Vp=O ON THE STEADY-STATE SOLUTIONS OF THE NAVIER- STOKES EQUATIONS, III

ON THE STEADY-STATE SOLUTIONS OF THE NAVIER- STOKES EQUATIONS, III

BY

ROBERT FINN Stanford University, Calif., U.S.A. (1)

C o n t e n t s

Page

I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 7

. N o t a t i o n a n d d e f i n i t i o n s ; p r e l i m i n a r y e s t i m a t e s ; t h e r e p r e s e n t a t i o n f o r m u l a . . . . 201

2. A p r i o r i e s t i m a t i o n of t h e D i r i c h l e t I n t e g r a l . . . . . . . . . . . . . . . . . . 205

2 a) E s t i m a t i o n of t h e D i r i c h l e t I n t e g r a l i n a b o u n d e d r e g i o n . . . 208

2 b) E s t i m a t i o n of t h e D i r i c h l e t I n t e g r a l i n a n e x t e r i o r r e g i o n ; c a s e of zero o u t f l u x 211 2 c) E s t i m a t i o n of t h e D i r i c h l e t I n t e g r a l i n a n e x t e r i o r r e g i o n ; g e n e r a l c a s e . . . . 213

3. A p r i o r i e s t i m a t i o n of t h e s o l u t i o n . . . . . . . . . . . . . . . . . . . . . . 215

3 a) E s t i m a t i o n of t h e s o l u t i o n i n a b o u n d e d r e g i o n . . . 218

3 b) E s t i m a t i o n of t h e s o l u t i o n i n a n e x t e r i o r r e g i o n . . . 219

4. B e h a v i o u r a t i n f i n i t y ; t h e r e p r e s e n t a t i o n f o r m u l a . . . 223

5. E x i s t e n c e t h e o r e m s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

5 a) E x i s t e n c e of a s o l u t i o n i n a f i n i t e r e g i o n . . . . . . . . . . . . . . . . . 226

5 b) E x i s t e n c e of a s o l u t i o n in a n e x t e r i o r r e g i o n . . . 227

6. R e m a r k s o n t h e p r e c e d i n g s e c t i o n s ; a n e x a m p l e . . . . . . . . . . . . . . . . 228

7. T r a n s i t i o n t o z e r o R e y n o l d s ' N u m b e r . . . . . . . . . . . . . . . . . . . . 231

7 a) T r a n s i t i o n t o z e r o R e y n o l d ' N u m b e r ; c a s e of a b o u n d e d r e g i o n . . . 231

7 b) T r a n s i t i o n t o zero R e y n o l d s ' N u m b e r ; e x t e r i o r r e g i o n . . . 234

7 e) T r a n s i t i o n of t h e f o r c e e x e r t e d o n a f l u i d i n t e r f a c e . . . 240

8. U n i q u e n e s s a n d c o n t i n u o u s d e p e n d e n c e . . . . . . . . . . . . . . . . . . . . 241

I n t r o d u c t i o n

In this work we study the relations connecting a solution of the Navier-Stokes equations

,u A w - - @ w . V w - - V p = O (1)

T " w = O ,

(~) This investigation was s u p p o r t e d b y t h e Office of N a v a l Research.

198 ~OB~RTFr~N

with the values achieved b y the solution on the b o u n d a r y of the region of definition, and with the magnitudes of certain energy integrals which are associated naturally with the solutions of (1).

The notation in (1) is the usual one of vector analysis. E a c h of the quantities which appears admits a simple physical interpretation. The solution w (x) can be inter- preted as the velocity field of an incompressible fluid motion, and p (x) is then the associated pressure. The constant ~t is the viscosity coefficient of the fluid, and the t e r m /x A w denotes accordingly the shearing force on a unit volume due to relative motion a t a fluid interface. Q denotes the density of the fluid, Q w . V w the inertial reaction of a unit volume, and V p is the force per unit volume acting normal to a fluid interface. The first equation expresses the equilibrium of these forces at points of the flow; the second expresses the assumption t h a t ~ is constant in the motion.

Because of the difficulty in integrating (1) in a general case, it is natural to consider the linear equations satisfied b y the perturbations of a particular solution.

The simplest of these are the Stokes equations

# A W - V p = 0 (2)

V " w = O

which correspond to the identically vanishing solution of (1). A m a j o r task of the present work will be to examine the connection between the solutions of (2), and the solutions of (1) which correspond to small b o u n d a r y data.

The system (2) has been studied in considerable detail b y Odqvist [11], who proved the existence of a Green's Tensor for an a r b i t r a r y region. Odqvist used this tensor to obtain an integral equation for the solutions of (1), and this equation led in t u r n to a proof of existence of a solution of (1) in a finite region 6, corresponding to prescribed d a t a w* on the b o u n d a r y ~ which satisfy the (necessary) condition

w * . n d S = 0 (3)

(n = unit exterior direct normal), provided only t h a t ]w*] is everywhere sufficiently small ([11], see also [10]).

The first general s t u d y of (1) for a r b i t r a r y prescribed d a t a is due to L e r a y [8].

L e r a y d e r i v e d general a priori estimates on the solutions of (1), depending only on and on the b o u n d a r y data. H e applied these estimates, using a device which is now classical b u t which was a t t h a t time not yet clearly formulated, to prove the exist-

ON T H E S T E A D Y - S T A T E S O L U T I O N S OF T H E N A V I E R - S T O K E S EQUATIOI~S~ I I I 199 enee of at least one solution of (1) in ~ corresponding t o a r b i t r a r y (sufficiently smooth) d a t a w* on ~. The solution is o b t a i n e d b y a continuous d e f o r m a t i o n in f u n c t i o n space, starting with t h e solution of (2) given b y Odqvist, a n d essential use is m a d e of O d q v i s t ' s integral e q u a t i o n a n d of his estimates on t h e Green's Tensor.

F r o m a physical p o i n t of view t h e problem just discussed has little meaning, since the n a t u r a l b o u n d a r y condition is w * ~ - 0 , a n d in this case one sees easily t h a t the only solution of (1) in ~ is w = 0. Of more interest is the exterior problem, in which a solution is sought which assumes d a t a w* on Z a n d which tends to a given c o n s t a n t v e c t o r w 0 at infinity. I n this case, however, new difficulties arise. Experi- m e n t a l evidence indicates t h a t , at least for large prescribed data, t h e solution either does n o t exist or is unstable. F o r the linear s y s t e m (2), it is k n o w n t h a t no solution exists in two dimensions. (1) I n three dimensions, the solution exists b u t is k n o w n to violate, in a neighborhood of infinity, the assumptions u n d e r which the equations were derived (see, e,g. [12, p. 165]). Also for the strict equations (1) there is evidence t h a t solutions m a y exhibit pathological behavior at infinity, A n example in t w o di- mensions is discussed in w 6 of this paper. Nevertheless, L e r a y succeeded in constructing, f o r a r b i t r a r y prescribed d a t a in three dimensions, a s o h t i o n of (1) in the exterior E of Y: which equals w* on ~, for which the Dirichlet I n t e g r a l is finite, a n d which tends t o w 0 in t h e sense of an integral norm. L e r a y also o b t a i n e d a priori estimates o n the solution which are valid in a n y c o m p a c t subregion of E.

The behavior of t h e solution at infinity has been discussed in some detuil in [1]

a n d in [2]. I n [1] we h a v e p r o v e d t h a t the solution of L e r a y (more generally a n y solution with finite Dirichleb Integral) necessarily tends to a limit in tile strict sense as x--> oo, a n d a representation of the solution b y means of an integral e q u a t i o n is obtained. I n [2] we discuss solutions which need n o t have finite Dirichlet Integral.

