Qualitative Properties of Navier-Stokes Equations

G.M. de La Penha, L . A . Medeiros (eds.) Contemporary Developments i n Continuum Mechanics and P a r t i a l D i f f e r e n t i a l Equations @North-Holland P u b l i s h i n g Company (1978)

QUALITATIVE PROPERTIES OF NAVIER-STOKES EQUATIONS

R.

TEMAM

DGpartement d e Math6matiques U n i v e r s i t 6 d e P a r i s Sud

91405 - Orsay, France

The p u r p o s e o f t h i s p r o p e r t i e s o f t h e s e t of

l e c t u r e i s t o p r e s e n t some new

s t e a d y - s t a t e s o l u t i o n s t o t h e Navier-

S t o k e s e q u a t i o n s of a v i s c o u s i n c o m p r e s s i b l e f l u i d .

I t i s known t h a t f o r s m a l l Reynolds numbers, i f a steady e x c i t a t i o n i s applied t o the f l u i d then there i s a u n i q u e s t a b l e s t e a d y s t a t e which a c t u a l l y a p p e a r s f o r

(t +

If

m).

Joseph

171 )

Benjamin

A s f a r as we know, v e r y l i t t l e

h a s b e e n p r o v e d c o n c e r n i n g t h e s e t of the equations.

( c f . B.T.

t h a t new s t e a d y s t a t e s a p p e a r some being

s t a b l e and some b e i n g u n s t a b l e .

of

large

t h e Reynoldsnumber i n c r e a s e s , t h e n i t i s

c o n j e c t u r e d and e x p e r i m e n t a l l y well-known

[ 1,21, D.

t

a l l s t a t i o n a r y solutions

I n j o i n t works w i t h C .

Foias (cf

[ 4 1 [ 5][ 61)

t h e a u t h o r has a t t e m p t e d t o f i n d some q u a l i t a t i v e i n f o r m a t i o n o n t h i s s e t , and we a r e g o i n g t o summarize t h e main r e s u l t s of

C51[61. S e c t i o n 1 c o n t a i n s t h e d e s c r i p t i o n of N a v i e r - S t o k e s

e q u a t i o n s and t h e i r f u n c t i o n a l s e t t i n g . the description of the r e s u l t s . 1. S t e a d y - s t a t e N a v i e r - S t o k e s 2.

P r o p e r t i e s of 2.1

S(f,v,v)

General p r o p e r t i e s

Section 2 c o n t a i n s

The p l a n i s t h e f o l l o w i n g : Equations.

486

R.

TEMAM

2.2

Generic p r o p e r t i e s

2.3

Generic b i f u r c a t i o n .

.

-

1. S __t e a d y S t a t e Navie r- S t olce s equa t i o n s

R

Let

4 = 2 o r 3.

be t h e domain f i l l e d by t h e f l u i d ,

W e assume t h a t

62

62

C

RL,

i s bounded w i t h a smooth

F.

boundary

,...,u , ( x ) ) ,

u ( x ) = (u,(x)

Let

velocity of the p a r t i c l e pressure a t

x

(x

and

p(x)

x

of f l u i d a t point

E 0);

u

then

and

p

be the

and t h e

s a t i s f y the

equations

where of

f

r

r e p r e s e n t s volumic f o r c e s ,

i s t h e g i v e n velocity -1

CQ

which i s assumed t o be m a t e r i a l i z e d and s o l i d , V = Re

i s t h e i n v e r s e of a Reynolds number. g i v e n , t h e problem i s t o f i n d

u

f , cp,

For

and

p

v

(and

n)

s a t i s f y i n g (1.1)-

(1.3)I n t h e f u n c t i o n a l s & t t i n g of

lL2(n) = L 2 ( n ) &

t o i n t r o d u c e t h e space

2

orthogonal decomposition o f

[8]

or R.

Temam

IL.

2

G = [vp

H =

v

(n)

and t o c o n s i d e r t h e

( c f O.A.

