have the following

Corollary 5.1. If '~rn+ l

2--~-> c~ Gr 2

(5.1)

then the number of determining modes of type (2.6) is, for any solution of (3.6), not larger than m.

Remark 5.1. We will write (see (2.21) and (3.4)) F = lim sup If(t)l,

so that Gr =

t~c¢

F

(5.2)

V2 T ~'---"

and recall that c2 = x/~ in case (2.3).

Proof of the corollary. Let us assume that (5.1) is valid and let u be any solution of (3.6). Then l dlul 2

d----i-+vllull'=(f,u) lfllul,

(5.3)

,>0,

from which we obtain immediately d

dt lul + v211ul < l f l '

t >0,

and hence (by (5.2)) lim, ~sup

[u(/)[ <---1 =v21 F

(5.4)

=vGr.

On the other hand from (5.3) we deduce also t+T

i'lu(t

t+T

= + Z)}Z+v t I}ull=d~<½[u(t)[2+ l

f lfl lul t

for all 0 < t, T < m. It follows easily that t+T

lim sup 1 I~CO

T

f

V

IIu II2 dz -< ~-~ Gr 2 + F Gr

t

and therefore t+T

c2,,.nmt~oosuP T1 f "-72

< c~ Gr2 + e l 2 ' Gr2' Ilull2dT =2-~v

I

for any 0 < T < oo. By virtue of (5.1) we can choose T so large that c~2 G r 2 + c 2 2 1 G r 2 < 2 m + l .

2Tv

(5.5)

C. Foias et al.lAsymptotic analysis of the Navier-Stokes equations

171

For this T, (5.5) and the above inequality imply (4.13) and therefore, by corollary 4.1, the modes (2.6) are determining for u; since their number is < m and the solution u was chosen arbitrarily, corollary 5.1 is proved. 5.2. A priori estimates It is appropriate to introduce here two estimates which will be useful in the sequel. The first one is a basis fact concerning the operator A, namely that

IlUI!H~
(5.6)

The proof of relation (5.6) for the case (2.3) is very difficult and may be found in [15, 17, 18, 19]; for the case (2.4) it is very simple (see for instance [13]). The second estimate is also easy: Lemma 5.1. Let u be a solution of (3.6), in the case (2.4). Then t+T

IAul ~ dr <

lim sup lim sup

(5.7)

t

t+T

limsup(limsup1 f

)<_F.

(5.8)

I

Proof. By (3.14) we obtain from (3.6)

1 rill. II~ + ~laul ~ =

2

(f, Au)

(t > O)

(5.9)

dt

from which we infer that, for all 0 < t < T < oo, t+T

t+T

=

t+T

2

t

=

t+T

2

t

' t

t

so that t+T

71 f IAuI~ dr <± =.r t

t+T

f li(')i d" t

We obtain (5.7) by first letting t--+ oo, using (3.10) and the definition of F and then, letting T--+ oo. Obviously, (5.8) is a direct consequence of (5.7). 5.3. The heuristic argument

Comparing (5.1) with (2.22) we see that the rigorous estimate provided by (5.1) is, for large generalized Grashof numbers, much weaker than the heuristic one given by (2.22) (see (2.19) and the sections 2.2-2.4). In this section we shall prove that this heuristic estimate is "almost true". The meaning of this statement is the following:

C. Foias et al./Asymptotie analysis of the Navier-Stokes equations

172

The classical Sobolev inequalities (see [20, 21]) yield r - s u - IIu

II

,o,2

(5.10)

for all 1 < p < oo, but the constant Fp < oo is not dimensionless; in fact its dimension is

Irol = (length) 2/p. Obviously, in the limiting case p = oo, the constant F® would be dimensionless, but unhappily (5.10) is not valid anymore in that limit [20, 21]. However, from a different point of view (see in particular section 7), the relation (5.10) is in some sense "almost true" for p = oo. Hence for the remainder of this section we make use o f the incorrect statement that

sup IlvjllL ,o,

sup

j = 1,2

-c6
(5.11)

and then in the next two sections, we will present two approaches to mitigate the consequences of the fact that strictly speaking c6 = + o0. Assuming (5.11), we have

sup

OUj

j,k = 1,2 ~X k LC~(12)

= sup sup

k = 1 , 2 j = 1,2

OUj L~(Q) <

~X k

sup c6

= k = 1,2

On ~X k

< crllu 11,2(o,=
=

-~-

(5.12) '

by (5.6). Introducing (5.12) in (4.14) we infer from corollary 4.2 that if, for some 0 < T < oo, t+T

(5.13)

- csc6 lim s u p ~ t

then the modes (2.6) are determining for u. But by (5.8) if T is chosen large enough, (5.13) follows from

'~m+ 1 > ~ C5C6F , V" or ~m+l Al

> N//2

c5e6 G r .

