Finite Parameter Approximative Structure of Actual Flows

Nonlinear Problems: Prerent and Future A.R. Bishop, D.K. Campbell, 8. Nicolaenko (eds.) 0 North-HollandPublishing Company, 1982

317

FINITE PARANETER APPROXIMATIVE STRUCTURE OF ACTUAL FLOWS C i p r i a n Foias and Roger Temam Analyse Numerique e t Fonctionnel l e CNRS e t U n i v e r s i t e Paris-Sud Bdtiment 425 91405 Orsay Cedex (France)

-

INTRODUCTION

Most approaches t o the mathematical understandinq o f the onset o f turbulence (e.q. 161, [4J , [lO] , [l]) assume t h a t the comnlexity o f the dynamical system {S(t)}t>o associated t o the i n i t i a l value problem f o r the Navier-Stokes equations i n some adequate f u n c t i o n a l space H (formed by divergence f r e e v e l o c i t i e s f i e l d s ) i s increasina i n an almost universal monotonic way, w i t h sudden jumps a t some threshold values of a p h y s i c a l l y s i g n i f i c a n t parameter. However t h i s universal p a t t e r n (although displayed by adequately constructed models) was n o t established f o r actual flows, for which even the d e f i n i t i o n o f {S(t))t.o i s n o t y e t s a t i s factory. The o n l y known f a c t , which i s r e l e v a n t t o t h i s problem, i s t h a t f o r Dlane flows ( w i t h s t a t i o n a r y boundary conditions and d r i v i n q forces) a l l i t s t r a j e c t o r i e s S(t)uo converqe, for t -+ m , t o a s e t X m a x c H o f f i n i t e Hausdorff dimen-

, f o r which a r a t h e r e x p l i c i t , u m e r estimate d ( l / v ) , as f u n c t i o n o f the kinematic v i s c o s i t y v , e x i s t s ([9], c3J5.6). Althouqh d ( l / v ) f m f o r sion dmax

.

l/vpm i t i s n o t y e t known whether dmax --t m f o r l / u --c On the o t h e r hand the change from a laminar t o a t u r b u l e n t f l o w o f a f l u i d i n motion i s o f t e n so f a s t t h a t i t i s hard t o conceive t h a t i t s mathematical understanding l i e s o n l y on the qsymptotic behaviour of u ( t ) f o r t j. m and thus on the s t r u c t u r e o f a ( s t i l l hypothetical, i f the flow i s n o t plane) a t t r a c t i n g s e t X c H A way by which nature could s a t i s f y a l l the guested approaches i s given i n the f o l l o w i n g

.

Conjecture. Any t r a j e c t o r y u ( t ) i s "very r a n i d e l y " converging ( i n a manifold M of f i n i t e dimension d such t h a t d t m f o r 1/v j.

-. H)

to

To prove t h a t t h i s conjecture i s " g e n e r i c a l l y " t r u e seems t o be outside the grasp o f the present mathematical techniques. Strangely enough, we can g i v e a rigourous p o s i t i v e answer t o an approximative form o f the conjecture. I n order t o s t a ' e t h i s answer l e t us r e c a l l t h a t H (see the next 9.1) i s a r e a l H i l b e r t

I

space and t h a t 1/2 o f the square U )* o f the norm o f u e H represents t h e ( A c t u a l l y we assume t h a t t h e density o f t h e k i n e t i c a l energy o f the f l o w u e H f l u i d i s = 1). This i s our Answer (see 9.3 below). manifold M of dimension i n i t i a l data (1)

l u ( 0 ) l 2 G Eo

.

For given

d<-

, the

O
and O
such t h a t f o r a l l t r a j e c t o r i e s set o f the

distance ( i n H) from u ( t ) t o M o f length = 1 i n t e r v a l c (0,m)

.

< E:12

t's

u(t)

with

f o r which i s o f measure >, 1-T on any

318

C. FOIAS. R. TEMAM

A c t u a l l y t h i s answer i s n o t unexpected since on any s e t f i e l d s such t h a t

VRc H

o f velocities

j R ( c u r l u (2 dx c R the map S ( t o )

(namely the deplacement

u(0)

~--t.u ( t o )

along the t r a j e c t o r y

u ( t ) ) i s a w e l l defined one t o one compact diffeomorphism ([2], we s h a l l show t h a t d (for

Eo, T

S(to)

, where

and v

Ch. 111). However

( l o g l/El)n

'L

f i x e d ) , which depends on a more p a r t i c u l a r s t r u c t u r e o f f o r plane flows and n = 3 otherwise.

