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VueHpmM
m1^m29
O<0<1,
(le)m1 +
Actually (cf. S. Agmon [1]), the inequality (2.21) is also valid if (2.22)
0^m1<^<m2,
( l  d ) m 1 + 0m2 = ^ ,
2
i.e., 6=
2
m2 — m1
In particular, if n = 2,
(2.23)
O lul 1/2 lul 1/2 S V 6H
W™ KU&
"
P^)>
and because of (2.16)
(2.24)
i"^>={ c c uriA«r
E(aAiaa:,kip),*Q:1i*iipv*'. P=I/«. p'=i/a«).
—
Vu€D(A);
n
6m2>.
12
PART I.
QUESTIONS RELATED TO SOLUTIONS
if n = 3 r IIJI 1 / 4 IMI 3 / 4
{,
1 S «
C U!
(2.26)
\U\2
l^«tNP^
Vu€H2p(Q), VM£D(A)

The form b. We now show how to apply these properties of Sobolev spaces to the study of the form b. Let 0 be an open bounded set of Un which will be either ft or Q. For u, v9 weV(C), we set (2.27)
b ( u, u , v, u,w ) = 2w)= X
MDi%Wiix,
i,j = l •'O
whenever the integrals in (2.22) make sense. In particular, we have: LEMMA 2.1. Let 0 = ft or Q. The form b is defined and is trilinear continuous on H m <0)xH m * + 1 (0)xH m <0) where m ^ O , and ml + m2+m3^
ifm^,
mi + m2+m3>
ifm{
i = l,2,3,
(2.28) for some i.
Proof. If mi < n / 2 for i = 1,2, 3, then by (2.18) H m <(0)c L q '(0) where l/q4 = \  mjn. Due to (2.28), (1/q! + l/q 2 + l / q 3 ) ^ 1, the product ui(PiCj)wi is integrable and b(u, u, w) makes sense. By application of Holder's inequality we get n
\b(u, v, w) ^ £ IUIII/MO) lA^li/w) IW/ILW), b(u, l), w)glCi Mm, I^U+l w m3 .
If one or more of the m^s is larger than n/2 we proceed as before, with the corresponding qt replaced by +<*> and the other q,'s equal to 2. If some of the m t 's are equal to n/2, we replace them by m'i<mi, mi — m'i sufficiently small so that the corresponding inequality (2.29) still holds. • Remark 2.2. i) As a particular case of Lemma 2.1 and (2.29), b is a trilinear continuous form on V m t xV m 2 + 1 x Vm3, m, as in (2.28) and (2.30)
b(u, v, w)\^c% \u\ni \v\m2+1 \v\m3
V U G Vmi, ve Vm2+1, w e Vmy
In particular, b is a trilinear continuous form on V x V x V and even on VxVxV1/2. ii) We can supplement (2.29)(2.30) by other inequalities which follow from (2.29)(2.30) and (2.20)(2.26). For instance the following inequalities combining (2.30) and (2.20) will be useful. (2 31)
b(U
' *
W)1
*
C 2 W  1 / 2 lU 1/2 I  U   1 / 2 ! A U P / 2  W 
'
"
V U € V , U G D ( A ) , weH,
ifn = 2,
FUNCTIONAL SETTING OF THE EQUATIONS
13
\b(u, v, w) ^ c 3 u N 1 / 2 1Au 1 / 2 \w\ (232)
V u e V , veD(A),
w e H , ifn = 3.
iii) Less frequently, we will use the following inequalities. We observe that UiiDiV^Wj is summable if (for instance) ut e V°(0)t D{v^ w, € L2(C), and
11
UiDiVjWj dx
klL^lA^IIWyl.
In this manner, and using also (2.16), we get in the case n = 2 Au1/2WIH I(t.wV2AwW
\h( M< \D(u,v,w)\_c
x(l"l
1/2
V u e D ( A ) , t ) € V , weH, V u e H , i)6 V , w e D ( A ) ,
and in the case n = 3 \b(u v ^ c x f
w f
"
 A u l i / 2 M W
VueD(A),t,eV, weH,
Finally we recall a fundamental property of the form b: (2.33)
b(u, t>, w) = b(u, w, u) V u, u, w € V.
This property is easily established for w, u, w e V (cf. [RT, p. 163]) and follows by continuity for u,v,we V. With v = w, (2.33) implies (2.34)
b(u,v,v) = 0
\fu,veV.
The operator B. For u, u, w € V we define B(u, u) € V and Bu e V by setting (2.35)
(B(u, v), w) = b(u, v, w),
Bu = B(u, u).
Since b is a trilinear continuous form on V, B is a bilinear continuous operator from V x V into V. More generally, by application of Lemma 2.1 we see that (2.36)
B is a bilinear continuous operator from V ^ x V ^ + i (or from Hmi(0)xW^+1(0)) into V_m3, where ml9m2,m3 satisfy the assumptions in Lemma 2.1.