(5.14)

Thus we have proved t h a t / f the relation (5.14) (which formally coincides with (2.22), but for the reservation that e6 < ~ is only "'almost true")is valid then the number of determining modes of type (2.6)for any solution of (3.6) is < m. The fact that the heuristic estimate established in section 2 is "almost true" (within the above meaning) suggested to us the existence of rigorous upper estimates based on very slightly modified forms of the inequality (5.14). These will be treated in the next two sections.

c. Foias et al,/Asymptotic analysis of the Navier-Stokes equations

173

6. A rigorous shape independent estimate 6.1. Preliminaries In this section we shall need the following inequality of type (3.11), (3.12): IB(u, ~ )1 _-< e~lu { tl~ It"~{A ':~ I"~ ,

u ~ ~,

~ ~ ~ ~,:~,

(6.1)

where A 3/2 can be readily defined as

/l~/2v = ~ ,~2(v, w.)w., m~l

and v ~ ~A3;2~

~

~ ~ml(V,w.)l 2 < ~

m=l

Therefore we shall assume from now on that in the case (2.3) the boundary OI2 o f I2 is o f classe C 3. This condition is sufficient for the validity of the inequality

Ilv IIH3(a,2<<-csll hv Ilrv,a,2 ,

(6.2)

for all v e~A such that Av 6Hi(Q) 2 (see [15, 17, 18, 19] for the case (2.3), while for the case (2.4) this is an elementary result [13]). Therefore we have also I]v 11r~3(~)2-< cscgllAv l] = CsCglA 3/2vl,

(6.3)

for all v ~ ~3:2.

Proof (of (6.1)). As already noted, we have

for all u, v, w ~ NA. Therefore,

Ovj

[B(u, v)l < 2 j , ksup {u[ , = 1,2 ~ X k Lco(f~)

(6.4)

for all u, v e ~a- But a classical inequality of Agmon [22] (see also the appendix) yields lift liL°°(a)< e~0[l~b I{[/~a)llq~ ll~(a),

V~b eH2(g2),

(6.5)

and therefore (6.4) becomes

{a(u, v)] <=

2c,0 sup

ob '/:

~vj '/:

j,k = 1,2 II~Xkll L2(fl)lI 0XklIH2(.Q) lul

< 2Cl01lv II"~Jlo ll~,o,21ul •

(6.6)

Inequality (6.1) is now obtained from (6.3) and (6.6) by setting c7 = 2Clo(C:9) '/2. The main mathematical result of this paragraph is the following lemma, in which the assumptions and the notations are those of section 2.4.

C. Foias et al./Asymptotic analysis of the Navier-Stokes equations

174

Lemma

6.1. I f for some m = 1, 2 , . . .

and 0 < T < oo,

t+T

lim supf- 1 f

,,o~ \ T

IAul2&)'/2<[c~ + c~(l + eV'~m +t + e/2 log 2,,/21] 1/2

(6.7)

t

(e = base o f natural logarithm) and

lim]Pm(v(t) -

(6.8)

u(t)l = O,

then lim]v(t) l~oo

Proof.

u(t)] = O.

(6.9)

F o r 0 = e 2 we define the set 1 = kt < kl_ 1 < ' • " < kl < k0 = m by the condition

2kj < 0 -JAm < 2kj+ ~,

(6.10)

if 0 -J2,~ > 21.

It is clear that (6.10) and the condition k+

I(B(q, u), q)l = (B(q,

(t

1 uniquely define the set {kj}J=0. F o r q = q~ we have

'-'

[

y=0

I

- Pro)U), q) + ~ (B(q, (Pkj -- Pkj +,)U), q) + (B(q, PIu), q) +--I

< ](a(q, (T - P~)u), q)[ + ~ [(B(q, (Pkj - ekj +,)u), q)[ + [(B(q, P,u), q)l, j=0

whence, by (3.11) and (6.1),

[(n(q, u), q)[ < c~lq [ I[q [[ I[(I -

P~)u11

l-1

+ E cl}ql:ll(P+-

-

Pkj +,)Ul 1/2 -I-cl}ql2HPlul[l/21A3/2Plull/2.