Ladyzhenskaya

1141): L (62)

with

the equation, i t i s usual

I {U

= H @ G,

2 P E L (R),

3P ax. E

L

2

(n),

1 s

i s L]

1

E

2

IL

(62)

I

V U = 0,

u.vlr. =

t h e u n i t outward normal v e c t o r on

r.

01, We d e n o t e by

487

QUALITATIVE PROPERTIES O F N A V I E R - STOKES EQUATIONS P

the If

u

l y regular,

and

p

u

then

(1.4)

t o modifying satisfies

H.

clearly satisfies

Pf = f ,

+ ( u - v ) ~ )=

f

in

0,

which i s a l w a y s p o s s i b l e and amounts

181) t h a t i f

Conversely i t i s c l a s s i c a l ( c f

p.

(1.4)

t o g e t h e r with ( 1 . 2 ) p

exists a scalar function

(1.3).

onto

s a t i s f y (1.1)-(1.3) and a r e s u f f i c i e n t -

P(-WAU

assuming t h a t

u

a2((n)

projection i n

and (1.3),

such t h a t

u, p

Therefore the equations ( 1 . 2 ) - ( 1 . 4 )

then t h e r e

s a t i s f y (1.1)for

u

are

e q u i v a l e n t t o t h e o r i g i n a l problem.

N o w we w r i t e function

cp

+

u =

@

0.

inside

CP

where

i s some e x t e n s i o n

It i s convenient t o d e f i n e

as t h e s o l u t i o n o f t h e nonhomogeneous S t o k e s problem

I n t h i s case

1

-A@

vCP

+

VTI

= 0

= 0

R,

in

H = ep

n,

in

r.

on

satisfies

VU

=

o

R,

in

W e i n t r o d u c e t h e l i n e a r unbounded o p e r a t o r

A

in

H,

whose domain i s D(A)

= ~

~ ( n0 ~((61) )

n

H,

and AV = -PAv,

here

Hm(h2)

v

i s t h e S o b o l e v s p a c e of

E

D(A); order

. m,

H”(n) =[H”’(n)l”

485

R.

H:(n)

and

TEMAM

u E H1(n)

i s t h e s p a c e of

the theory of

J.L.

Sobolev spaces cf

[141)

I t i s known ( c f

that

A

Lions-E.

H,

(n

F

(for

Magenes 1111).

i s a self-adjoint

p o s i t i v e and i n v e r t i b l e o p e r a t o r i n

i s compact

v a n i s h i n g on

strictly

and t h a t

A-'

E e(H)

bounded).

B

W e a l s o introduce the operators

= P[

B(u,v)

such t h a t

(~*V)vl,

B ( u ) = B(u,u).

4, = 2 o r 3 ,

For

)

that

B(',

D(A)

x D(A

we i n f e r from t h e S o b o l e v imbedding theorems

H2(n) x k12(n)

maps

I

(1.9)

(1,6)-(1.8)

G

To f i n d VAu

+

and i n p a r t i c u l a r

H.

into

The e q u a t i o r i s p r o b 1em

H,

into

D(A)

in

are now equivalent t o the

such t h a t

B(G+Acp) = f.

T h i s i s t h e f u n c t i o n a l f o r m of

t h e s t e a d y - s t a t e Navier-

S t o k e s e q u a t i o n s which we had i n v i e w and on which i s b a s e d the f o l l o w i n g s t u d y .

6 E D(A)

2.

satisfying

Properties of

W e d e n o t e by

S(f,cp,v).

B 3 / 2 ( 1 - ) = [H3'*(r)]'.

cp

the s e t of

(1.9).

W e assume t h a t

(1.2),

S(f,cp,V)

f

i s given i n

H,

cp

i s given i n

By t h e S t o k e s f o r m u l a , and b e c a u s e o f

must v e r i f y

W e w i l l impose a s l i g h t l y s t r o n g e r c o n d i t i o n o n

cp:

Q U A L I T A T I V E : P R O P E R T I E S O F NAVIER-STOKES

q3.v

(2.1)

489

EQUATIONS

1 s i s N,

dr = 0 ,

‘i

rl,.. . , r N

where

N=l

and

cp

s e t of

r

i f

a r e t h e c o n n e c t e d components of

i s connected).