It is clear that various estimates for the norm of the bilinear operator B(, •) can be derived from the above estimates for b. 2.4. Variational formulation of the equations. As indicated before, we are interested in the boundary value problem (1.4), (1.5), (1.8), (1.11), when the average of u on O is 0; u 0 and / are given, and we are looking for u and p. Let T > 0 be given, and let us assume that u and p are sufficiently smooth, say u e ( g 2 ( R n x [ 0 , T]), p e ^ f l T x[0, T]), and are classical solutions of this problem. Let u(t) and p(t) be respectively the functions {x€(Rn H* U(X, f)}, {x eUn »> p(x, r)}. Obviously u G L 2 (0, T; V), and if v is an element of V then by multiplying (1.5) by v9 integrating over Q and using (1.4), (1.11) and the
14
PART I.
QUESTIONS RELATED TO SOLUTIONS
Stokes formula we find (cf. [RT, Chap. Ill] for the details) that u v jat («• ) + ^
(2.37)
v
» +b ^
M u
> ) = & u)«
By continuity, (2.37) holds also for each veV. This suggests the following weak formulation of the problem, due to J. Leray [ U [2], [3]. Problem 2.1 (weak solutions). For f and u 0 giuen, feL2(0,T;V),
(2.38) (2.39)
u0eH,
find u satisfying U€L 2 (0, T; V)
(2.40) and
(2.41)
4at( M ' u ) + ^ M ' ^ ) + b ( M ' u ' i; ) = ^ i ; > Vu€V,
(2.42)
u(0) = wo2
lf u merely belongs to L (0, T; V), the condition (2.42) need not make sense. But if u belongs to L 2 (0, T; V) and satisfies (2.41), then (cf. below) u is almost everywhere on [0, T] equal to a continuous function, so that (2.42) is meaningful. We can write (2.41) as a differential equation in V by using the operators A and B. We recall that A is an isomorphism from V onto V and B is a bilinear continuous operator from V x V into V . If ueL 2 (0, T; V), the function Bu:{t*+ Bu(t)} belongs (at least) to L\0, T; V). Consequently (2.41) is equivalent to d — (u, v) = (f  vAu  Bu, v), dt and since /  v A u  B u e V ( 0 , T; V ) , u' = du/dt belongs to 1/(0, T; V ) , and (2.43)
^+vAu
+ Bu=f
in V .
Furthermore, (cf. [RT, Chap. Ill, § 1]) u is almost everywhere equal to a continuous function from [0, T] into V , and (2.42) makes sense. We refer to [RT] (and the sequel) for more details and in particular for the relation of Problem 2.1 to the initial problem (1.4), (1.5), (1.8), (1.11). One can show that if u is a solution of Problem 2.1, then there exists p such that (1.4), (1.5), (1.8), (1.11) are satisfied in a weak sense (cf. [RT, p. 307]). Of course a weak solution of the NavierStokes equations (Problem 2.1) may or may not possess further regularity properties. For convenience we will introduce a class of more regular solutions which we call strong solutions.
FUNCTIONAL SETTING OF THE EQUATIONS
15
Problem 2.2 (strong solutions). For f and u0 given, (2.44)
/eL2(0,T;H),
(2.45)
u0eV,
find u satisfying (2.46)
u € L2(0, T; D(A)) n L°°(0, T; V)
and (2.41)(2.43). Further regularity properties of strong solutions are investigated in § 4 (for the space periodic case) and § 6 (for the bounded case). Let us observe here that, by application of (2.32) and (2.41)(2.42), (2.47)
B(ifc * )  ^ c 3 U\\3'2 A
Hence if u is a strong solution, then for almost every t, \Bu(t)\^c3\\u(t)\\3/2\Au(t)\l/2,
(2.48)
and the function Bu: {t >* Bu(t)} belongs to L 4 (0, T; H). Since f and Au belong to L 2 (0, T; H), u' = /  M u  B u belongs to this space too, and u'eL2(0,T;H).
(2.49)
The two conditions u e L 2 ( 0 , T; D(A)), u'eL 2 (0, T; H) imply by interpolation (cf. J. L. LionsE. Magenes [1] or [RT, Chap. Ill § 1.4), that u is almost everywhere equal to a continuous function from [0, T] into V: (2.50)
u e « ( [ 0 , T ] ; V).
2.5. Flow in a bounded domain. Although up to now we have concentrated on the space periodic case, all of what follows applies as well to the case of flow in a bounded domain with a fixed boundary (1.4), (1.5), (1.8), (1.9) with= 0, O bounded, provided we properly define the different spaces and operators. In this case (cf. [RT], Y = {vec€~(Q)n,
divu = 0},
V = the closure of Y in Hl0(£l) = {veHl0(a),
divu=0},
H = the closure of V in L2(fl) = {ve L2(ft), div v = 0, yvv = v • v r = 0}, H ^ i n L2(ft)) = G  {v, v = Vp, p e D(A) = VHH2(n) = {veHj(fl)
Au = Pku
H\0)},
HIHI2(n), div v = 0},
VueD(A),
where P is the orthogonal projector in L2(H) onto H. Most of the abstract results in § 2.2 remain valid in this case. We cannot use
16
PART I.