(6.11)

j=O

But II(P+ - P k j + ,)u }11/2]h 3[2(p+ _ Pkj+ 1)[/[ 1/2 <~ (JL+/~kj + I + 1)1/4]A (P+ -- rkj + 1)1~1

and

IIPIU 111/21h 3/2p1ul I]2 = IaP,ul , so that (6.10), (6.11) and (6.12) yield I-1

I(B(q,u),q)] < c21q[ ]lqll I1(1 - Pm)ull+ £

j=0

cllq]20]/alA(ekj

--

Pkj+l )u] "b c, IqHAP1ul

< c2]q] Ilqll I](1 - em)u ]1+ c7(1 + l°"2)a'~lql2 u=0 F'~l IA(P+ - p~j÷ ')u]2 ÷ IAelul2l 1/~'

(6.12)

C.Foiaset al./Asymptoticanalysisof the Navier-Stokes equations

175

that is

(6.13)

I(B(q, u), q)[ <= c,lq [ IIq II II(I -- P,~)u II+ ct(l+ to~/2)~/2lq[~[AP.ul ,

where (by virtue of (6.10)) 1 /~m ..[_ l-<~log21 1 and

(6.14)

01/2 --

-e.

Using (6.13) and (6.14) we obtain t+T

liminf I f vllqll2+(B(q'u)'q)dz -

Iql ~

t

t+T

>=litminfl ; Iv ]lq]!------~2c2HqI_~[[(I-Pr,)U 11- c,(1+ ,O'/2)"zlAPmU[]dz t

t+T

> lim inf 1 t~oD

J"

f EV-~ I[q[12-c2 []q[llhlq[ (1-'~1/2+ "1Pm)u[ Cl(l+]Ol/2)l/2l'4Pmu] ]d~ t t+T

> liminf~vf 1

t+T

111~]!2dz) t

( f,q,2

Ilqll2dz~ '/2c2

t

t+T

t+T

t

t

f ~dz)-[(-~ f~dz)c~,

t+T

>liminf)'v( 1

1

t+T

+c2(1+101/2)]'/2

t

t+T

I

Therefore if (6.7) holds we conclude the above computation with t+T

limt~® inflT f v IIqII2 +lql 2(B(q'u), q) d~ I

t+T

=> (c~ + c~(1 + e + e/2 10g 2./2,)) '/2 limt~®sup:\-~l ]Aul 2 dz t

t+T

q2 2 × lim inf~(lt.oo LkT f~zd~)~c-~--2+c~(l+lO'/2)] t

~-~V~m÷

I --

[ (

t+T

( ;

. C2"~-C2 l + e + ~ l eo g 2m 1/2 hmsup -~1

t

]Aul 2dz

:]

>0

C. Foias et al./Asymptotic analysis of the Navier-Stokes equations

176

where we used several times (6.14) and the fact that t+T

T

--d'c

> 2,,+1 •

I

Note that the chosen value of 0 = e 2 is best in the following sense: for ln(2~/20 >> 1, this value minimizes [1 + ln(Rm/20/ln 0]01/2, which is the least upper bound on 10'/2 Concluding, we remark that all the assumptions of lemma 4.2 are satisfied and therefore (6.9) follows directly from lemma 4.2 (compare (6.8) and (6.9) to (4.9) and (4.10)). 6.2. The main result In the sequel we set c~, = max{c 2 + (1 + e)c~, ~c1 1 2 e}.

(6.15)

With this notation we can now state one of the main rigorous result of our paper:

Theorem 6.1. If /~m+ ,/'~, > Cll G r , (1 + log 2,,/2,) m

(6.16)

then, at least in the case (2.4) i.e. periodicity in the space variables, the number of determining modes of type (2.6) is, for any solution of (3.6), not larger than m.

Proof Indeed from lemma 5.1 (see (5.7); see also (5.2) and (6.15)), we deduce that if (6.16) is satisfied then for any solution u of (3.6), the condition (6.7) is also satisfied. Thus the conclusion follows directly from lemma 6.1. Remark. Obviously, (6.16) is satisfied if /~m+l >

21 = c,2

Gr(log(c, Gr)) m

(6.17)

(where c12 = e c11) whenever c,1 Gr > e; therefore, unlike (5.1), (6.17) constitutes a very slight modification of the heuristic estimate (2.22).

7. A rigorous shape-dependent estimate

7.1. A priori estimates In the sequel we shall need the following inequality which is a variant of inequality (2) in Brezis-Gallouet

[23]: sup Ouj


j,k=l,2 OX k LOO(O)

where R > 1 is an arbitrary constant.