n = 2

might represent the unavoidable experimental e r r o r i n the The q u a n t i t y El estimation o f the energy o f a f l o w o r even might represent a much more basic e n t i t y comparable w i t h the mean k i n e t i c a l energy o f a molecule o f t h e given f l u i d a t r e s t . I n t h i s case i t i s c l e a r t h a t d i f f e r e n t i a t i n g two flows u, v such t h a t l u - v I 2 < El

, although mathematical$meaningful i s v o i d o f any physical basis. But

because o f ( 2 ) even mical f u n c t i o n o f

t h i s microscopical

,T

Eo

.

and v

El

w i l l y i e l d a reasonable non astrono-

The c o n s t r u c t i o n o f M , given i n 5.3, a l s o provides some supplementary information on i t s p o s i t i o n i n H allowing us t o show ( i n 5.4) t h a t the behaviour o f d , as f u n c t i o n o f l / v , i s c o n s i s t e n t w i t h the conjecture. 5.1.

PRELIMINARIES (

1

We s h a l l consider t h e Navier-Stokes equations (1.1)

ut

+

( u . 0 ) ~ = v Au

+

Vp

+

f

,

V.u = 0

in R

x

(0,~)

where

R i s an open connected bounded s e t c IRn (n=2 o r 3) such t h a t i t s 9 boundary r i s a compact manifold o f class C' o f dimension n-1 , and R i s l o c a l l y located on one side o f an I n order t o s i m p l i f y the e x p o s i t i o n we s h a l l f i r s t consider o n l y the boundary problem

.

As usual, l e t

H

and V

(see L3J .5.2 f o r more d e t a i l s ) denote t h e closure

i n L2(Q)n and H l ( ~ 2 ),~respectively, o f

v = Iv (')

See [31.55.1-3

:

V €

c p ) " : v.v = 0)

for more d e t a i l s .

,

FINITE PARAMETER APPROXIMATIVE STRUCTURE OF ACTUAL FLOWS

319

and AU

(1.3)

where

=

-

PA^

3 u G ~ ( A d) s f ~n H 2 (0)

for

B(u,v) = P[(u.v)v]

Sobolev L -space o f order

.

the orthogonal p r o j e c t i o n o f { ~ ~ l of i = H~ such t h a t

a(A)

u,v E

2

~ = 1 , 2 ) denote the

H'(a)

for

R and P denotes

L2(n)n onto H There e x i s t s an orthonormal basis A wm = A, wm (n;=1,2 ), O
,...

( 1 -4) (where

c1

as the others

c.'s

J

(j=2,3, ...) denote p o s i t i v e

i n the sequel

constants depending o n l y on 9 ) . The orthogonal p r o j e c t i o n i n H onto Rwl t + IRwm w i l l be denoted by Pm (m=1,2 ; Po = 0) We s e t

...

.

,...

IuI w i l l be t h e norm on H , IIvI( t h a t on V and (u,,~,) and ((v1,v2)) w i l l denote the corresponding s c a l a r products. The operator B i s continuous from d)(A) x V and V x a ( A ) t o H as w e l l as from V x V i n t o the dual V ' o f V. For many more f i n e estimates on B we r e f e r t o [3],5.1. Also B enjoys the f o l l o w i n g basic o r t h o g o n a l i t y property B(u,v),v)

(1.5)

= 0

whenever the l e f t hand s i d e makes sense. (1.2) can be With the above notations the i n i t i a l value problem f o r (l.l), w r i t t e n i n H , namely

(1.6) where

uOcH

du t v Au t B(u,u) = P f

for

t>O

,

u(0) = uo

i s given. For the o n l y sake o f s i m p l i f y i n g the presentation i n the

sequkl we s h a l l assume t h a t

Pf = 0

, i.e.

t h a t the forces are p o t e n t i a l .

The f o l l o w i n g are w e l l known c l a s s i c a l r e s u l t s (see, f o r instance, [5], o r [ll]) :

(')

The problem (1.6) has a (weak) s o l u t i o n

u( .)

A c t u a l l y we s h a l l use o n l y the f a c t t h a t where c2 i s s u i t a b l y chosen, O
A,,

.

[6]

such t h a t

>, c2 m2'n

for a l l

m = 1,2,

...

C. FOIAS, R. TEMAM

320

A c t u a l l y , because o f our assumption on

I f uo= V

f

we have

then there e x i s t s an i n t e r v a l

[O,t(u,))

, where

on which a l l (weak) s o l u t i o n coincide w i t h a ( s t r o n g o r r e g u l a r ) s o l u t i o n , i . e .