QUESTIONS RELATED TO SOLUTIONS
Fourier series any more, so that neither can A~1f be explicitly written, nor can the eigenfunctions of A be calculated. Still, A is an isomorphism from V onto V and from D(A) onto H, but this last result is a nontrivial one relying on the theory of regularity of solutions of elliptic systems (cf. S. AgmonA. DouglisL. Nirenberg [1]), L. Cattabriga [1], V. A. Solonnikov [1], I. I. VorovitchV. I. Yodovich [1]). The spaces Va = D(Aa/2),a>0, (which will not be used too often) are still closed subspaces of H"(ft), but their characterization is more involved than (2.15) and contains boundary conditions on T. All the inequalities in § 2.3 apply to the bounded case, replacing just H™(Q) by Hm(fl) (assuming that (1.7) is satisfied with an appropriate r). The proof of (2.20) 4 , which was elementary, relies in the bounded case on the theory of interpolation, as well as on the definition of H m (ft), meIR\N (cf. J. L. LionsE. Magenes [1]). Finally, with these definitions of A, V, H , . . . , § 2.4 applies to the bounded case without any modification. 4
In the bounded case, (2.20) becomes
3
Existence and Uniqueness Theorems (Mostly Classical Results)
In this section we derive basic a priori estimates for the solutions of NavierStokes equations, and we recall the classical existence or uniqueness theorems of weak or strong solutions. The only recent result is the theorem of generic solvability of NavierStokes equations, given in §3.4 and due to A. V. Fursikov [1]. 3.1. A priori estimates. We assume that u is a sufficiently regular solution of Problems 2.12.2, and we establish a priori estimates on u, i.e., majorizations of some norms of u in terms of the data u0, / , . . . . i) By (2.41), for every fe(0, T) and (3.1)
ve.V,
(«'(!), t>) + v((u(t), v)) + b(u(t\ u(f), v) = (f(t\
v).
Replacing v by u(t) we get, with (2.34), + v\\u(t)\\2 =
(u'(t\u(t))
(f(t),u(t)).
Hence (3.2)
\ ± \u(t)\2 + v \\u(t)f = (f(t), u(t))S /(0v lu(r)
z at
s>(0l!2+^/(0ll2v., 2 (3.3)
2v
2
«(r) 4vu(0lPg^/(0 2 v..
By integration in t from 0 to T, we obtain, after dropping unnecessary terms, •T
f \\u(t)\\2dt^Ku
(3.4)
Jo
(3.5)
K^K^UoJ,
v, T) = ^ (l«o2 + ; [ ll/(0H2vdf).
Then by integration in t of (3.3) from 0 to s, 0 < s < T , we obtain u( S ) 2 guo 2 + ^ S   / ( 0 l  2 v ' ^ (3.6)
sup
\u(s)\2^K2,
se[0,T]
(3.7)
K2 = K2(u0,f,v,T)
= vK1.
ii) Assuming again that u is smooth, in (3.1) we replace v by Au(t): (u'(t), Au(t)) + v((u(t), Au(t))) +17b(u(t), u(t), Au(t)) = (f(t),
Au(t)).
18
PART I.
QUESTIONS RELATED TO SOLUTIONS
Since (3.8)
= ((*, i«)
V<J>,iAeV,
this relation can be written ~\\u(t)\\2 2 at
+
v\Au(t)\2^b(u(t),u(t\Au(t))
(3.9)
„ i (f(t),Au(t))^\Au(t)\2+\f(t)\2. 4 v
=
Now the computations are different, depending on the dimension, iii) Dimension n=2. We use the relation (2.31); (3.9) implies ^ M O f + ! v  A u ( t )  2 S  /(0 2 + 2c 2 u(0 1/2 Au(t) 3/2 u(0. The righthand side can be majorized by   / W  2 + ^  A u ( 0  2 + c;u(r) 2 u(0 4 , using Young's inequality in the form (3.10)
ab^ea* + cebp\
l,
e>0,
p'=~3T, p i
ce =
(
p ?~pi> p e
with p = I and e = v/2. We obtain
(3.ii)
^IU(OII 2 +VAU(OI 2 ^I/(OI 2 +C; M (OI 2 II«WII 4 . at
v
Momentarily dropping the term v Au(f) 2 , we have a differential inequality, y ' ^ a + 0y, (3 12)
7 a(f) = /(f) 2 , 6(t) = c[\u(t)\2\\u(t)\\2, v from which we obtain by the technique of Gronwall's lemma: y(r) = u(r)2,
j { (y(r) exp (  P y(t) H y(0) exp ( [
$(T)
0(T)
dr)) S a(t) exp (  P dr) + f a(s) exp (J
0(T)
0(T)
dr),
dr) ds,
or u(02==uo2exp ( £ ci
U(T) 2 U(T) 2
dr)
(3.13) + * P l/(s) 2 exp (Jf'ci
U(T) 2 U(T) 2
dr) ds.