(7.1)

C. Foias et aL/Asymptotic analysis of the Navier-Stokes equations

177

We show in the appendix that this inequality holds in the case (2.4) with c13= v/2]Tr and cl4 = l/rL Also we shall need the following:

Lemma

7.1. Let u be a solution of (3.6) in the case (2.4). Then t+T

limsup(limsupl r.oo \ ,.oo -T f [A3/2ui2dt )

18c~ 4 + 2~limsuplifl[ 2,
(7.2)

t t+T

limsup(limsupl

f

iAmu ]d')
A1

+ ~2 _lira sup ][fl12) '/2"

(7.3)

t~oo

Is

t

Proof

These a priori estimates are derived in several steps, using in particular (5.7) and (5.8) in lemma 5.1. The majorization (7.3) is a simple consequence of (7.2) so that, actually, we only need to establish (7.2). i) From the scalar product of (3.6) with A2u, w e deduce

l dlAul 2

2

de

(7.4)

+vlA3/2ul2=--(B(u'u)'A2u)+(f'A2u)"

In the case (2.4), (3.13) to get

Au = - Au. Thus we integrate by parts

the term involving B and we use (3.11) and

g u, ~ u ), Au )] I('(u, u ), A2u )l = (B(Au' u ), Au ) + 2 j~, ( B (-~xs

,+

l uI 2

I g

--u

3c21A3/2ul IAul II~ II < v iA3/2ul2+ _9c~ _ iiu ii2 iAui2.

i ,

12x~1/2/ 2

~

gu

2N, 1[2

--

--4

(7.5)

v

Similarly,

(f,

A2u)

= (A'/2S, A3:2u) < IIs II IA~:2ul <=4

IA3/2"12+ ~ IIf II2 ,

so that from (7.4) we obtain, for t > 0 d d-7

I~ul 2 + ~la"2u12 = 18c~ ~ Ilull~lAul2+ !

IITV-

(7.6)

C. Foias et al.lAsymptotic analysis of the Navier-Stokes equations

178

ii) It is easy to deduce from (5.9) the following: d dt llu ]12+ 2viAul2 < 2]fl IA"I ___
Ifl2'

dtJJull2+v21jlult:<

1

+-v I f i z '

t>o,

and, by integration from to to t, 0 < to < t,

'i

Ilu(t)tl 2 < e x p ( - v2,(t - to))llu(to)ll 2 + -"v

If(s)l 2 e x p ( - v21(t - s)) ds.

to

For 6 > 0 arbitrarily small, we choose to > 0 so large that

If(s)l
for s => t0,

and thus, for t > to,

llu(t)ll2 < e x p ( - v 2 , ( t -

to))[lU(to)l[ z q

(F + 6)2 v22,

Letting t ~ or, we find lim sup Iiu(t)(l: < (F + 6) 2 t~oo

~

I,' 2,~1

and since 6 > 0 is arbitrarily small,

lim supHu(t)ll2 < t~ov

~

F2

.

(7.7)

~'2~ 1

iii) We infer from (7.6) that for 0 < t < s, and T > 0, t+T

]Au(t + T)p <=IAu(s)p 2 + [ o(<,) e x p ( - v2,(t + V - a)) d a , I

where e denotes the right-hand side of (7.6),

0 -- ~8 c~llu II2 IA.I ~ +v

2 v

II/112.

Then by integration with respect to s from t to t + T, we get t+T

t+T

t

t

IA.(t + T)p ___< which implies with (5.7) and (7.7) that lira suplAu(t)l z < oo. t~oo

c. Foias et al./Asymptotic analysis of the Navier-Stokes equations

179

Using this fact, and (7.6) once again, we obtain that t+T

IA3/2u]Zdz) -<-

l i r~oo m s u pk( l i ,~® m s u pTV f

18c~F2 V

V2,~l

t

t+T

lim sup( im s u p -

IAul 2 dz

lira sup Ilfll 2 .

+

t

Hence (7.2) follows, because of (5.7). Finally, as mentioned before (7.3) is a simple consequence of (7.2). The proof is now complete. 7.2. The main result Now we define the shape factor Sh of the driving force f in (3.6) as lim sup''~ I[f 112 lim s u p ' ~ Ilfll2 Sh = 2, lim sup,,~lf} 2 = 2,F 2

(7.8)

Evidently, the shape factor is a measure of the importance of the higher order modes in the driving force,f. With this notation we have, by (7.1), (5.8) and (7.8) the following basic inequality for any solution u of (3.6) (case (2.4)) t+T

limsup(limsupl T~oo

\

t~oo

T

f

sup ~

j,k = 1,:

L®(a)dz

)

t

1/2F x/~ <= -~- f 2 ( l + l o g R ) u v + ~ R

-- / 9c2 -4 1 2 ~1/: - 1) ~v-~21/'" + ~ 21F Sh)- ,

(7.9)

for all R > 1, and ez = (1/4rr 2 + x/~/n + 2) la (see the appendix). We state now one of the other main rigorous results of our paper, Theorem 7.1. If 2m+l

> 2 _ ~ Gr{1 + [1 + log(1 + (9c~ Gr z + Sh)1/2)]'/2}

(7.10)

then, at least in the case (2.4), the number of determining modes of type (2.6) is, for any solution of (3.6), not larger than m. Proof. Once we have shown that the condition (4.14) is satisfied, the theorem follows directly from corollary

4.2. But because of (7.9) this condition is satisfied if, for some R > 1 2,.+, 2 fx/~ 'z F ~ 2--T-> v-~l [-~ - ( 1 + l ° g R ) / v "t re(

F(9e22F 2

),/z}

1)v k v-~212+Sh

or

2°+' >

Gr (1 +logR)'

2+ 1

(9c2 Gr 2 + Sh)~/2}.