For such a strong s o l u t i o n t h e d e f i n i t i o n S(t)uo = u ( t )

(1.11)

makes sense. Moreover i n case has

t(uo) =

interval

[tO,m)

Ogt
,or

i n case

n=3 b u t if IIuo1/4_c c4v4

, one

and any weak s o l u t i o n i s unique and obviously i s r e g u l a r on any (to>O)

.

Among the other r e g u l a r i t y p r o p e r t i e s o f S(t)uo l e t us mention o n l y t h e f a c t t h a t S(t)uo i s a a(A)-valued a n a l y t i c f u n c t i o n o f t E (O,t(uo)) such that

(see [3],5.3).

5.2.

THE SQUEEZING PROPERTY

I n the sequel, a basic r o l e w i l l be played by the f o l l o w i n g squeezing p r o p e r t y of S ( t ) , which we obtained i n [3] ,§.5 :

321

FINITE PARAMETER APPROXIMATIVE STRUCTURE OF ACTUAL FLOWS

whenever

where

0 < ClY C Z y C3 <

A c a r e f u l perusat the estimates

where the constants

m

do n o t --

c8 and c9

or

m

.

are n o t dimensionless. The estimates (2.4) are

.

n=3 ( A c t u a l l y f o r n=2 2 and 4 , r e s p e c t i v e l y ) .

n=3 i t can be shown t h a t

some dimensionless

t,u,v

o f the p r o o f (given i n [3],5.5) o f t h i s property y i e l d s

v a l i d f o r both cases n=2 and 5 i n ( 2 . 4 ) can be decreased t o I n case

depend on

constants

coly

cll

the exponents 5 / 2 and

c8 3 0cl Xi/* and c9 d c Introducing the variables

.

with

the r e l a t i o n (2.3) can be w r i t t e n under the form

(2.6)

5 K

'R; Kd)

5 c12

(with

c12 dimensionless)

which can be i n t e r p r e t e d as showing t h a t thefrequencies K f o r which the squeezing property o f S ( t ) holds are l y i n g f a r i n s i d e the d i s s i p a t i v e spectrum o f the flow. This s t r o n g l y suggests t h a t the exponents i n ( 2 . 4 ) must be much nearer t o 1/2 and 1, respectively. 5.3.

THE APPROXIMATIVE STATIONARY MANIFOLD

L e t 0 < El << 1 << Eo

(3)

ci =

2 exp(2cg

Ci").

0<~<<, 1 be such t h a t

A c t u a l l y we can drop these conditions, b u t t h i s

would bake the expression (3.2) much more complicated.

C. FOIAS, R. TEMAM

322

Then we have the f o l l o w i n g r e s u l t , l o s e l y Theorem

1 - There e x i s t m p r o p e r K

and a

--

J, :

s t a t e d i n the I n t r o d u c t i o n : PmH

IJ,(PmU)-$(PmV)l C IPmu-PmV I

(3.3)

and such t h a t for any weak -----{ t c (0,m)

(3.4)

satisfying

(U,VE H)

u(.)

solution

M c H -i s the graph o f J, J, interval c (0,m) o f length = 1 , of R

)

to

set M

i s o f measure >/ 1

and

.

the

,

of (1.6)the

: distance ( i n H) from u ( t )

(where

-

(I-P,)H

I+

J,

-

,

< ]":E T

on any

i s the dimension

n ( = 2 o_r 3)

Before passing t o the proof, l e t us n o t i c e t h a t MJ, i s obviously a L i p s c h i t z manifold o f dimension m , which can be c a l l e d an approximative s t a t i o n a r y manif o l d o f (1.6). --

Proof-of-Theorem-lL e t u(.)

Thus f o r

any

be a (weak) s o l u t i o n o f (1.6)

r>O ,

and consequently i f (3.5) then

Let

to = t o ( r )

and choose a subset

M(m) o f

S(to)Cuo=V:l~uol~c r l maximal under the property

such t h a t

luol

2

< Eo

. Then

FINITE PARAMETER APPROXIMATIVE STRUCTURE

where

m

OF ACTUAL FLOWS

323

satisfies

(3.8) Then, by v i r t u e o f the squeezing property (see 5.2), we have t h a t t h e distance ( i n H) o f u(s+to) = S(to)u(s) t o M(m) i s bounded by

whenever

Ilu(s)ll 4 r

so t h a t i f to < 1

. But f o r

- 9 , then

~ { S C (t,t+l) :

any

, we

tat,

have obviously

by (3.6

t30 :

for all

\Iu(s)II 4 r } ] 3 1 - T / Z

-

to

.