EXISTENCE AND UNIQUENESS THEOREMS (MOSTLY CLASSICAL RESULTS)
19
With (3.4)(3.7), (3.14)
sup
\\u(t)f^K3,
le[0,T]
(3.15)1
K3 = K3(u0, f, v, L) = (u 0  2 +^ f l/(s) 2 ds) exp (cJ^JCj),
We come back to (3.11), which we integrate from 0 to T: v\T\Au(t)\2dt^\\u0\\2+\T\f(t)\2dt (3.16) (3.17)
+ c'1 sup \u(t)\2\\u(t)\\4,
\Au(t)\2dt^K4,
f
K4 = K4(u0,f, v,L) =  (u 0  2 + [T\f(i)\2dt
+ c[K2K2X
iv) Before treating the case n = 3, let us mention an improvement of the preceding estimates in the periodic case, for n = 2. This improvement, which does not extend to the case of the flow in a bounded domain or if n = 3, is based on: LEMMA 3.1. In the periodic case and if n = 2, (3.18)
fc(
)  b
Ve fr{A) = H*(Q) H V.
Proo/. In the periodic case A = A, and 2
= I i,j,k = l
^(D^Dl^dx, J
Q
and, by integration by parts, using the Stokes formula,
(3.19) M,A<M=£ f <MWVM* + I f i,Uk JQ
U,k
D^D^D^dx.
JQ
Now both integrals vanish, the first one because
and the second one because the sum Zu,k = i DkPi Pk i vanishes identically (straightforward calculation). • 1
c 2 (and therefore c[) depends on the domain 0, i.e., on L if 0 = Q.
20
PART I.
QUESTIONS RELATED TO SOLUTIONS
This lemma allows us to transform (3.9) into the simpler energy equality i j t \\u(t)\\2 + ν \Au(t)\2 =
(3.20)
Au{t)\
from which we derive, as for (3.4)(3.7), (3.21)
at
^\\u(t)\\2+V\Au(t)\2S\f(t)\2, ν
sup \\u(t)\\2^K^\\u0\\2+
(3.22)
/(02
te[(),T]
£ \Au(t)\2dt^K'4
(3.23)
,
V Jo
= ^K^.
v) Dimension η = 3. In the case η = 3, we derive results which are similar to that in part iii) (but weaker). After (3.9) we use (2.32) instead of (2.31), and we obtain ^   u ( 0   2 +  v  A u ( i )  2 â   / ( t )  2 + 2c 3 u( ( ) 3/2 A M (t) 3 ' 2 at 2 ν S/(r) 2 4^Au(i) 2 + ^u(i) 6 v 2 (by Young's inequality), £ ii(t)2 + ν \Au(t)\2^\f(t)\2 + c^ u(t)6. at ν
(3.24) But (3.25)
H^LAu
VveD(A\
and    u ( i )   2 + νλ, \\u(t)\\2^ \f(t)\2 + c^ at ν
(3.26)
6
·
This is similar to (3.11), but instead of (3.12), the comparison differential inequality is now y'^y3, (3.27)
y(i) = l + u(f)2,
<£ = J a x ( c £ ,  sup /(i) 2 ). V
V te[0,T]
We conclude that y(t) =
(0)
Vl2y(0) 2 cir
/
EXISTENCE AND UNIQUENESS THEOREMS (MOSTLY CLASSICAL RESULTS)
21
as long as r
3.2. There exists a constant K6 (== 3/8ci) depending only on f, v, Q, T
such that u(f) 2 ^2(l + u02)
(3.28) for (3.29)>
rsTi(Mo1) =
_ ^ _ .
It follows that if n = 3 and u is a sufficiently regular solution of Problems 2.12.2 then (assuming T ^ T ) (3.30)
sup  M (r) 2 ^K 7 = 2(l + !u02), teCO.T,]
7\ given by Lemma 3.2, and from (3.24), \T,\Au(t)\2dt^K8, (3 31)
*
W
2f
\
V \
V J0
/
3.2. Existence and uniqueness results. There are many different existence and uniqueness results for NavierStokes equations. The next two theorems collect the most typical results, obtained in particular by J. Leray [1], [2], [3], E. Hopf [1], O. A. Ladyzhenskaya [1], J. L. Lions [1], J. L. LionsG. Prodi [1], and J. Serrin [1]. THEOREM 3.1 (weak solutions). For f and u0 given, (3.32)
/GL2(0,T;V),
u0eH,
there exists a weak solution u to the NavierStokes satisfying (3.33)
equations (Problem 2.1)
u e L 2 (0, T;V)Pi L°°(0, T ; H)
as well as (2.39) (or (2.43)) and (2.40). Furthermore, if n = 2,u is unique and (3.34)
u€<£([0,T];H),
(3.35)
u'eL2(0,T; V).
• t ^ T^IIUQII) and obviously t ^ T.
22
PART I.
QUESTIONS RELATED TO SOLUTIONS
If n = 3, u is weakly continuous from [0, T] into H: (3.36)
ue«([0,T];Hw),
and (3.37) THEOREM
(3.38)
U'GL4/3(0,T;V).