(7.11)

C. FoRts et al./Asymptotic analysis o f the Navier-Stokes equations

180

Now conclude the proof by letting R = 1 + (9e~ Gr 2 + Sh) In .

Remark 7.1. Although the inequality (7.10) involves the shape factor Sh, the contribution of this factor in (7.10) is not significant since it appears in the argument of (log(.)) 1/2. Also in comparison with (6.16) and (6.17), the relation (7.10) has the advantage that the constants involved are explicit numbers. Further we suspect that the constant e12 occurring in (6.17) is larger than the constant 2x/~/~ occurring in (7.10), so that for a moderate shape factor Sh, (7.10) will yield a lower estimate for the number of the determining modes. Finally it is also obvious that (7.10), like (6.16) or (6.17), is a very slight modification of the heuristic estimate (2.19) provided that the generalized Grashof number may be identified with the usual Grashof number.

8. Improved asymptotic convergence 8.1.

Shape-dependent improved asymptotic" convergence

In section (2.1) it was mentioned that (2.9) often implies (2.10). The obvious significance of that is that when (2..10) holds for t ~ @ , not only are the two functions, u and v, equal to one another in the mean square sense, but, more strongly, asymptotically in time their actual values are equal to one another. When this is so the determining modes are capable of describing the asymptotic behavior of the given flow within the limits of the spatial resolution imposed by the cutoff at the highest retained mode. In this subsection we discuss the conditions under which (2.10) holds; in the next subsection we present a somewhat weaker result which obtains when those conditions are not met. Formally, we ask the following question: if the forces f and g satisfy

]f(t)--g(t)[~O,

for t ~ ,

and if some solutions respectively satisfy

u(t) and v(t) of the Navier-Stokes equations, with the driving force f and g

I.(x,t)-v(x,t)12dx

[.(t)-v(t)l=

(8.1)

-.0,

for t ~ ,

(8.2)

t2

then can the convergence be improved? An affirmative answer is easily obtained for the case (2.4) when the shape factors (7.8) o f f and g are finite, i.e. when lim supllf(t)] I , lim sup IIg(t)ll < ~ .

(8.3)

Indeed by virtue of (8.3) lim suplAu(t)l < oo

(8.4)

C. Foias et al./Asymptotic analysis o f the Navier-Stokes equations

181

as well as

(8.5)

lim suplAv(t)l < t~

(cf. proof of lemma 7.1(ii)). Hence on using inequalities (6.5) and (5.6) we obtain

llu(t) -- v(t)llL~,a,2 < CsC,olU(t) -- v(t)ll/:lA(u(t ) -- v(t))l'/2 . Therefore, it follows from (8.2), (8.4) and (8.5) that as t - ~ suplu(x, t ) -

v(x, t) I = I l u ( t ) - v(t)ilLO¢(o)2---~O.

(8.6)

XEI]

Evidently, the convergence in (8.6) is a significant improvement over (8.2). 8.2. Shape-independent improved asymptotic convergence At present we do not know if the improved convergence (8.6) is valid when no assumptions are made concerning the shape factors o f f and g, nor have we been able to extend the proof of (8.6) to the more general case of boundary conditions (2.3). Nevertheless we note the following improved forms of (8.2): Hu(t)-v(t)ll~0,

for t ~ o o ,

(8.7)

and t+T

1 f i~u(~)_Av(~)(~d~_,0 for t--,ov,

(8.8)

t

where T > 0 is arbitrary. Note too that (8.8) leads to t+T

for t ~ oo,

(8.9)

t

a result similar to, but weaker than (8.6).

Proof of (8.7) and (8.8). For w = u - v and h = f -

g, we have

l drwl 2

2 dt +vllwll2<=lhllwl+l(B(w'u)'w)l 5 Ihl Iwl +

c=[lu LILIwII Iwl,

so that

dlwl =

dt + v[lw[12
and, consequently, for t ~ o c t+T

t+T

Ilwll~dz
since lim sup,~oo Ilu II < 0% while

21hI[wl+-~ ][uiI2(w[:. d~--,0, t

lim,.~lw[ =

0 and

lim,.o~lh[ =

0.