We d e n o t e by

~ ” ~ ( sr a )t i s f y i n g

in

F

(P

if13’2(r)

F,

=

the

(2.1).

General p r o p e r t i e s .

2.1

F o r every the s e t

H

given i n

f

i~s-nonempty.

S(f,Cp,V)

and

Lions

1101

equations.

[S],

Ladyzhenskaya

w i t h s t r o n g e r a s s u m p t i o n s on f,cp E H x k3’ * (T ),

t h e weaker a s s u m p t i o n

S(f,cp,v)

The s e t

k3j2(r),

T h i s i s a n e x i s t e n c e theorem

f o r t h e s t e a d y s t a t e Navier-Stokes r e s u l t a p p e a r s i n O.A.

given i n

J.

f

This existence Leray [ 9 ] , J . L .

and/or

cp;

for

c f . C.Foias-R.T.[6].

i s reduceed t o a s i n g l e p o i n t i f

J!

i s s u f f i c i e n t l y l a r g e , more p r e c i s e l y i f

where t h e f u n c t i o n

Uo:

R+xR+-R+

i s increasing with respect

t o e a c h of i t s t w o a r g u m e n t s . Now l e t

consisting of adjoint i n H

w.,

J

j 2

1,

b e t h e orthomormal b a s i s i n

the eigenvectors of

H).

Let

Pm

(A”

A-1

H

i s compact s e l f -

denote the orthogonal projector i n

o n t o t h e s p a c e s p a n n e d by

wl,

...,wm.

We have t h e follow-

ing:

(2.3)

For

m

sufficiently large,

one t o one mapping on

rn

S ( f ,Cp , W

a r e a l compact @ - a n a l y t i c s e t .

2

)

m*(f,rp,V), and

Pm

pm S(f,cP,V)

is a is

R.

490 T h i s means t h a t

TEMAM

( a n d i n some s e n s e

Pm S ( f , c p , V )

S ( f , c p , ~ ) i t s e l f ) i s a f i n i t e u n i o n of p o i n t s ,

regular

a n a l y t i c c u r v e s , r e g u l a r a n a l y t i c manifolds of h i g h e r

[ 31 ) .

d i m e n s i o n s ( c f Bruhat-Whitney

(2.4)

Either

S(f,Cp,V)

points o r

A s a c o r o l l a r y we g e t :

i s t h e u n i o n o f a f i n i t e number o f

~ ( f , c p , v ) c o n t a i n s a t l e a s t an a n a l y t i c

curve.

2.2

Generic P r o p e r t i e s . We a r e now g o i n g t o d e s c r i b e g e n e r i c p r o p e r t i e s of A r e s u l t which i s t y p i c a l of

S(f,cp,V). ed i n

[4][5]

Theorem 1

-

F o r every

V

>

is finite,

and

0

8

1

C

cp E

H,

&3/2(r) f i x e d ,

and f o r e v e r y

c a r d S(f,cp,V)

The p r i n c i p l e o f t h e p r o o f t h e non l i n e a r mapping

N;

N~

such t h a t

N1,

= f).

Whence t h e

i s compact,

v a l u e s of

N1

The f a c t t h a t

we c o n s i d e r

H:

the Fr6chet derivative

?its

i s o l a t e d and t h e r e i s a f i n i t e number of

Nyl(f)

into

i s r e g u l a r a t every preimage p o i n t

N1(;)

E Q1,

@.

i s as follows: D(A)

from

is a regular value of

f

of

N~

f

there

i s odd, and c a r d S(f,cp,V)

i s c o n s t a n t on e v e r y c o n n e c t e d component of

When

establish-

i s the following

e x i s t s an open d e n s e s e t S(f,cp,V)

the r e s u l t s

in

N;'(f)

such

the s e t

(i.e.