3.2 (strong solutions, n = 2, 3). i) For n = 2, f and u0 given, /<EL°°(0,T;H),
U0GV,
there exists a unique strong solution to the NavierStokes equations (Problem 2.2), satisfying3: (3.39) (3.40)
u G L 2 (0, T; I?*A)),
"' e L 2 (0, T; H),
u e « ( [ 0 , T ] ; V).
ii) For n = 3, f and u0 given, satisfying (3.38), there exists T*=T*(u0) = min (T, T^HuolD), T^Huoll) given by (3.29), and, on [0, T J , there exists a unique strong solution u to the NavierStokes equations, satisfying (3.39), (3.40) with T replaced by T*. Remark 3.1. i) The theory of existence and uniqueness of solutions is not complete for n = 3: we do not know whether the weak solution is unique (or what further condition could perhaps make it unique); we do not know whether a strong solution exists for an arbitrary time T. See, however, § 3.4. ii) We recall that as long as a strong solution exists (n = 3), it is unique in the class of weak solutions (cf. J. Sather and J. Serrin in J. Serrin [1], or [RT, Thm. III.3.9]). iii) Due to a regularizing effect of the NavierStokes equations for strong solutions, those solutions can have further regularity properties than (3.39)(3.40) if the data are sufficiently smooth: regularity properties will be investigated in § 4 for the space periodic case and in § 6 for the bounded case. Let us also mention that if n = 2 and in (3.38) we assume only that u0eH, then the solution u is in L 2 ^ ((0, T ] ; D(A)) and «((0, T]; V). Remark 3.2. The strong solutions (and the weak solution if n = 2) satisfy the energy equality (3.2). For n = 3 we know only that there exists a weak solution which verifies the energy inequality ~ l « ( « )  2 + v « u ( r ) f S on (0, T). 2 at It is not known whether all the weak solutions satisfy this inequality or whether this inequality is actually an equality. Remark 3.3. Let us assume that the conditions (3.38) are verified. Then for n = 2, if u is a weak solution to the NavierStokes equations (Problem 2.1), u is automatically a strong solution by uniqueness and Theorem 3.2. For n = 3, 7 G L 2 (0, T; H) is sufficient for n = 2.
EXISTENCE AND UNIQUENESS THEOREMS (MOSTLY CLASSICAL RESULTS)
23
the same is true on (0, T*(u0)), but u will be a strong solution on the whole interval [0, T] if u satisfies some further regularity property. For instance, if a weak solution u belongs to L 4 (0, T; V), then u is a strong solution: indeed u is a strong solution on some interval (0, T*), T*^kT, Let us assume that the largest possible value of T* is V < T. Then we must have i*
II
/ Ml
limsupu(0 = + 00 . t—T'O
On the other hand, the energy inequality (3.24), which is valid on (0, T'e) all e > 0, implies d_ (l+u(r)2)^c(l+u(r)2)3, dt
for
and by the technique of Gronwall's lemma, lhu(r) 2 ^(l4u 0  2 )exp (c£ X (l^u(5) 2 ) 2 ds),
0
Thus i*(f) is bounded for t  * T 0, contradicting the assumption that T < T. 3.3. Outlines of the proofs. The proofs of Theorem 3.1 and 3.2 can be found in the original papers or in the books of O. A. Ladyzhenskaya [1], J. L. Lions [1], [RT]. We finish this section with some outlines of the proofs which we need for the sequel. i) We implement a Galerkin method, using as a basis of H the eigenfunctions wj9jeN, of the operator A (cf. (2.17)). For every integer m, we are looking for an approximate solution um of Problems 2.12.2, (3.41)
Um=L Zjm(t)Wj, J=
l
um : [0, T]»> Wm = the space spanned by w^ . . . , w„ The function u^ satisfies, instead of (2.39)(2.40), (3.42)
(i4m,v)
+ v((um,v)) + b(um,um,v)
= (f,v)
VveWm,
(3.43) where
u m (0) = P m u o ,
(3.44) 4
P m is the orthogonal projector in H onto Wm.
The semiscalar equation (3.42) is equivalent to the ordinary differential system (3.45)
^ + vAUn + PmBum = Pmf.
4 Due to the definition of the Wy's, P m is also the orthogonal projector on V^, in V, V , D(A),....
24
PART I.
QUESTIONS RELATED TO SOLUTIONS
The existence and uniqueness of a solution II*, to (3.42)(3.45) defined on some interval (0, T m ), T m > 0 , is clear; in fact the following a priori estimate shows that Tm = T. ii) The passage to the limit m —» °o (and the fact that Tm = T) is based on obtaining a priori estimates on u^. The first estimate, needed for Theorem 3.1, is obtained by replacing v by u^ (=1^(1)) in (3.42). Using (2.34) we get
(3.46)
  \um(t)\2 + v IK«)! 2 = (/, 1^(0).