W e show there t h a t whenever w - + w 0 at infinity, t h e n necessarily all first order derivatives of w t e n d to zero. If, in addition, I w - w 0 l < C r ~-~ for some e > 0 , t h e n w (x) has the same a s y m p t o t i c s t r u c t u r e at infinity as t h e corresponding solution of t h e system obtained b y linearizing (1) a b o u t the solution w - - w 0 . I n particular, [ w - w ~ [ < C r -1 a n d there exists a paraboloidal " w a k e " region outside of which I w - w0] < Cr .2. I t is n o t k n o w n however whether there exist solutions which exhibit t h e assumed rate of d e c a y to w o at infinity.

(1) See, e.g., [4]. An improved discussion of this phenomenon will appear in a forthcoming work

~3f I. D. Chang and the author.

2 0 0 ROBERT F I N N

T h e crucial step i n t h e m e t h o d of L e r a y consists i n o b t a i n i n g a n a priori b o u n d for t h e Dirichlet Integral(1) of a n y possible solution, d e p e n d i n g o n l y o n b o u n d a r y d a t a . L e r a y p r o v e d t h e existence of such a b o u n d , a n d also gave a n i n d e p e n d e n t d e m o n s t r a t i o n w h i c h y i e l d e d a n explicit e s t i m a t e (2). I n w 2 of this p a p e r we o b t a i n a b o u n d for t h e Dirichlet I n t e g r a l b y a m e t h o d which derives c o n c e p t u a l l y from t h a t of L e r a y . O u r r e s u l t is a slight i m p r o v e m e n t o n t h a t of Leray, i n t h e sense t h a t we do n o t i n s i s t t h a t t h e o u t f l o w i n t e g r a l (3) v a n i s h , b u t m e r e l y r e q u i r e i t to b e s u f f i c i e n t l y small. T h e d e m o n s t r a t i o n we give uses a t e c h n i c a l device d u e to E. H o p f [5]

which, we believe, simplifies a n d clarifies t h e r e a s o n i n g considerably.

I n w 3, we a p p l y t h e b o u n d s o n Dirichlet I n t e g r a l i n order to o b t a i n a-priori e s t i m a t e s o n a n y possible s o l u t i o n a n d o n its first d e r i v a t i v e s , d e p e n d i n g o n l y o n p r e s c r i b e d d a t a . I n t h e case of a f i n i t e region ~ , these b o u n d s are essentially those of L e r a y . F o r t h e region ~ exterior to ~, we i m p r o v e t h e r e s u l t s of L e r a y b y g i v i n g e s t i m a t e s which are u n i f o r m l y v a l i d t h r o u g h o u t t h e flow region.

W e show i n w 4 t h a t solutions w i t h f i n i t e Dirichlet I n t e g r a l are necessarily con- t i n u o u s a t i n f i n i t y . W e p r e s e n t here a proof which' is m o r e e l e m e n t a r y t h a n t h e o n e we h a v e g i v e n i n [1]. I n w we prove t h e existence of a s o l u t i o n c o r r e s p o n d i n g t o p r e s c r i b e d b o u n d a r y d a t a . A g a i n t h e r e s u l t is e s s e n t i a l l y t h a t of L e r a y w h e n t h e region is finite. T h e n e w features i n t h e o t h e r case are t h a t t h e s o l u t i o n is shown to a t t a c h c o n t i n u o u s l y to t h e prescribed v a l u e a t i n f i n i t y , t h a t some o u t f l u x is per- m i t t e d , a n d t h a t u n i f o r m b o u n d s are a v a i l a b l e for t h e s o l u t i o n a n d its d e r i v a t i v e s . T h e p r i n c i p a l n e w results of this p a p e r are p r e s e n t e d i n w 7. t t e r e we s t u d y t h e m a n n e r i n w h i c h t h e solutions of (1) t r a n s f o r m i n t o those of (2) as t h e p r e s c r i b e d d a t a t e n d to zero. Precisely, we consider d a t a of t h e form 2w*, X w 0, 0 < ~ < 1. (a)

(1) This integral can be interpreted physically as half the SUln of the rate at which energy is converted into heat by the fluid, and the total vorticity in the flow.

(~) Another proof of the existence of a bound, based on an inequality of Sobolev, has been given by O. A. Ladyzhenskaia [7]. The method of Leray, besides yielding an explicit estimate, is intrinsically simpler and more elementary.

(a) Equivalently, we could keep the boundary data fixed and let ju --> co or Q --> 0. I n the former ease we would find [ w (x; ~t) - W 0 (x) [ < C/l~ in a bounded region, and

l w(x; ~)- Wo(x) l<C(~t- 89 -1)

in an exterior region. The estimate for a bounded region can be obtained also from the work of Od- qvist [11]. The emphasis in the present paper is on the behavior of the solution in a neighborhood of infinity, to which the methods of Odqvist do not seem to apply.

ON THE STEADY-STATE SOLUTIONS OF THE NAVIER-STOKES EQUATIOlqS~ III 201 W e p r o v e t h a t if w ( x ; 2) is a s o l u t i o n of (1) w i t h t h e s e d a t a a n d if W 0 ( x ) is t h e s o l u t i o n of (2) w i t h d a t a w*, w 0 t h e n 12 l w (x; 2 ) - W 0(x) l = O(2) w h e n t h e r e g i o n is finite, a n d

I -lw(x;2)-W0(x)l=o(V r 1+ )in

t h e case of a n i n f i n i t e region.(1) A n a l o g o u s e s t i m a t e s for t h e d e r i v a t i v e s a r e also given. Thus, w e see t h a t t h e solu- t i o n s of (2) in a n e x t e r i o r r e g i o n a r e u n i f o r m l y close t o t h e c o r r e s p o n d i n g s o l u t i o n s of t h e N a v i e r - S t o k e s e q u a t i o n s (1), e v e n t h o u g h t h e p e r t u r b a t i o n f r o m t h e l a t t e r s o l u t i o n s t o t h e f o r m e r is s i n g u l a r i n tilts region. These c o n s i d e r a t i o n s a r e a p p l i e d t o a discussion of t h e force" e x e r t e d on ~ b y t h e fluid, a n d a n e s t i m a t e is g i v e n for t h e error i n c u r r e d b y using t h e s o l u t i o n of (2) t o c a l c u l a t e t h e force. T h e d e m o n s t r a - t i o n s a r e s t r a i g h t f o r w a r d , b u t l e a n h e a v i l y on t h e d e v e l o p m e n t s in t h e e a r l i e r sections of t h e p a p e r .

I n t h e final section w e i m p r o v e t h e classical u n i q u e n e s s t h e o r e m for (sufficiently small) s o l u t i o n s in a f i n i t e r e g i o n ~ b y showing t h a t t h i s r e s u l t can be g i v e n in a n a p r i o r i f o r m u l a t i o n , d e p e n d i n g o n l y on b o u n d a r y d a t a . (The classical r e s u l t a s s u m e s a k n o w l e d g e of one s o l u t i o n in t h e e n t i r e region, see, e.g. [16].) W e o b t a i n t h i s t h e o r e m as a special case of a m o r e g e n e r a l r e s u l t , t h a t t h e difference of t w o suffi- c i e n t l y s m a l l s o l u t i o n s wx (x) a n d w 2 (x) can b e b o u n d e d u n i f o r m l y in ~ in t e r m s of t h e s o l u t i o n of t h e l i n e a r e q u a t i o n s (2) w i t h b o u n d a r y d a t a e q u a l t o w l ( x ) - w 2 (x).