I f we assume now t h a t

then i t f o l l o w s t h a t the s e t o f

t's

i s Q E:/'

on any i n t e r v a l

M(m)

has measure 3 1

-

T

such t h a t the distance o f

i s the graph o f the f u n c t i o n $,

: Pm M(m)

M(m) By (3.7),

u(t)

.

to

(t,ttl) c ( O p ) -(I-P,)H defined by

which moreover s a t i s f i e s the c o n d i t i o n (3.11)

lJI0(Pmu)

- $o(pmv)l <

IPmU

-

P,Vl

(Pmu, Pmv€ P, M(m))

.

can be extended on the By v i r t u e o f the Kirszbaum extension theorem (see [8]) $ whole HP, t o a f u n c t i o n JI s a t i s f y i n g (3.3). Thus i t 6Lemains o n l y t o v e r i f y (3.2). To t h i s aim we f i r s t note t h a t due t o the r e l a t i o n (1.9) and the f i r s t r e l a t i o n (3.1), the second c o n d i t i o n (3.10) i s a d i r e c t consequence o f (3.5). S i m i l a r l y , due t o the second c o n d i t i o n (3.1), (3.5), and the f i r s t c o n d i t i o n (3.10) imply (3.8). F i n a l l y we must have t o impose (3.5) and the f i r s t c o n d i t i o n (3.10). We define r by r e p l a c i n g (3.5) by an e q u a l i t y . Replacing the f i r s t r e l a t i o n (3.10) by an e q u a l i t y and t a k i n g (1.4) i n t o account, we g e t (3.2).

P

We remark t h a t i n case n=2 the approximative s t a t i o n a r y manifold can be chosen such t h a t any s o l u t i o n e v e n t u a l l y becomes and remains w i t h i n d i s ance

E:12

of

Ma

.

C. FOIAS. R. TEMAM

324

5.4.

THE BEHAVIOUR FOR w --+

0.

We want t o study the behaviour o f the dimension m o f approximative s t a t i o n a r y manifolds f o r w --+ 0 I n order t o avoid very d i f f i c u l t boundary l a y e r problems we make the remark t h a t a l l the previous considerations remain

.

v a l i d i f R i s replaced by Q = [O,LIn whew L>O i s f i x e d ,and the boundary problem (1.2) i s replaced by t h e p e r i o d i c c o n d i t i o n s (4.1)

u(O,..)

... , u(..,O)

= U(L,..),

= u(..,L)

.

S i n c e the Navier-Stokes equations are Galllean i n v a r i a n t , we can assume, by a change o f the reference frame, t h a t t h e s o l u t i o n s v e r i f y a l s o

.

I Q u dx = 0 Therefore

H and V become t h e closure i n L2 (9)n and H 1

o f the space o f a l l

IRn-valued trigonometric polynomials w(x)

;1

v.w

dx = 0 and

E

0

, respectively, such t h a t

.

A l l the other d e f i n i t i o n s and p r o p e r t i e s remain e s s e n t i a l l y unchanged. A c t u a l l y i n t h i s case much more concrete a l g e b r a i c s t r u c t u r e i s present. For instance the X,

are now o f the form 4a2 L-21k[

with

k c Zn

, k#O

and wAu

+

B(u,u)

has the f o l l o w i n g property. For any rn = 1, 2,

_.-

... t h e r e e x i s t s an i s o m e t r i c operator U

: H e(I-P,)H

such t h a t --

where X -

i s some adequate constant such t h a t

0<

(4.4) One defines

U w

j F i r s t choose t h e f i r s t

x 4 C14 x,2 .

( j = 1, 2, ...)

according t o t h e f o l l o w i n g scheme :

!2(=1,2, ...) such t h a t ~ 4 ' "

A . = 4rr L-21k12 chose J such t h a t AJ = 4a L-21Q'k12

L-2 > A,

, then if

and s e t Uw. wJ J J (Here some supplementary care i s necessary i n order t o avoid the complication due t o the f a c t t h a t the m u l t i p l i c i t y o f h o r XJ i s not simple. Since we i n t e n d J t o study t h i s s e l f s i m i l a r i t y phenomena elsewhere, we d o n ' t i n s i s t w i t h the d e t a i l s o f t h e proof.) F i n a l l y , f o r u0= V the s o l u t i o n s o f (1.6) s a t i s f y , i n case n=2 , besides the r e l a t i o n s (1.8), t h e r e l a t i o n s

FIN1TE PARAMETER APPROXIMA TI VE STRUCTURE OF ACTUAL FLOWS

--

o f (1.6). rf E1/Eo and v1/3 and El/Eo < 1/36 (4.6)

Proof _----

are s u f f i c i e n t l y small

(for instance

then

= dimension o f I$

--f

m

for

+

v

.