This relation is the same as (3.2), and we deduce from it the bounds on um analogous to (3.4)(3.7):
jjMrtll 2 dt^~ («0m2+^ [T/(0ll2v dt), sup  u m ( 0  2 ë (  u O m  2 + te[0,T]
\
\T\\f(t)\\l.dt).
V J0
/
Since w 0m  = P m u 0  = u 0 , we get for u^ exactly the same bounds as (3.4)(3.7), from which we conclude that (3.47)
um remains in a bounded set o/L 2 (0, T; V)nL°°(0, T; H).
Then, as usual in the Galerkin method, we extract a subsequence u^, weakly convergent in L 2 (0, T; V) and L°°(0, T; H ) , , . ,Q* (3.48)
Jin L 2 (0, T; V) um>»u<. r f l e m _ „ . (in L (0, T ; H )
weakly and weakstar.
The passage to the limit in (3.42), (3.43) allows us to conclude that u is a solution of Problem 2.1 and proves the existence in Theorem 3.1. However, this step necessitates a further a priori estimate and the utilization of a compactness theorem; we will come back to this point in a more general situation in § 13. Finally, we refer to [RT] for the proof of the other results in Theorem 3.1. iii) For n = 2, the only new element in Theorem 3.2 is (3.39)(3.40). We obtain that U G L 2 ( 0 , T; D(A))nL°°(0, T; V) by deriving further a priori estimates on i v , in fact, a priori estimates similar to (3.14)(3.17). We obtain them by taking the scalar product of (3.45) with Au^. Since P m is selfadjoint in H and PnAUn = APmum = Au^, we obtain, using (3.8), (3.49)
i    u m   2 + i ,  A u m  2 + b(u m ,w m ,Au m ) = (/,A Mm ).
This relation is similar to (3.9). Exactly as in §3.1, we obtain the bounds (3.14)(3.17) for i^, with u0 replaced by u0m. Since u 0m =Pmu0 and P m is an orthogonal projector in V, (3.50)
u 0m HP m Uo^uo,
EXISTENCE AND UNIQUENESS THEOREMS (MOSTLY CLASSICAL RESULTS)
25
and we then find for u^ exactly the same bounds as for u in (3.14)(3.17): (3.51) um remains in a bounded set o/L 2 (0, T; D(A)) n L°°(0, T; V)
(n = 2).
By extraction of a subsequence as in (3.48), we find that u is in L 2 (0, T;D(A))nL°°(0, T; V). The continuity property (3.40) results from (3.39) by interpolation (cf. J. L. LionsE. Magenes [1] or [RT]). The fact that u ' e L 2 ( 0 , T;H) follows from (2.43), (3.52)
u' = fvAuBu,
and the properties of B;f and Au are obviously in L 2 (0, T; H ) and for B we notice in the relation (2.31) that (3.53)
\Bu(t)\ ^ c2 \u(t)\1/2 II(f) IAu(r) 1/2 ,
so that Bu belongs to L 4 (0, T; H ) and u' belongs to L 2 (0, T;H). The proof of Theorem 3.2 in the case n = 3 is exactly the same, except that we use the estimates (3.30), (3.31) valid on [0, Tjflluoll)] instead of the estimates (3.14)(3.17) valid on the whole interval [0, T]. Also, instead of using (2.31) and obtaining (3.53), we use the relation (2.32) which gives us (3.54)
\Bu(t)\^c3\\u(tW/2\Au(t)\1/2;
Bu is still in L 2 (0, T; H ) , and u' is too. Remark 3.4. Except for Lemma 3.1 and (3.20)(3.23), the a priori estimates and the existence and uniqueness results are absolutely the same for both the space periodic case and the flow in a bounded domain with u = 0 on the boundary (cf. §2.5). If u==£0 on the boundary and/or ft is unbounded, similar results are valid. We refer to the literature for the necessary modifications. 3.4. Generic solvability of the NavierStokes equations. We do not know whether Problem 2.2 is solvable for an arbitrary pair u0, /, but this is generically true in the following sense (A. V. Fursikov [1]): THEOREM 3.3. For n = 3, given v90 (= Q or ft) and u0 belonging to V, there exists a set F, included in L 2 (0, T; H) and dense in (3.55)
L«(0,T;V)
Vq,
lMq<$,
such that for every feF, Problem 2.2 corresponding to u0,f, possesses a unique solution (strong solution). Proof, i) Since L 2 (0, T;H) is dense in L q (0, T; V ) , l ^ q < i it suffices to show that every feL2(0, T,H) can be approximated in the norm of L q (0, T; V) by a sequence of fm% fm e L 2 (0, T; H ) , such that Problem 2.2 for Wo>/m possesses a unique solution. For that purpose, given /, we consider the Galerkin approximation um described in § 3.3 above (cf. (3.41)(3.45)). It is clear that u „ is in L 2 (0, T; Wm) and hence in L 2 (0, T; D(A)) and that um is continuous from [0, T] into Wm and hence into D(A). Now for every m we consider also the solution vm of the
26
PART I.