To our k n o w l e d g e , t h i s is t h e first r e s u l t o n c o n t i n u o u s d e p e n d e n c e of t h e s o l u t i o n s of (1) on b o u n d a r y d a t a t o be p u b l i s h e d .

T h e chief c o n c e r n of t h i s p a p e r is w i t h s o l u t i o n s of t h e s y s t e m (1) i n t h r e e di- mensions. Those of our r e s u l t s w h i c h p e r t a i n to s o l u t i o n s in a f i n i t e r e g i o n are pre- s u m a b l y v a l i d also in t h e c o r r e s p o n d i n g t w o d i m e n s i o n a l case, b u t a rigorous p r o o f r e q u i r e s c e r t a i n g e n e r a l e s t i m a t e s w h i c h are n o t y e t a v a i l a b l e . T h e b e h a v i o r of a t w o d i m e n s i o n a l s o l u t i o n a t i n f i n i t y a p p e a r s to p r e s e n t difficulties of a m o r e p r o f o u n d n a t u r e , a n d a p r e c i s e discussion m u s t p r o b a b l y a w a i t t h e d e v e l o p m e n t of n e w m e t h o d s .

l . N o t a t i o n a n d d e f i n i t i o n s ; p r e l i m i n a r y e s t i m a t e s ; t h e r e p r e s e n t a t i o n f o r m u l a W e consider a v e c t o r field w (x), w = (w 1, w 2, w~), w h i c h is d e f i n e d i n a r e g i o n of t h r e e d i m e n s i o n a l E u c l i d e a n space, x = (x 1, x 2, %). S u c h a field is s a i d to be a solution o/ the N a v i e r - S t o k e s equations,

(1) The origin of coordinates is assumed interior to ~. The result implies, in particular, the uniform inequality [ W (x; 2) - W o (x) [ < C l/~ in ~ + Y,.

2 0 2 ROBERT FINN

~ A w - - p w . V w - - V p = O (1)

V - w = O

in ~ whenever there exists a scalar field p (x) in ~, such t h a t (1) is satisfied through- out ~ b y the pair (w, p). I t is assumed t h a t w (x) and p (x) are sufficiently smooth t h a t all quantities entering in (1) are defined and continuous throughout ~. The vector field w (x) and scalar p (x) have the physical significance of velocity and pres- sure, respectively. We have found these interpretations helpful in providing motiva- tion and suggesting methods, but they are of course unnecessary for the formal mathematical developments. I n order to simpliy notation we shall normalize (1) so t h a t # = ~ = 1. This can always be achieved by multiplication of w and of p by ap- propriate constant factors. Equations (1) then take the form

A w - w . V w - V p = 0 (4)

V . w = 0 .

Since most of our results are valid for every value of the Reynold's number, (1)this normalization entails no loss of generality. I n w 7 we shall permit the Reynold's number to vary, but we shall effect this I by varying the boundary values of the velocity field and keeping all other parameters constant.

A particular solution of (4) is the uniform flow, w ~ w 0 = const. The perturbations of this solution are solutions of the linear system,

A W - - W o- V w - - V p = O (5)

V . w = 0 ,

the equations o/ Oseen. I n the case w 0 = 0 , we obtain the equations o/ Stokes,

w - V p=O (6)

~7 . w = 0 .

We shall need a fundamental solution tensor ~ (x, y) associated with (5). Such a tensor has been determined explicitly b y 0seen [12, p. 34]. I t can be obtained from the relations,

(1) The Reynold's number is defined by the relation R=QUL/~, where U and L denote a characteristic speed and length in the flow. For a discussion of the role played by this quantity in the theory of (1) and in experimental observation, see, e.g. [6].

O~T T H E S T E A D Y - S T A T E SOLUTIO~TS O F T H E ~ T A V I E R - S T O K E S EQUATIO~TS~ I I I 203

~20 ($is = {1,

i = j

Z i s = ( ~ s A O

8xi~x s'

0, i # j (7)

~s = - ~ [ A (I) + Wo" v

O]

vxt

0

1 f~ 1-e-~do:

8~a

Iw.I

W 0 9

(y

X) a = ~ 8 - r x y - ~

~ IWol

The tensor X = (Z~s) and vector tp = (Fs) become singular at x = y in such a w a y t h a t

(s)

where ~r denotes the surface of a sphere of radius r a b o u t x as center and n = (nj) is the unit exterior directed normal on ~ . For fixed j, the column vectors Z,j de- fine, as function of x, a solution of (5) with ~s as COiTesponding pressure. As func- tion of y, Z,s defines a solution of the adjoint system,

A w + w o"

V w - - ~ T p = 0

(9)

V . w = 0 .

I n the case w o = 0 , the tensor (Zis)" takes a particularly simple form. We then have

- l l(~,s ~ (xi-y~) (xs-ys)}

Z~s = - ~ t r ~ r~y '

x j - yj (10)

~fs 4~r~y"

We define the

stress tensor T w

b y the relation

(ew~ ~wj~

( T w ) , j = - p ~ j + \~xj + ~ ] " ( l l )

Formal integration by parts leads to the relations> valid for any divergence-free vector fields u(x) and v(x) and associated scalars p(x), q(x) defined in a region ~ with b o u n d a r y ~,

fq u . ( A u - - V p ) d V + 2

f~

(def

u)2dV = ~r~ u. T u d S

fo r-.(A v q ) - v . ( A

^{u -}

v

_Z

(12)

2 0 4 ROBERT F I N N

where def u = 8 9 ( ~ u ~ / O x j + ~ u j / S x i ) is the de/ormation tensor associated with the mo- tion.(1) These identities are to be understood in the sense t h a t u - T v = u ~ (Tv)~jnj and summation is extended over repeated indices.

From (8) and (12) we obtain the representation, valid for a n y solution of (4) in 6,

w(x)= f0X'(w-w0)'vw V,, (13)

where T X is formed by interpreting the components of ~ as pressures. One sees easily that, conversely, any vector field w (x) which satisfies (13) is a solution of the Navier-Stokes equations (4). We find similarly a relation for the pressure,

p ( x ) = ~ { w . T , - , . T w + ( , - w ) ( w 0 . n ) } d S y § f q , . ( w - w 0 ) . v w d V y ' (14)

r 1 1 ~ 1

,v,,ero wo ,,avo in ro uood corresponding to the vector

L \ ' A y / /

(x, y).

We collect here some elementary properties of the tensor X (x, y) for later re- ference. I n what follows, we assume (without loss of generality) t h a t the vector w 0 is directed along the positive xl-axis. We denote by ~ the (polar) angle made b y a r a y which starts from the point x, with the positively directed xl-axis , and b y r the distance from x to a point y of this ray. We present the estimates for X(x, y) as function of y for fixed x. Considered as function of x, all estimates for X (x, y) re- main true if ~ is replaced b y ( g - ~ ) . Since ~ is a function only of ( y - x ) , we m a y assume that x is the origin of coordinates. Letting I~1 denote an upper bound for the magnitudes of all components of ~, we then have for some positive constant C,

C 1 1 - e -"8 i) as r - ~ ,

Ixl<

_{r a s}

Ivxl<C

1-- e os _ a s e-~S al

(as)~

# ' where s = r + y l = r ( l § and a = l l w o ] ,

ii) for a n y integer N > 0, the N t h derivative ~(N) of ~ in a n y direction satisfies the inequality I X(N)I < C r 89 uniformly in ~ for sufficiently large r, (1) Throughout this paper, we denote volume integrals by S "'" d V, and integrals over closed surfaces by ~ ... dS.