(8.10)

C. Foias et a l . / A s y m p t o t i c analys& o f the N a v i e r - S t o k e s equations

182

On the other hand we have

1 d[lw II~ + vlAwl2 < [h f [Awl + i(g(w, u), Aw) I + I(g(v, w), Aw) I

2

dt

1

v

2

< -)hi = + ~ IAw) + c, lwl"=ll u II I"wl 3j= + c3lv)'/=l)v 1t'/2tlw )l'JZl~wl 3j= 1; < - 1 [hi2+ 3 v tAwt =4 = v ~

c~l[u l['lwl= -k c~lvl=ll~ll=llwll 2 v~

v~/~

where ~ and fl are some positive numbers. Thus for t > T we infer dllw 1t2 dt

1 2 2 +~vlAwl <-IhP + ~,Jwl ~ + ~llw fl2 = v '

(8.11)

where 7J and 72 are some convenient positive constants depending on v, sup,~_rllull and sup,_~rNv JlEvidently, 6=-21hl=+TllW12~o ' v

fort~.

(8.12)

As a result we deduce from (8.11) that for 1 < to < t @

[lw(t)ll2 <

e~"-'°'llw(t°)ll2+ i er:U-~)6(z)dr. it/

to

Now integration with respect to

J[w(t)][2 <= eysrlT i

toe[t - T, t]

[[W(to)][2dt0 + e~sr-1T

t-T

yields for t > 2T

i6(z)dz.

(8.13)

t-T

Therefore it follows from (8.10) and (8.12) that IIw It~0 as t---,~, thus proving (8.7). On integrating (8.11) from t to t + T, with t > T, we obtain t+T

1 1

t+T

iAwl2dz
t+T

adz t

+7z~

f [Iwl?dz, t

whence by virtue of (8.10), (8.12) and (8.13) t+T

--T1 f IAwlS dz ~O ,

for t---,oo

(8.14)

t

Now (8.8) follows obviously from (8.14). 9. C o n c l u d i n g remarks

We have shown rigorously that the number of modes which determine the solutions of the twodimensional Navier-Stokes equations is no larger than a number slightly in excess of the generalized

C. Foias et aL/Asymptotic analysis of the Navier-Stokes equations

183

Grashof number. In fact the departure from strict proportionality to the Grashof number is of less than logarithmic order. An important consequence of this work is that a heuristic estimate of the determining modes for two-dimensional Brnard convection is very close to the precise bound. Because of the potential practical consequences of the results presented here we have to make some qualifying remarks. To begin with we reiterate an earlier observation, viz. that the bound obtained here is in the nature of a sufficient condition. Indeed the nature of the proofs used here is such that any improvement in the estimates of the various norms of interest will automatically decrease the bound derived here. Moreover it must be recalled here that the basis functions from which the determining modes are drawn are the eigenfunctions of the related Stokes problem. In practice the Galerkin approximation may be based on some other eigenfunctions which are only required to satisfy the imposed boundary conditions. In this case as has been demonstrated elsewhere [2], the sufficient bound will probably exceed the bound derived in the present work. Given that in practice the interesting computational cases are those in which the Grashof number is relatively large, as in turbulent flow, the preceding remarks show the imperative need for necessary and sufficient determining modes. It is to be hoped that their number will be significantly lower, thus helping to validate the current computational fluid dynamics efforts. That this hope is realistic is easily substantiated by the observation that the heuristic estimated itself is based on an exaggerated bound for the maximum fluid velocity in a Brnard cell. Another aspect not yet contained in the determining modes, but no doubt very important, is the role of the initial conditions. It is well known both on experimental and computational grounds that even in the nonturbulent regime of convective and rotational flows, the final states of the flows depend critically on the initial conditions. Indications are that different final states require different numbers of modes for their description. Finally, from the practical point of view it is important to extend all these and related results to three dimensions. This is of special concern in the context of computational simulation of turbulent flows which are intrinsically three-dimensional phenomena. That this may be feasible is suggested by the following heuristic considerations. Recalling corollary 4.2 we see that it has a simple physical interpretation: on comparing the inertial and dissipative terms in the Navier-Stokes equation we see that it yields the length scale at which the effects of dissipation begin to dominate the inertial effects, i.e. U "V/J ,-~ ~'VEv ,~, V,~,n+ 11)

or equivalent, speaking very loosely,

v;~.+1>~IVvlL~. Pursuing this thought furthcr, it is evident that IV /) [L ~ ~ Vmax.~n+ , ]1/2 1

since, with sufficient regularity, the spatial variation of the velocity should not exceed significantly the ratio of the largest velocity in the field of flow to the smallest length of interest. Thus it follows that 2.+1 > 2 2, ~__/)max/~ which to within numerical and geometrical factors is precisely the same estimate as that presented in section 2.3. Since these qualitative arguments are independent of the dimensionality of the given problem, there is a prospect that corollary 4.2 and its consequences may be extended to the case of three dimensions.