Q1

;Is,

every are since

of r e g u l a r

i s d e n s e i s t h e l e s s t r i v i a l r e s u l t and follows

from t h e i n f i n i t e d i m e n s i o n a l v e r s i o n o f S a r d ' s theorem due

t o Smale [13].

ii

The o t h e r p r o p e r t i e s a r e c o n s e q u e n c e s o f t h e

QUALITATIVE

PROPERTIES

OF

NAVIER-STOKES

491

EQUATIONS

o f N1.

i m p l i c i t f u n c t i o n t h e o r e m and some s p e c i f i c p r o p e r t i e s F i n a l l y t h e o d d n e s s of

f o l l o w s from a

c a r d S(f,ep,V)

t o p o l o g i c a l d e g r e e argument. A s i m i l a r r e s u l t when

f , Cp,

a r e simultaneously

V ,

allowed t o var y i s t h i s one:

-

Theorem 2

x R+,

T h e r e e x i s t s a d e n s e open s e t f,cp,V E Q 2 ,

and f o r e v e r y

and odd. ____

Furthermore card ___________

c o n n e c t ed component o f

82

(r) x

i n Hxk3'2 -

c a r d S(f,cp,V)

is finite

i s c o n s t a n t on e v e r y

S(f,cp,V)

82'

Same p r o o f as Theorem 1. We may n o w t h i n k of a r e s u l t

s y m m e t r i c a l t o Theorem 1

i n the sense of a generic r e s u l t w i t h respect t o and

are fixed.

V

Theorem

a (-a )L

3

n

-

(a >

C

k2'a(r)

+

tp

E Q3,

V

> 0

??& (9

f

8

3

C

i s f i n i t e and odd,

card S ( f , c p , V )

c a r d S(f,cp,V)

in

f

f i x e d , f o r every f i x e d

t h e r e e x i s t s a d e n s e open s e t

such t h a t

(l),

component o f

0),

when

We h a v e :

For every

H

cp

i s c o n s t a n t on every connected

3.

The p r o o f author ( c f [ 1 2 ] )

of

t h i s r e s u l t due t o J . C .

S a u t and t h e

involves d i f f e r e n t technics,

I n particular

theorem i s r e p l a c e d by a t r a n s v e r s a l i t y t h e o r e m

Sard-Smale's

A s a t o o l f o r t h i s p r o o f we a l s o

d u e t o Abraham and Quinns.

need t h e f o l l o w i n g uniqueness

t h e o r e m f o r a Cauchy problem

a s s o c i a t e d t o Stokes e q u a t i o n s : For

(1)

*

Cs(r)

b

given i n

H

n

H1'm(a)t,

i s t h e s e t of f u n c t i o n s

satisfy (2.1).

cp

if

in

v

@*(T')'

and

q

satisfy

which

R.

492

+

-Av

I v = 0

then Remark

1

-

and

2.3

-av = 0

and

+

Vq = 0

hl

in

r

on

av

i s a constant.

The f a c t t h a t

c o n j e c t u r e d by B.T.

(v*V)b

hl

in

0

v = O q

+

(b.V)v

vv =

I

TEMAM

i s g e n e r a l l y f i n i t e was

S(f,rp,V)

Benjamin.

Generic B i f u r c a t i o n . W e now d e s c r i b e a r e s u l t o f g e n e r i c b i f u r c a t i o n f o r

the equation ( 1 . 9 ) .

S i m i l a r r e s u l t s a r e proved i n

C6l f o r

t h e c l a s s i c a l T a y l o r and B h a r d p r o b l e m s . Theorem 4 exists

-

Q4(rp)

f E G4(ep),

a dense

h3'*(T')

cp E

W e assume t h a t

s u b s e t of

G

i s fixed.

There

and f o r e v e r y

H

t h e manifold

s

=

u

S(f,cp,V)

V 70

h a s t h e f o l l o w i n g form: ( i ) I t i s c o n s t i t u t e d of i s o l a t e d p o i n t s and i s o l a t e d

a n a l y t i-c. _ ma n i f o l d s which l i e above s_ o l a t e d v a l u e s of _____ ____________ - _iThe number of s u c h v a l u e s o f

v

2

V.