O)7 T H E S T E A D Y - S T A T E S O L U T I O N S O F T H E N A P I E R - S T O K E S E Q U A T I O N S ~ I I I 2 0 5

iii) for a spherical surface ~n of radius R and center at x, there holds

txldS<C, .ITxldS<O,

iv) in a neighborhood of the singular point x = y ,

[:xI<Cr-', I V:xI<Cr -~.

P r o p e r t y iv) follows immediately from the definition ( 7 ) o f X(X, y). P r o p e r t y if) is obtained from a generalization of p r o p e r t y i) to higher derivatives. P r o p e r t y iii), except for the estimate on T X, follows easily from p r o p e r t y i). P r o p e r t y i) is obtained b y tedious b u t formal computation, starting from (7). We omit details. The analogous estimates for ~b (x, y) are obvious. We have, in fact, ~b (x, y ) = V (r-l). The estimate in iii) for [ "

I T x I d S

follows from this and from the estimate for

JE R

f ~ IV X[

d S.

Some further estimates on X(X, y) are included in [2]. We have given

R

here only those which are necessary in the present context.

We shall deal with solutions w (x) of (4) which are defined in a bounded region 6, and with solutions defined in a region ~ which contains a neighborhood of infinity.

I n either case we denote the boundary of the region b y ~. ~ is to consist of a finite n u m b e r of closed, connected component surfaces. B y a

smooth sur]ace ~

we shall m e a n a surface which admits in a neighborhood of each of its points a parametric representation b y means of functions which have continuous derivatives of all orders entering in the context. Although in m o s t cases slightly weaker assumptions will suf- fice, we will be safe to assume t h a t these functions are of class C (8). Correspondingly, we will usually assume t h a t the b o u n d a r y values w* of w are of class C (3) when considered as functions of the same parameters.

Throughout this paper the symbol C will be used to denote a positive constant, the value of which m a y however change even within a given context. Thus, from the relation fl < C (1 + a2) we m a y conclude fi < C a~ for all ~ > 1.

2. A priori estimation of the Diriehlet integral

I n this section we derive the bounds on the Dirichlet integral of solutions of (4), which are basic to the subsequent developments. We prove first a preliminary

1 4 : - - 6 1 1 7 3 0 5 1 . A c t a mathematica. 105. I m p r i m 6 le 30 j u i n 1961

206 ~OB~RT ~IUN

result concerning the possibility of c o n s t r u c t i n g solenoidal extensions of prescribed v e c t o r fields. (1)

LEMMA 2.1. Let W* be prescribed data on a smooth surface ~, which are o/ class C (3) on ~ and which satis/y the out/low condition ~ ( w * - n ) d S = 0 /or each component

J zi

~i o/ ~. Then there exists an in/inity o/ vector /ields ~b (x)which are de/ined througho~lt space, which vanish outside a neighborhood o/ ~, and are such that curl ~b ( x ) = w* on ~.

The /ield ~ (x) can be chosen to be bounded together with all its partial derivatives up to third order, the bounds depending only on the corresponding derivatives o/ w* with respect to suitable sur/ace parameters and on the smoothness o/ ~.

The v e c t o r field v ( x ) = curl ~b t h e n provides a solenoidal extension of t h e given d a t a w*.

L e m m a 2.1 is t r u e in a n y n u m b e r of dimensions. W e present here a proof for three dimensional case. W e consider first a representative c o m p o n e n t ~0 of ~, a n d introduce a p a r t i t i o n of u n i t y over ~0. T h a t is, we cover ~0 b y a finite n u m b e r of neighborhoods ~(0 k), k = 1 . . . /V, a n d define n o n - n e g a t i v e functions fk) of class C ~ on

N

~0 such that, a) each /(k) vanishes outside ~(0 k) and, b) ~ / ( ~ ) = 1 a t all points of ~0.

(For details of t h e c o n s t r u c t i o n see, e.g., De R h a m [15].) W e m a y assume t h a t t h e

~k) a n d /(k) are chosen in such a w a y t h a t each such n e i g h b o r h o o d a d m i t s a repre- s e n t a t i o n of class C (a) onto the interior of a plane u n i t disc F (k) a n d t h a t each /(k)= 0 in the a n n u l a r region consisting of all points in the disc whose distance f r o m t h e origin exceeds ~. Finally, we e x t e n d each representation to a m a p p i n g from a (thin) cylinder Z (~) of which F (k) is the mid-section, o n t o a n e i g h b o r h o o d of ~k), b y m a p p i n g t h e normals to the disc isometrically onto the normals to ~0 a t corresponding points, a n d we e x t e n d t h e p a r t i t i o n functions b y c o n s t a n c y along t h e normals.

N

A t each point of ~0 we h a v e w * = ~/(k) w*. D e n o t e b y ~1, ~2, the rectilinear

k = l

coordinates of the disc F (k) a n d b y ~3 the distance along t h e n o r m a l to t h e disc.

L e t A(1 k), A(~ ), A (~)~ be the c o m p o n e n t s of /(k)w*, a n d let

(1) This lemma has been used by several authors, but we know of no proof in the literature previous to a demonstration we have given in [2]. The proof presented here is due to Professor C.

Loewner (oral communication). It is more elementary than the one in [2], and it has the advantage that estimates on the extension field can easily be found from a knowledge of the corresponding esti- mates for w*.

O N T H E S T E A D Y - S T A T E S O L U T I O N S O F T H E N A V I E R - S T O K E S E Q U A T I O N S ~ IIl 207

where the quantities in parentheses denote J a e o b i a n s . D e n o t e b y p ( k ) t h e v e c t o r , p(k>= (p(k>, .(k) _~ , p(~)), p(k) is defined on F (~) a n d vanishes outside a circle of radius 1.

We n o w seek a v e c t o r field to = (co:, e%, o)3) defined in Z (k), such t h a t curl to = p(k) on F (k) a n d to = 0 whenever ~ + ~ >~ ~. I n general there is no such solution, for o n e sees easily t h a t a necessary condition is ( jr(k)P8 (k) d a : d o % = O . We therefore m o d i f y P(3 k) in order to achieve this condition. To do this, observe first t h a t each point.

where 0 < / ( ~ ) < 1 in ~ ) is interior to one of t h e other covering neighborhoods, s a y

~ ) , a n d 0 </o) < 1 at this point. We select a n e i g h b o r h o o d N of such a point w h i c h lies interior t o b o t h neighborhoods, a n d m o d i f y P(~)in N so t h a t ( P(~)doq do: s = O.