184

C. Foias et a l . / A s y m p t o t i c analysis o f the N a v i e r - S t o k e s equations

After completion of this paper we have learned about similar results obtained on the B6nard problem [24, 25]. However these results are based on a rather strong hypothesis regarding the initial temperature distribution and the asymptotic behavior of the solutions.

Acknowledgement This paper was supported in part by the U.S. Department of Energy under Contracts n ° DE.AC02.82ER12049 and n ° DE.AS03.78ET53072.

Appendix A Inequalities Our aim in this appendix is to determine the value of c2 in (3.11) in the case of periodic boundary condition (2.4), and to establish (6.5) and (7.1) in this case.

A. 1. The value of c2 in (3.11) Let v = v(t) be a periodic function of period L whose mean value over a period vanishes, i.e.

L f v(t) dt = O. 0

Then for t and s arbitrary,

t v(t) = v(s) + f v'(z) dz, s

and integration with respect to s over a period L yields

Lv(t) =

v'(Q dz ds = t--L s

(~ - (t - L))v'(O dr, t--L

so that

i

L

t-L

0

(A.1)

If u e~A, then u is continuous on N2 periodic with period L in xl and x2, and for every x2 we can apply (A. I) to the function

L V(Xl) =

lu(x,,

x2)l 2 - Zl

f [U(XI, X2)]2 dx 1 0

185

C. Foias et al./Asymptotic analysis o f the Navier-Stokes equations

which gives L

s~p{,,,(~l, x2)}2=
L

~u

f {u(x,,x2)}2dx,+2 f (U-~x,)(x,,xg{dx, 0

0

L

L

Similarly,

supltA(Xl' X2)} ~ Z

}/g(xD x2){2

dx2+ 2

0

u

(x,, x2) dx2.

0

Thus L

L

f{ui'dx<--(I~up{'{2dx2l(~=;sup{'{~dx'),a ~,

g2

/,a

0

0

<[~ f u2dx+2(f, 2dx"l';2f ~'OU'2dx)"2] / UI~I

=

O

Q

D

ljl
Since So u dx = O, for 2,

'f lu{2dx__<

L--5

O

=

4lg2/L 2, we

O

have

(A.2)

Vul 2dx

O

and

f {u{4dx<(f,ulEdx)'[(-~ = f{u,Edx)'/z-F2(f ~Ou2dx)\'/2-}l-1 J[-~ f ,u[2dx)'/2_F2(f I~.~x2[Odx)u2 ]'/2 ~1,,1~ 11,,11+2~

~1{,,11+{0x~{/

('+~- ~ + 2 ){~,I:I{,,{I ~,

s\~

f l,,{~dx~_( ' + ~ Q

2){,,{~{{,{{~

(A.3)

186

C. Foias et al./Asymptotic analysis of the Navier-Stokes equations

Now we estimate B (compare to lemma 3.4, ch. III of [13] for the case (2.3)):

I(B(u'v)' w)l =
: id= l

12

~'~'i dx) till IMil~4(O,) tjEl.=[wA~4(I')2)

< (by (a.3))

47~2+-+7r

<

2

v(I {ulii2(iullii21wlii21lwil

I,

which proves (3.11), i.e.

c2 =

~5~ + - -

7Z

•

+ 2

A.2. Proof of (6.5) Agmon's inequality may be found in [22] in the case of the boundary condition (2.3). We prove it here in the case (2.4). For u e~A, we write

lu(x)l<= E I< --<[k]E~ R la~l+[k]E> R %, where (for k = (k~, k2)), [k] = max@t[, [k2[), and R real > 1 is arbitrary. Hence 41t 2 L2 [u(x)l ~ ,~12=<.lakl + Ny'>. ---~kl:}ak}4"21k]2 <= (keg 2E

L-2)lx~1/2 -~-(k~2 16~4 1/2 I~ \lkl=R -~-}k}'}ak}2) (tk~,

\1/2/ L21akl2)

/

1

,12

I'~(x)(-<-- Z lu( t~ . l

L

+ ~

IAul

~k . ikl, )

Now we have to estimate the two sums

Z1 Ikl~ R

I

and ,~.lkt'

I

Z 4 1/2 16-~-k4)

(A.4)

C. Foias et al./Asymptotic analysis of the Navier-Stokes equations

187

Define the integer M by M < R < M + 1. Then

Z

1 = {number o f k ' s with 1 < [k] < R}

tel =
=Z

n u m b e r o f k ' s with [k] = m

M

= E 8m=4M(M+I) rn=l

<=4R(R + I). Similarly,

m = M + l [kl=m ~ - ~

<8

m=M+l

m3

ct~

<8fd~ 4

=

M 2

M

4

(R

=

--

I) 2.