& s f i ? i . t e on e v e r y semi a x i s

V

vo > 0.

( i i ) It; i s c o n s t i t u t e d of one ( o r more) a n a l y t i c manifold(s)

of d i m e n s i o n 1, whose p r o j e c t i o n on t h e __ i-n_ t e_ r v_ al

]O,+m[.

V

a x i s i s t h e whole

The s e t o f s i n g u l a r p o i n t s o f t h i s manifold

i s f i n i t e i n every r e g i o n

V

A s a C o r o l l a r y we g e t

2

V

>

0.

QUALITATIVE PROPERTIES OF NAVIER-STOKES EQUATIONS

(2.6)

493

Generically, the set of (primary and secundary) hi-

v

furcating values of only accumulate at Remark 2 .___

-

V

for (1.9) is countable and can = 0.

As far as we know, this is the first information

available concerning all the primary and secundary bifurcating points of a nonlinear equation, Remark 3

-

The methods used for the proof of the above results

are quite general and probably apply to the equations of nonlinear elasticity.

References [l] T.B.

Benjamin, Applications of Leray-Schauder degree

theory to problems of hydrodynamic stability, Math. Proc. Camb. Phil. SOC. 79, 1976, p.373-392. [2]

T.B. Benjamin, Bifurcation Phenomena i n Steady flows of a viscous fluid, Part 1 Theory, Part 2 Experiments, Reports no 83, 84, University of Essex, Fluid Mechanics Research Institute, May

-

[ 3 ] F. Bruhat

1977.

H. Whitney, Quelques proprietes fondamentales

des ensembles analytiques Rgels, Comm. Math. Helv. 3 3 ,

1959, p. 132-160.

[4] C. Foias

-

R. Temam, On the stationary statistical

solutions o f the Navier-Stokes equations and Turbulence, Publication Math6matique d’orsay, no 120-75-28, Universit6 de Paris

-

Sud, Orsay, 1975.

494

R. TEMAM

[ 5 ] C. Foias

-

R. Temam, Structure o f the set of

stationary

solutions o f the Navier-Stokes equations, C o m m .

Pure

Appl. Math., XXX, 1 9 7 7 , p. 149-164.

[ 6 ] C. Foias

-

R. Temam, Remarques sur les kquations de

Navier-Stokes stationnaires et les ph&norn&nes succesifs de bifuraction, Annali di S c .

Norm. Sup. di Pisa, vol.

d6di6 & J. Leray, & para?tre.

[ 7 ] D. Joseph, Stability of fluid motions, vol. I and 11, Springer-Verlag, New-York-Heidelberg, 1976. [81 O.A. Ladyzhenskaya, The mathematical _ _ theory ~ o f viscous incompressible flow, Gordon and Breach, New-York, 1969.

191 J. Leray, Etude de diverses Qquations int6grales lin6aires et
quelques probl&mes que pose l'hydro-

dynamique, J. Math. Pures Appl. XIII, 193 [lo] J.L.

non

,

p.

1-82.

Lions, Quelques m6thodes de rbsolution des probldmes

aux limites n o n linGaires,Gauthier-Villars-Dunod, Paris 1969. [ll] J.L.

-

Lions

E. Magenes, Non-homogeneous boundary value

problems and applications, Springer-Verlag, Heidelberg/New-York, 1 9 7 3 . [12]

J.C. Saut

-

R. Temam, Propri6tQs de l'ensemble des

solutions stationnaires ou p6riodiques des Qquations de _ Navier-Stokes: _

g6nQricit6 par rapport aux donn6es __

- -

aux limites, Comptes Rendus Ac. S c . Paris,

[I31

S.

1977.

Smale, An infinite dimensional version o f Sardts Theorem, Amer. J. Math., 87, 1965, p. 861-866.

1141 R. Temam, Navier-Stokes equations, Theory and Numerical Analysis, North-Holland-Elsevier, Amsterdam, New-York,

1977

rn