J r ^(k)

N

Simultaneously, we m o d i f y the corresponding t e r m in ~})~) so t h a t w * = ~/(~) w* re-

k = l

mains unchanged. W i t h this new function P(~) we determine a v e c t o r field ~ = (~5i, ~52, 0}

so t h a t curl ~ = P ( a ~) on F (z). We begin b y defining this field on F (~). W e m a y , for example, set 6 2 = 0 in F (~), s on the semicircle a ~ + a ~ = 1, a s < 0 , and de- termine iS: in F (z) b y t h e condition 8 r s = - P ~ ) . T h e n in the a n n u l a r region where P(~) = 0 we have 8 ~5:/~ a s - 8 t5~/8 a : = 0, hence there exists a function q (a:, ztz) such t h a t ~ = V ~. (The condition [ P(a ~) d ~x d zr s = 0 shows t h a t ~v is necessarily single

,]1~(k)

valued.) We m a y n o w e x t e n d ~ ( ~ , ~s) to the entire disc F (~), a n d define t o = ~ - Vcp

on F (k). Finally, we e x t e n d to to t h e entire cylinder Z (~) b y setting o) a - 0 a n d ex- tending to:, cos so t h a t D ws/~a~ = - p(k) a n d ~ t o l / ~ 2 =p(k) on F (k). W e m a y clearly also arrange t h a t to = 0 on the b o u n d a r y surface of Z Ck). The resulting field ca (al, as, as) t h e n satisfies the desired relation curl to = p(k) on F (~).

Since 70 is b y a s s u m p t i o n connected, a n y covering of the sort described h a s the p r o p e r t y t h a t for a n y two of the neighborhoods ~J) a n d ~.(0 k), there is a c h a i n of neighborhoods ~J), Z(0 ~ . . . 7(o ~), such t h a t each a d j a c e n t pair intersects a t p o i n t s

2 0 8 R O B E R T F I N N

where neither partition function vanishes. Thus, starting from a n y given neighborhood Z(01), it is possible by repetition of the above argument to construct, in a finite number of steps, corresponding fields to for each of the ~s) with the possible excep- tion of one last one, which we denote b y ~(0 N), i n which we can not modify the func- tion ~aD(N) without affecting the previous construction. But b y assumption,

N

while b y the nature of the construction,

(w*" n) d S = 0,

hence

~ , ( ~ l / ( j ) w* ) \ ] = 1 . n d S = 0 ~

f

z, (](N)w*).ridS=0,

see t h a t no modification of this last function is necessary. Thus a vector and we

field to(J) can be defined in a neigborhood of each of the images of the ~ ) such t h a t curl to(J)= p(s) in F (j), and to (s) vanishes on the boundary of Z (j).

We now transform these fields into a neighborhood of the original surface ~0- To do this, set

2 ~ ) = co~ j) ~ ~ ~(J) =

(~), ~), ~'))

~x~'

in a neighborhood of ~'), with summation extended over repeated indices, and set

N j = l

A simple calcu]ation then shows t h a t t~ is a field of the type sought, i.e., curl t~ = to*

on ~0. Repeating the entire procedure for each component of Z completes the con- struction of the field. (We must of course arrange what is easily d o n e - - t h a t the field constructed over each component vanishes over all o t h e r . ) T h e estimates on I t~]

and on its derivatives can be obtained directly from the method of construction.

2 a. Estimation o f the Dirichlet Integral in a bounded region

We consider first the interior region bounded b y a single closed surface ~. We suppose t h a t Z is smooth, and t h a t prescribed data w* of class C (3) are given on which satisfy condition (3). Applying Lemma 2.1, we obtain a vector field ~ ( x ) de-

O N T H E S T E A D Y - S T A T E S O L U T I O N S O F T H E N A V I E R - S T O K E S EQUATI01~S~ IYI 209 fined t h r o u g h o u t space, such t h a t curl t~ = w* on ~. W e m a y a s s u m e t h a t t~ (x) a n d its d e r i v a t i v e s u p t o third order are b o u n d e d , the b o u n d s depending only on ~ a n d

o n W * .

L e t 2 0 be chosen smaller t h a n a n y of the radii of c u r v a t u r e a t p o i n t s of ~ , a n d also so small t h a t all points on a n o r m a l line of length 2 4 originating f r o m a n a r b i t r a r y p o i n t P of ~ are closer to P t h a n t o a n y o t h e r b o u n d a r y point. T h e n in t h e shell region A~ d e t e r m i n e d b y the inequality 0 ~ s ~< 5, a non-singular c o o r d i n a t e s y s t e m is defined b y the n o r m a l s to ~ a n d the surfaces ~s of c o n s t a n t distance s f r o m along the normals, w i t h local surface coordinates on ~ o b t a i n e d f r o m those of ~ b y c o n s t a n c y along the normals.

) LEMMA 2.2 ( H o p f [5]). For any prescribed e > 0 , there is a real /unction ,~ (x with the /ollowing properties:

a) ~ (x) is de/ined in a neighborhood o/ 5 and has continuous derivatives up to the third order which are bounded, the bounds depending only on ~, on ~ ( x ) and

o n E .

b) 2 ( x ) = l on ~, 2 ( x ) = O outside A~, c) v ~ ( x ) = 0 on Y,

d) I curl ~ ~ l < s s 1 throughout .,4~.

Such a ~t(x) can be o b t a i n e d as a n o n - n e g a t i v e f u n c t i o n of s alone. A possib]e construction is as follows:

L e t M = m a x ] d~], M 1 = m a x ] curl ~k]. Choose 4 0 < 4 a n d sufficiently small t h a t

Adt A~

2 M 14 0 < s, a n d define

f ( ;o;

e ' 1 1 - d(r

for all s~> s 0. H e r e s o is the unique value of s d e t e r m i n e d b y the conditions

~ t ( s ) = l - - - - , S O < s < ~ o.

~o

I t is t h e n clear t h a t for s ~< So, ~ (s) can be defined w i t h continuous t h i r d d e r i v a t i v e s in such a w a y t h a t ~ (0) = 1, 2' (0) = 0, 0 ~< I~(s) l ~< 1, a n d ] ~' (s) l < 2--~s, in the inter- v a l 0 ~< s < s 0. I n particular, I~t ' (s)] < 2 ~ s E in the entire r a n g e 0 < s < 4 o, and, using t h e i d e n t i t y curl 2d~ = ~ curl ~ - d~• V 2, we easily o b t a i n the desired estimates.

2 1 0 ROBERT FINN

Consider now a solution w (x) of (4) defined in the region ~ bounded b y a smooth closed surface ~, such t h a t w ( x ) = w * on ~.

T ~ E O R E ~ 2.3 (Leray [8]). Let w* be o/ class C (3) on ~, and let w(x) be a solu- tion o/ (4) in ~ such that w ( x ) = w * on ~. Then n [ w ] = I ' I V w [ 3 d V is bounded by

ig

a quantity which depends only on ~ and on the derivatives o/ w* up to second order, and not on the particular solution considered.

Proo[. Corresponding to the data w* we introduce a field v ( x ) = curl ~ ~ with t h e properties indicated in L e m m a s 2.1 and 2.2. L e t ~ l = w - v . We rewrite (4) in the

f o r m

A ~ - ~ ' V ~ I - V p = - A v + ~ l " V v + v " V v i + v " V v, (16) V "v 1 = 0 ,

multiply the first equation b y v l, and integrate over 6. We obtain, since Yi = 0 on and since v = 0 outside A~,

B y L e m m a 2.2 and the Schwarz Inequality, we find

(18)

where K denotes a m a j o r a n t for quantities which depend only on X, on w*, and on e- Integrating along a normal to X, we find

7 d s . . . . s 2 ~ s d s

~12ds)89 ( f ~'2ds) 89

hence ~ d s < 4 I V ~ le d s,

from which we conclude immediately

f ,4 ~-~2 d V ~ K l f ,4 I V ~q [ 2 d v , ⁽¹⁹⁾

ON TILE S T E A D Y - S T A T E SOLUTIONS OF T H E I~AVIER-STOKES EQUATIOI~S, I I I 2 1 1

K 1 depending only on the maximum curvature of ~ (and not on w*). Inserting this result in (18) and choosing ~ = 1 / 2 K ~ , we obtain

ol v ~l~dV <4K~.