QUESTIONS RELATED TO SOLUTIONS
linearized problem (3.56)
^=+vAvm=0,
(3.57)
um(0) = ( I  P m ) u o .
It is standard that the linear problem (3.56)(3.57) possess a unique solution satisfying in particular u m eL 2 (0, T;D(A))nL°°(0, T; V).
(3.58)
We then set wm = um + vm and observe that wm satisfies (3.59)
w m e L 2 ( 0 , T ; D ( A ) ) n L ~ ( 0 , T ; V),
and by adding (3.56) to (3.45) and (3.57) to (3.43), (3.60)
wm(0) = uo,
(3.61)
^+IAwm+Bwm=/+gm,
(3.62) gm =  ( I  P m ) / + ß ( u m ) + ß(i; m ,u m ) + B(u m ,t) m ) + ( /  P m ) ß ( w m ) . The proof will be complete if we show that, for m —*• °°, (3.63)
gm»0
in L"(0, T; V),
lSq<§.
ii) Since  ( I  P m ) / ( f )   » 0 for m >°o for almost every t, and (JP m )/(t)S /(t), it is clear by the Lebesgue dominated convergence theorem that (I— P « ) /  * 0 in L 2 (0, T; H) for m — ». Multiplying (3.56) by Av m we get ^\\vm(t)\\2+2v\Avm(t)\2 dt
= 0,
l k , ( 0 I P + 2 v £ \Avm(s)\2 dsë(/P m )u 0  2 , from which it follows that, for m —» », (3.64)
um^0
in L 2 (0, T; D(A))
and
L"(0,T;V).
Using (3.64), the estimate (3.47) for u„, and Lemma 2.1 (with m ^ O , m 2 = 1, m3 = l), we find } o !B(«m, vm)\\2v. dt S c  T liUOl 2 A üm (t) 2 dt  0. In a similar manner we prove that BO^O
and
B(um,i;m)^0
inL2(0,T;V)
for m —> oo, and the proof of (3.63) is reduced to that of (3.65)
(IPJB0O+0
inLq(0,T; V),
form^oo.
EXISTENCE AND UNIQUENESS THEOREMS (MOSTLY CLASSICAL RESULTS)
27
iii) The proof of (3.65) is technical and we give only a sketch of it. The sequence i*™ converges to some limit u (cf. (3.48))5. Since Bue 4/3 L (0, T; V')6, and since, by Lebesgue's theorem, (IPm) Bu^0 in L 4/3 (0, T; V ) , it suffices to show that (3.66)
Bun + Bu
strongly in L q (0, T; V ) .
Due to (2.33) and Lemma 2.1, \b(, , )\ ^ c ^ /4 U\\
v, * ;
therefore B is a bilinear continuous operator from V3/4 x V 3/4 into V (see also (2.36)) and
(3.67)
lB+vSc+g» V<*>.
It follows from the proof of Theorem 3.1 (cf. the cited references) that u^ converges to u strongly in L 2 (0, T; Vi«) and L 1/e (0, T; H) for all e > 0 . By (2.20) with m1 = 0,m2=le,Om2 + (le)ml = e(le) = l, and the Holder inequality, lUlifc4*s(^'l«tM
1  l
*y
C 1
e )
([
T
"m"lts^)
2 e
,
l/p = e ( l  0 ) + 0/2. We conclude that i ^  ^ u in L p (0, T; V 3/4 ), and with (3.67), that Bu^^Bu in L p/2 (0, T; V). Finally, since p < 8 / 3 , p  * 8 / 3 , as e —> 0, we choose e sufficiently small so that p/2 = q. D Remark 3.5. As indicated in Remark 3.1, the theory of existence and uniqueness of solutions is not complete when n = 3, while it is totally satisfactory for n = 2. It was Leray's conjecture on turbulence, which is not yet proved nor disproved, that the solutions of NavierStokes equations do develop singularities (cf. also B. Mandelbrot [1], [2]). It seems useful to study the properties of weak solutions of NavierStokes equations with the hope of either proving that they are regular, or studying the nature of their singularities if they are not. The results in §§ 4, 5 and 8 tend in this direction. Of course they (as would Theorem 3.3) would lose all of their interest if the existence of strong solutions were demonstrated.
5 6
In (3.48) a subsequence um, of um converges to u, but this is sufficient. Cf. [RT]; this is how (3.37) is proved.
LL New A Priori Estimates and Applications In this section we establish some properties of weak solutions of NavierStokes equations. The results are proved for the space periodic case, but they do not all extend to the bounded case (cf. the comments, following § 14). We assume throughout this section that n = 3 and the boundary condition is space periodicity. 4.1. Energy inequalities and consequences. We derive formal energy inequalities assuming that u0, /, u are sufficiently regular. LEMMA 4.1. If u is a smooth solution of Problems 2.12.2 (space periodic case, n = 3), then for each f > 0 , for any r ^ 1,
(4.1)
u(0?+vM(t)? +1 SL r (l + «(r)lu(or (2r  1) )
where the constant L, depends on the data, v, Q and Nr_x(f) = l/ltto/riV,»)Moreover, for any r ^ 3 , we have (4.2)
j f u(t)?+» \u(t)\U ë L'r(l + u(0?) 2r+1 ,
where L'r depends also on v, Q and Nrx(f). Proof, i) We take the scalar product in H of (2.43) with Aru, and we obtain ~  u  ? + v  u  ? + 1 = (/,w)r(Bii,A^). By the définition (2.42) of B and as Au = Au in the space periodic case (cf. (2.14)), we can write (4.3)
^jt\u\^v\u\f+1
(f,u)r(lYb(u,u,Aru).