0

:

Mv

Let

I f u,v

in

mv

T

325

E

M

=

M

J,V

where

J,,

= J,

and t h e i r distance t o

H

enjoys the p r o p e r t i e s given i n Theorem 1.

$,

i s Q El

, then

there e x i s t

with

J,V

u*, vx

so t h a t

( u - v l & 6E:"

(4.7) Therefore i f satisfying v( . ) )

u(.)

t 21Pm ( u - v ) ) V

and v ( . )

luo19 l v o l \< Eo

.

are s o l u t i o n s o f (1.6) w i t h

i t f o l l o w s from (4.7) and (3.4)

u(0) = uo v(0) = vo ( f o r u ( . ) and

that I{tE ( 0 , l ) : l u ( t ) - v ( t ) I

.<

6E:"

t

ZIP,

(u(t)-v(t))l)l > 1 V

-

2.r

Since f o r v( .) we can take v ( t ) 0 , we o b t a i n t h a t f o r any s o l u t i o n of (1.6) such t h a t luol < Eo the f o l l o w i n g holds

(4.8)

I ( t e ( 0 , l ) : lu(t)l

,c 6E;l2

u(t)ll >

t 21P,

Now we assume t h a t there e x i s t s a sequence v L j a l l j=lYZ9. L e t now v be any o f these v j

O

operator considered i n (4.3) and l e t uo = UE:/2

w1

.. .

such t h a t

and l e t U

1

-

T

u( .)

.

mv

.(. m < 03 for j be t h e i s o m e t r i c

. We denote by

;(.)

the

C. FOIAS, R. TEMAM

326

s o l u t i o n o f (1.6) w i t h i n i t i a l data

Ei/2wl

dUu" t ~ [ U A U G+ B(UG,Uii)] and therefore

u ( t ) = (UG)(t/X)

.

. Then by

(4.3)

o

=

i s a s o l u t i o n o f (1.6) w i t h i n i t i a l data

.

Thus (4.8) i s v a l i d f o r t h i s u ( . ) But because uo(= E A j 2 Uwl) , luol = EA/2 UH i s orthogonal on PmH i t f o l l o w s t h a t P, u ( t ) E 0 and t h e r e f o r e (4.8) V

becomes

But, by (1.8) and (4.5),

so t h a t (4.9) i m p l i e s :

(4.10)

l{tc(0,l)

: Eo

Q

36E1

+

2 v t Eo

From (4.10) i t f o l l o w s t h a t there e x i s t s a Eo 6 36E1

+

L2Xm

t = (0,Z-r)

3 1- T

.

such t h a t

LLX, 2 v t Eo 71

and consequently E, .< 36E1

+

L2Xm 4 v ~Eo 71 f o r a l l

We conclude w i t h the c o n t r a d i c t i o n Eo

<

36E1

v = vj

(j=1,2 ,...)

.

. This proves t h e Theorem.

A c t u a l l y the Theorem i s a l s o v a l i d i f n=3 b u t t h e p r o o f must be modified i n order t o work w i t h small s o l u t i o n s i n V , and consequently the c o n d i t i o n Eo/E1 < 1/36 has t o be replaced by Eo/E1 < c15 w i t h an adequate constant c15 depending on t h e constant

c4

(see 5.1).

REFERENCES

[l] M.J. Feigenbaum, J . Stat. Phys. 21, 669 (1979); Phys. L e t t . 74J, see a1 so these proceedings.

375 (1979);

FINITE PARAMETER APPROXIMATIVE STRUCTURE OF ACTUAL FLOWS

S o l u t i o n s s t a t i s t i ues des 6 u a t i o n s Cours au c h a n c e

327

& Navier-Stokes

[2]

C. Foias,

[3]

C. Foias and R. Temam, Some a n a l y t i c and geometric p r o p e r t i e s o f t h e

s o l u t i o n s o f t h e e v o l u t i o n Navier-Stokes equations J. Math .pure e t appl , 58 ( 1978) pp.339-369.

.

141 E. Hopf, A mathematical example d i s p l a y i n g f e a t u r e o f turbulence Com.pure and appl. Math., 1 (1948) , pp.303-322. Ladyzenskaya, The mathematical theory o f viscous incompressible f l o w s Gordon-Breach , New York ( 1--

C5-l

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