B u t from w = ~q + v we find

f ,vwi2dV<~2 f lv~l~dV+2 f ivvl~dV

from which Theorem 2.3 follows immediately.

2 b. Estimation o f the Dirichlet Integral in an exterior region; case o f zero outflux

I t cannot be expected t h a t an estimate on Dirichlet Integral in an exterior re- gion depending only on prescribed data can be obtained, even for solutions which tend to a limit at infinity and for which this integral is finite, cf. the example in w 6.

Under suitable assumptions, however, such an estimate does exist. We consider here two such cases. The region of definition for the solution is assumed to be the exterior of a smooth closed surface ~, ~ to consist of a finite number of connected com- ponents carrying prescribed data w* of class C (~). Denote b y ER the region bounded b y ~ and b y the surface of a sphere 2R centered at the origin and with radius R sufficiently large t h a t ~ lies interior to En. Let w 0 be a prescribed (constant)vector.

TH]~OR~M 2.4 (Leray [8]). Let w(x) be a solution o/ (4) in ER such that w ( x ) = w * on 5 and w ( x ) = w 0 on ~R. Then D [ w ] = f [ V w [ 2 d V is bounded by a

d ^ER

quantity which depends only on w*, on ~, and on w 0 (and not on R).

Theorem 2.4 is true in a n y number of dimensions. To prove it, we introduce a new field u (x) = w (x) - w 0. I n terms of u (x), equations (4) become

A u - u . V u - w o. V u - V p = O (20)

V . u = 0 and the boundary conditions become

u = u * = w * - w 0 on u - 0 on ~R.

The remainder of the proof follows very closely the proof of Theorem 2.3. We choose v ( x ) = c u r l ~ d ? so t h a t v ( x ) = u * on ~, v ( x ) = 0 outside As, and set v l= u - v , The

2 1 2 R O B E R T F I N N

r e l a t i o n s (16) become

A ~ i - v i . V ~ i - W o . V~ i - V p = - A v + ~ I . V v + v . V ~ i + v . V v + w 0- V v , V "~l =O,

a n d (17) becomes, since ~q = O o n ~ a n d o n ~R,

f ,v 12dV=-f v 'vvdV§ f v' 'v dV

+ /A v ' v ' v ~ d V + /A v ' W o V ~ d V .

W e o b t a i n (18) a n d (19) as before. C o m b i n i n g these i n e q u a l i t i e s a n d choosing a g a i n e = 1 / 2 K 1 ~, t h e r e q u i r e d e s t i m a t e follows.(1)

T h e sense i n which T h e o r e m 2.4 applies to solutions defined t h r o u g h o u t t h e ex- terior of ~ is i l l u s t r a t e d b y t h e following corollary, which we shall a p p l y i n w 5.

COROLLARY. Let (Rj} be a sequence o/ values tending to in/inity and let wRj denote a sequence o/ solutions o/ (4) in ~Rj such that wRj= w* on ~ and wRj= w0 on ~aj.

We suppose /urther that the sequence {wRj}, together with all partial derivatives up to /irst order, converges at all points in the exterior ~ of ~, uni/ormly in any compact subregion o/ ~, to a vector field w(x). Then D [ w ] = t~ I V w l 2 d V < . M < co, where M depends only on w*, on ~, and on w 0.

L e t R 0 be a r b i t r a r y b u t fixed. B y T h e o r e m 2.4, we h a v e

f I V w R , 1 2 d V ~ f IVwR,12dV<M<oo for all R , > R o.

~Ro ,~Rj

B y u n i f o r m convergence i n Eno, there follows [ ] V w 12 d V < M. Since R o is a r b i t r a r y , J

we m u s t h a v e / ] V w l 2dV<~M, q.e.d.

J s

(1) Note added in proo]: I n order to obtain an indication of the suitability of this method for finding energy integral estimates in a practical case, Mr. Paul L. Patterson has used the method to estimate the Dirichlet Integral for the explicitly known solution of (6) which vanishes on a sphere of radius a and which tends to a limit w 0 at oo. From the choice,

= (0, - w0 x3, 0), (~0 = 4 a ,

;t = ~ 4 ( ~ o - s)8 (3 s + ~o),

he has obtained the estimate, f e [ V w ] ~ d V < 31 x~ a w02. The actual value, computed from the known solution, is 6 ~ a w~. We remark however, that in the non-linear case, the estimate tends rapidly to infinity as the Reynolds' number increases. We do not know in what sense this situation reflects reality.

ON T H E S T E A D Y - S T A T E S O L U T I O N S OF T H E : N A V I E R - S T O K E S E Q U A T I O N S , n I 2 1 3

The second case we consider is t h a t in which the solution is defined throughout the exterior E of ~ and tenlis to ~ limit at a suitable rate.

TI~EOREM 2.5. Let w(x) be a solution o/ (4) in g, and let w ( x ) = w * on 5. Sup- pose there exists an e > 0 such that as r--> oo, [ w ( x ) - w o I < C r - 8 9 -~ /or some constant C and constant vector w04:0. Then D [ w ] = t ' ] V w ] 2 d V < M < c ~ , where M depends only on ~, on w*, and on w 0.

The main burden for the proof of Theorem 2.5 rests on estimates given in [2]

for asymptotic behavior of solutions of (4) in E. The demonstrations are too com- plicated to reproduce here. I t is shown in [2] t h a t the hypotheses of Theorem 2.5 imply, in particular, the estimates

for u (x) = w (x) - w 0 and p (x) as R--> ~ , provided p (x) is modified b y a suitable additive constant. Using these estimates, we introduce a sphere ~n and apply the reasoning in the proof of Theorem 2.4 to w (x) in the region En. The only change t h a t occurs is in the surface integral over ~n, which no longer vanishes but assumes the value

u . ~ - n - l U l 2 ( u - n ) - l u ( w 0 . n ) - p ( u . n ) d S .

The above estimates show t h a t this term vanishes in the limit as R - ~ co. This fact established, the proof of Theorem 2.5 then coincides with t h a t of Theorem 2.4.

2 c. E s t i m a t i o n o f the Dirichlet Integral in an exterior region; general case We show now that in the case of an exterior region in three or more dimen- sions, the conditions (3) ~ (w*'n) d S = 0 which we have imposed on the boundary

d Z

data can be relinquished, and a bound on Dirichlet Integral obtained nevertheless for a n y possible solution of a suitable class, provided t h a t the net outflux through is sufficiently small. For simplicity, we restrict the discussion to the three dimen- sional case in which X consists of a single connected component. The method fails in two dimensions, but in every other respect these restrictions are unnecessary. L e t Q be the net flux across X,

Q= ~ ( w * . n ) d S . (21)

J

214 R O B E R T F I I q l i

Then since V .w = 0 in E, Q is necessarily the net flux across a n y surface ~R e n - closing ~, and hence if Q # 0 the boundary condition w = w 0 on ~a cannot be ful- filled b y a n y solution. The simplest choice available to us is obtained b y adjoining to w 0 the velocity field of a potential source flow whose singularity lies interior to Z.

T H E O R E ~ 2.6. Let w(x) be a solution o/ (4) in ER such that w ( x ) = w * on and w ( x ) = W o - ~ V on ~ . Then i/ Q is su//iciently small (depending only on ~), the Dirichlet Integral D [ w ] = { t ~7 w l 2 d V is bounded by a quantity which de-

al

pends only on w*, on ~, on Wo and on Q (and not on R).