Finally, for any R > 1,

< 4R(R + I)

~glu(x)l =

L

L

lul + £2(R _ ~)~IAul

< 201u I 8(R - 1) 2 lu I q L iAul = - L -I L ~2(R--- 1) 2 since u < 2 ~ -~Au =

<

< lo l.l,,~lAul,,=+

Au

8(R - 1)2

= •

lul +

L

L rr2(R - 1)2

IAu) "

The m i n i m u m o f the last expression with respect to R is attained for (R - 1)4= (L:/8n2)/([Aul/lul), and we find that

~l,,(x)l <

l0 + 4x/~

lull'21Aul''2.

(g.5)

A.3. Proof of(7.1) As in the case o f (6.5) we use the Fourier expansion o f u and obtain, for every R, an inequality similar to (A.4),

=

'

,,~,_~, i~)

+~

(

IAul ~>2 Ikl'/

188

C. Foias et al./Asymptotic analysis of the Navier-Stokes equations

Proceeding as above we write (M < R < M + 1) u

1 2

[kl~_R ~ ' ~

= <

EEp m = I [k]= m ¢8m

1

m=l m2 M <

8+8

fax-x- = 8 ( l + l o g M ) 1

< 8(1 + log R ) .

Hence,

plu(x)l =<

+ l o g m'/ Jlu tl q

(x )l =< ~X~(l + log

R)~/2IluII-~

L_

l)

JAil '

1

- l)''lAul

(A.6)

References [1] C. Foias and G. Prodi, Rend. Sem. Mat. Univ. Padova 39 (1967) 1. [2] C. Foias and R. Temam, Asymptotic Numerical Analysis for the Navier-Stokes Equations (I), MRC Technical Summary Report n° 2325, University of Wisconsin, 1981. [3] L. Landau and 1.M. Lifshitz, Fluid Dynamics (Addison-Wesley, New York, 1953). [4] D. Ruelle and F. Takens, Comm. Math. Phys. 20 (1971) 167. [5] E.N. Lorenz, J. Atm. Sci. 20 (1963) 130. [6] R.H.G. Helleman, ed., Non-linear Dynamics, Ann. N.Y.A.S., vol. 357 (1980). [7] V. Franceschini and C. Tebaldi, J. Stat. Phys. 25 (1981) 397. [8] O.P. Manley and Y.M. Trrve, Phys. Len. 83A (1981) 88. [9] C. Foias and R. Temam, J. Math. Pures Appl. 58 (1979) 339. [10] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, 2nd edition (North-Holland, Amsterdam, 1979). [11] J.L. Lions and E. Magenes, Nonhomogeneous boundary value problems and applications (Springer, Berlin, New York, 1972). [12] J. Ne~as, Les Mrthodes Directes en Th6orie des l~quations Elliptiques (Masson, Paris, 1967). [13] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis NSF/CBMS Regional Conference, series in Applied Mathematics (SIAM, Philadelphia, 1983). [14] J.L. Lions, Quelques M~thodes de R~solution des Problrmes aux Limites Non Lin~aires (Dunod, Paris, 1969). [15] I.I. Vorovich and V.I. Yudovich, Mat. Sbornik 53 (1961) 393. [16] O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Second English Edition, translated from the Russian by R.A. Silverman and J. Chu (Gordon and Breach, New York, 1969). [17] L. Cattabriga, Rend. Sem. Mat. Univ. Padova 31 (1961) 308. [18] S. Agmon, A. Douglis and L. Nirenberg, Comm. Pure Appl. Math. 17 (1964) 35. [19] V.A. Solonnikov, Dokl. Akad. SSRS 130 (1960) 988. [20] R.A. Adams, Sobolev Spaces (Academic Press, New York, 1975). [21] J.L. Lions, Probl~mes aux Limites dans les l~luations aux D~riv~es Partielles (Presses de l'Universit~ de Montrral, Montr6al, 1962). [22] S. Agmon, Lectures on Elliptic Boundary Value Problems (Elsevier, New York, 1965). [23] H. Br~zis and T. Gallouet, Nonlinear Schroedinger. Evolution Equations, Nonlinear Analysis Theory, Methods and Applications 4 (1980) 677. [24] E.M. Maschke and B. Saramito, Phys. Lett. 88A (1982) 156. [25] B. Saramito, Sur le comportement asymptotique de certaines solutions du probl~me bidimensionnel de la convection de B~nard, CEA-EURATOM (1981). [26] G. M~tivier, J. Math. Pures Appl. 57 (1978) 365.