We choose the origin of coordinates interior to ~, and let ~, ( x ) = - 4 ~ "

Then on ~, the d a t a u * - y satisfy condition (3), hence for a n y e > 0 t~here exists a field h ( x ) = c u r l ;tt~ of the type described in L e m m a s 2.1 and 2.2, such t h a t h (x) = u* - 1( on ~. L e t v (x) = 1( + h in ER. Then v (x) = u* on Y, and setting ~1 = u - v, we have ~ = 0 on ~ and on ~R. F r o m (20) we find

A v l - ~ . V ~ - w 0 - V ~ - V p = - A y + ~ l - V v + v - V Y l + v - V v + w 0 . V v, v .vl=O.

Multiplying b y ~i a n d integrating over E~ yields, since A I( = 0,

-- f~R Y.v.v~dV- f~R v'w~

We s t u d y the right hand side t e r m b y term:

(ef. the proof of Theorem 2.3), and an integration b y parts, using the relations V ' v I = O ' Y = - 4 ~

f,,, 1( v dV= f

,~ l (A ~" v * where V**l denotes the transpose of the m a t r i x V *1. Also,

ON T H E S T E A D Y - S T A T E S O L U T I O ~ I S Ol~ T H E ] f f A V I E R - S T O K E S EQUATI02~S~ I I I 215

(; )~

< K Iv l dr ,

~R

K depending only on prescribed data, since y4 is integrable over the exterior of ~.

Similar reasoning shows t h a t

f RV'Wo'V dVI<K(fe lvnI dV) +"

Collecting these estimates we obtain

D [Vl] ~< K (D [~l]) + + e K x D [~1] + ~ Q D[~I], (22) 4:r r 0

where r o is the shortest distance from the origin to X. From (22) we see t h a t an estimate for D [~], independent of R and the particular solution considered, can be obtained whenever Q < 4 : r r 0. Using this estimate, we can find a bound for D [u] from the inequality

D I n ] ~< 2 D [)]] + 2 D [ v ] ~< 2 D [~] + 4 D [ h ] + 4 D [ y ]

and the fact t h a t y has finite Dirichlet Integral over the region exterior to Z.

We remark finally t h a t the Corollary to Theorem 2.4 and the conclusions o / T h e o - rem 2.5 are valid also i n this case.

3. A priori estimation of the solution

The estimates on Dirichlet Integral obtained in the preceeding section are applied here to find pointwise estimates for a n y possible solution, depending only on pre- scribed data. We base these estimates on general properties of the Green's Tensor associated with the linearized equations

• w - v p = o (6)

A . w = 0

in a bounded region. We consider in detail only the three dimensional case. I t seems certain that similar estimates hold for two dimensional solutions, b u t the necessary estimates on the Green's Tensor have not y e t been formally verified in this case. I n higher than three dimensions, there appears to be an intrinsic difficulty in the method.

216 ROBERT FINN

T~EOREm 3.1 (Odqvist [11, p. 365]). For any interior region ~ bounded by a su/.

/iciently smooth closed sur/ace ~, there is a unique tensor G (x, y ) = (G~j), such that each column vector, considered as /unction o/ x, is a solution o/ (6) /or all x in ~ with x # y , such that G ( x , y ) = 0 /or x on ~ and y in 6, and such that at x = y , G(x,y) has the singularity o/ the ]undamental solution tensor X (x, y) de/ined in (7). Throughout + ~, the tensor G (x, y) and the (suitably normalized) associated pressure vector P (x, y) admit, as /unction o/ x, uni/ormly in y, the estimates

r I - e

I fi (x, y)[ < - - C I fi (x, y) - fi (x', y)[ < C xx" (23)

r x y r ~

[ V fi (x, y) [ <

r~-,C i e (x, y) i < r ~-c

x y x y

r 1 -e

I V (G (x, y) - fi (x', y))[ < r~

IP(x, y ) - P (x', Y)I < c - -

r 1 - e XX p

r~

/or any e > 0 , provided x and x' lie exterior to a sphere o/ radius r o centered at y.

G (x, y) has the symmetry property, G~j (x, y) = Gj+ (y, x).

We shall assume Theorem 3.1 and use it in what follows. Using the Green's Tensor, any solution of (6) in ~ which equals w* on ~ admits the representation

w ( x ) = ~ w * ' T G d S (24)

Z

p ( x ) = f w * ' T P d S ,

where the "pressure" used in forming the expression T P is the identically vanishing function. Conversely, the representation (24) leads to the construction of a solution in ~ which assumes boundary data w* on ~. We have, in fact:

T H e O r E m 3.2 (Odqvist [ii]). Let w*(x) be boundary data o/ class C (~) and saris- lying condition (3) on the smooth closed sur/ace ~. Then there is a unique solution w (x) o/ the linearized equations (6) in the region ~ bounded by ~, such that w (x)= w* on ~.

Throughout ~ + ~, w (x) and the suitably normalized associated pressure p (x) satis/y, /or any e > 0, the inequalities

Iw(x)l<c, I v w ( x ) l < c , Ip(x)l<v, (25)

~ r l - ~ IVW(X)--VW(y)I <Crl-~xy, IP(x)--P(Y)I < v x y ,

ON THE STEADY-STATE SOLUTIONS OF THE NAVIER-STOKES EQUATIONS, III 217 where C depends only on w*, on ~, and on e. A t any interior point o[ 6, the deriva- tives o/ w (x) o] all orders are bounded, depending only on Z, on w*, and on distance ]rom the point to ~.

Proo/. The function w (x) defined b y (24) is clearly a solution of (6) in 6. Ap- plying L e m m a 2.1, we m a y extend w* into the interior ~ as a divergence-free vector field v (x) with bounded second derivatives, depending only on the b o u n d a r y data.

L e t ~ q = w - v . Transforming (6) b y the identities (12) yields, for x in 6, f c, . A v dV

p(x)= f V. A v dV.

The assumption of boundary data b y w (x) and the estimates (25) then follow from the estimates (23) of Odqvist b y standard potential-theoretic arguments. To obtain the interior estimates, we represent w(x) b y means of the fundamental solution tensor (7), which is explicitly known,

w (x) = f ~ (w*" T X - X" T w*). (26)

Interior bounds then follow directly b y differentiation of (26) under the integral sign and use of the estimates (15).

We shall need also the following lemma, due to P a y n e and Weinberger [13]:

LEMMA 3.3. Let y (x) be a vector valued ]unction de/ined and piecewise continuously di//erentiableO) in a neighborhood N o/ in/inity. Then there exists a constant vector Yo such that /or any sphere ~R which lies, together with its exterior ER, in N,

F r o m L e m m a 3.3 we can obtain the following r e s u l t :

L~MMA 3.4..Let y(x) be a vector valued ]unction de]ined and pieeewise continuously di//erentiable throughout space. Then there exists a constant vector Yo such that, /or any choice o/ the origin o/ coordinates,

(1) That is, continuous in N and continuously differentiable except on a finite number of smooth surfaces.

218 I~OB EXiT ~ l q ~

fz 1u176 3 ( 1 2 d V , r 2 I v (2S)

the integration being taken over the entire space E.

We m a y assume t h a t . f l V 3 ( 1 2 d V < ~ , for otherwise L e m m a 3.4 is tri- Proo/.

vially correct. L e t 3(0 be the vector whose existence is asserted in L e m m a 3.3. Integra- tion b y parts in a sphere V~ of radius R and b o u n d a r y ~R yields

r~ d V = - v~ r r " V ( 3 ( - 3(o)~ d V + ~ ~ Using Schwarz' Inequality and L e m m a 3.3 yields

where e ( R ) - + 0 as R - - > ~ . L e m m a 3.4 then follows b y a passage to the limit.

Remark 1. I n the special case t h a t the function 3((x) vanishes outside a compact set, the surface integral in (29) vanishes for sufficiently large R, and it is unnecessary to a p p l y L e m m a 3.3 in the proof.

Remark 2. I n case it is known t h a t 3( tends to a limit a t infinity, this limit coincides with Y0- F o r otherwise we would have R = I ~

necessarily

J ZR

as R --> co, a contradiction.

3 a. E s t i m a t i o n o f the solution in a bounded region

We assume as given a solution w (x) in a finite region 6, which takes on values w* on the boundary ~ of 6. I t is then possible to extend w* to the exterior of in such a w a y t h a t the extension vanishes outside a c o m p a c t set and has finite Dirichlet Integral, depending only on w*. B y Theorem 2.3, w (x) has interior to ~ a finite Dirichlet Integral, depending only on w*. Denote the sum of these two inte- grals b y D. Using the identities (13, 14), we find for w(x), p(x) the representation,

W ( X ) : (30)

J Z Y~

O : ~ T H E S T E A D Y - S T A T E S O L U T I O N S O F T H E : N A V I E R - S T O K E S E Q U A T I O N S , I I I 219 The first term on the right is a solution of the linearized system (6) which assumes the boundary data w*. This solution, which we denote b y w 1 (x), satisfies the in- equalities (25) of Theorem 3.2. Using Theorem 3.1 and Lemma 3.4, we find (cf. the remarks at the end of w 1),

(f0 )~

[w(x)l<lw (x)l+c dV I vwl~dV

< C + D < C ,

constants depending only on prescribed data. Further,

y~ f~ v w(y)l

Ivw(x)[<lVWl(X)[+ Iw(y)[

_{r 2}

[vw(Y)ldVy<C+c

_{r 2}

dVy.

x y x y

(31)

Multiplying by rx~ and integrating with respect to x, we obtain

r2 dV <~ C

z x 6 r z y

by Schwarz' Inequality. Inserting this result in (31), we obtain

I vw(x)l<c.

Placing these results in the relation (30) for the pressure and using Theorems 3.1 and 3.2, we obtain immediately

Ip] <c,

and the estimates (23) for the Green's Tensor imply, b y standard potential theoretic arguments,

[ V w ( x ) - V w ( Y ) ] <Crl-~xr [ P ( x ) - p ( Y ) t < C r l x y ~"

Collecting these results, we obtain:

T H E O ~ E ~ 3.5 (Leray [8]). For any bounded region 6, the estimates (25) o/ Theo- rem 3.2 are valid also /or any solution o/ the Navier-Stolces equations (4), with constants independent of the particular solution considered.

3 b. E s t i m a t i o n o f the solution in an exterior region

We base the estimates for an exterior region on the following general property of solutions of the Navier-Stokes equations:

2 2 0 ROBERT FINN

T ~ n O R E M 3.6. Let w ( x ) be a solution o/ (4) in a region 6 (finite or infinite) which the Dirichlet Integral is finite, J ~ I V w l 2 d V < ~ . Suppose it i8 possible to /or

extend w (x) to a piecewise continuously di[]erentiable vector field defined in the entire space, possessing finite Dirichlet Integral and tending(1) to a limit w o as x --> oo. Then in every compact subregion o[ 6 , w (x) and the associated pressure p (x) satis/y the inequa- lities (25) o/ Theorem (3.2), with constants depending only on the Dirichlet Integral o]

w (x) and o[ its extension, on distance to the boundary ~ o/ 6, and on w 0. The estimate /or p (x) depends also on the choice o/ an additive constant, and on the particular sub.

region considered.

Pro@ L e t x be a point of 6 , let 4 d be the distance f r o m x t o the b o u n d a r y . Describe spheres S 1 a n d $2 of radii d a n d 2d, respectively, w i t h x as center. W e t h e n h a v e the r e p r e s e n t a t i o n

w(x)=f

_S~

w.T•dS+f

_V2

^-w.vwdV,

⁽³²⁾

where V 2 is the interior of S~ a n d G (x, y) is the Green's Tensor associated w i t h the s y s t e m (6) in V~. W e rewrite (32) in the f o r m

u ( x ) = ~ u . T G d S + f G . u . V u d V + f G ' w 0 " v u d V ,

8 2 V 3 V~

where we h a v e set u (x) = w (x) - w 0.

F o r x interior to $1, I T G ( x , y ) [ < C d -~ on S s. Thus,

L e m m a 3.3, ~ ( w - w o ) 2 d S < 2 d D , where D is the s u m of Dirichlet I n t e g r a l s B y

d S~

of w (x) a n d its extension. Also, b y T h e o r e m 3.1,

(f ; dV Ivwl dV C

V~ V~ V~]

b y L e m m a 3.4. Finally,

elw.lf Ivwl2dV <C.

_V,

Collecting these inequalities, we o b t a i n a n e s t i m a t e of the f o r m (i) I t is sufficient, tha~ w-~w o in the sense of Lemma 3.3.

ON T H E S T E A D Y - S T A T E S O L U T I O N S OF T H E N A V I E R - S T O K E S E Q U A T I O N S ~ I I I 221

which establishes the interior bound on I w (x)I.

Now we permit x to v a r y within S 1. We observe first that

+ u. TGdS

^can

J $2

be differentiated under the sign for x in

V1,

resulting in a uniformly bounded func- tion, depending only on the bound for l ul on s 2. We then obtain from (32),

[ V w ( x ) [ < c + c f

L~-~ IdV

⁽³³⁾

V2

for all x in V r Multiplication by rx2 and integration over V 1 yields

f ldV< c+cf f vwl2dV<C.

V~ V.

Repeating this reasoning with V 1 replaced b y V 2 and V 2 b y Va shows that

V~

Insertion in (33) yields the bound on IV wl. The estimates (23), together with (32) and the corresponding relation for p (x), imply by the usual methods of potential theory the remaining estimates (25). The function p (x) is, however, determined in a given sphere only up to an additive constant. If determinations in overlapping spheres are to coincide, one of them must be adjusted by a suitable constant. Thus (for example), along any path of length L covered b y spheres in ~ of radius bounded from zero, the determination of p (x) m a y conceivably change b y an amount C L. I n particular, Theorem 3.6 does not provide a uniform bound for p(x) in an infinite region.

We study next a particular case. The region 0 has as boundary component a smooth connected closed surface 2. I t is assumed t h a t on 2, w takes on data w* of class C (a). The same assumptions on Dirichlet Integral are made as in Theorem 3.6.

Let +40 be a neighborhood of 2 in ~ which contains no boundary points and which is bounded by ~ and b y a smooth closed surface ~0 in ~. Let ,"tl be another such neighborhood, such t h a t ~1 lies in M0. On 20, estimates on l wl are available from Theorem 3.6. Let x lie in , 4 r We m a y write

w(x)= ~ w.TGaS+f~

G ' w - V w d V ,

i!]+Z0 0

where G (x, y) is the Green's Tensor for the system (6) relative to M0-

1 5 - 61173051. A c t a mathematica. 105. I m p r i m g le 30 j u i n 1961