=
The first term in the righthand side of (4.3) is majorized by /(0,i  u ( 0 U ^  u ( t )  ? + i +  N r _ x ( / ) 2 . 4 v The second term is a sum of integrals of the type (4.4)
UiDiUjAUjdx or JQ
UiD^D^Dl^Dl^Uj
dx,
ateN,
a 1 + a 2 + a 3 = r.
JQ
We can integrate by parts using Stokes' formula; the boundary terms on dQ cancel each other due to the periodicity of u, and the integrals take the form (4.5)
I Da(uiDiui)DOLuidx,
Da = D^D%*D%\ 29
30
PART I.
QUESTIONS RELATED TO SOLUTIONS
With Leibniz' formula, we see that these integrals are sums of integrals of the form (4.6)
f uJDJD*up~uidx,
and of integrals of the form (4.7)
f O^SrkAu,DX
k = l,...,r,
JQ
where 8k is some differential operator Da with [a] = a 1 + a 2 + a 3 = k. The sum for i = 1, 2, 3 of the integrals (4.6) vanishes because of the condition div u =0. Then it remains to estimate the integrals (4.7). ii) Proof of (4.1). By Holder's inequality, the modulus of (4.7) is less than or equal to I S ^ ( O U Q ) I S ' ^  V O U Q ) DU(I). From the Sobolev injection theorems (H1(Q)C=L6(Q), HV2(Q) c L3(Q), cf. (2.18)), this term is less than or equal to c'i\u(t)\k+m\u(t)\r+2k\u(t)\„ C'I depending on fc, r, Q. We then apply the interpolation inequality (2.20) with m ^ l , m 2 = r + l , 0 = fc/rl/2r, ( l  0 ) m 1 + 0m2 = k+e and 0 = l  ( k  l ) / r , ( l  0 ) m 1 + 0m2 = r + 2fc, to get u(r) k + 1 / 2 ^c 2 u(r)ir k /  1 / 2 nu(r)^r 1 / 2 r , u(f) r+2  k Sci iMWß^lMWlr1^"1^. Therefore the modulus of the integrals of type (4.7) is bounded by an expression i,2 /2
cï\u(t)\\ '\u(t)\}:i '\u(t)i.
Hence 6(u, u, A'«)âc' 4 u(()}1/2r W(t)\)X\n' \u(t)\r
(4.8)
s  «(0l?+i+c5 tt(r)ß lu«!4"2'"1 (by Young's inequality).
This relation with (4.3) and (4.4) gives (4.1). iii) Proof of (4.2). We majorize b(u, u, Aru) in a slightly different way. We write (4.9)
\u(t)\r^cf5\u(t)\\/r\u(t)\l~lfr
NEW A PRIORI ESTIMATES AND APPLICATIONS
31
by application of (2.20) with mt•* 1, m 2 = m + 1 , 0 = 1  1/r. Then (4.8) gives 6(u, u, A'u)^ ci u(r)ir i/2r l"(02+x1/2r (4.10)
v
£u(*)? + l +c4 u(r)f +2
(by Young's inequality).
This relation combined with (4.3)(4.4) gives (4.2). D An immediate consequence of (4.2) is that u remains in Vr as long as \\u(t)\\ remains bounded. This is expressed in LEMMA 4.2. If u0eVr and feU°(0, T; V ^ ) , r ^ l , then the solution u to Problem 2.2 given by Theorem 3.2 ii) belongs toand since P w is self adjoint in H and P ^ ' i ^ = Arum9 we get
2 A Mml?+ " Mml ' +1 = * *•* " ( ~ i m U m ' * AfUm)This is similar to (4.3) and thus, exactly as in Lemma 4.1, we get the analogue of (4.2):
 te(t)t+v MOB* ^ tXl + l"m(0lf)2r+1; due to (3.51) we conclude that
liUOI^cl+liUO)!? forOsrsiT*, Jo
Now Pm is a projector in Vr too, and P m u 0  r ^u 0  r , so that u^ remains in a bounded set of L°°(0, T*; Vr) and L2(0, T*; VP+1), and (4.11)
ueL~(0, T*; V r )HL 2 (0, T*; V r+1 ).
We then check that Bu, and therefore u' = fvAuBuy belongs to L2(0, T*; Vr_x); the continuity of u from [0, T*] into Vr follows by interpolation (cf. J. L. LionsE. Magenes [1] or [RT, §111.1]). ii) For u0e V, we observe that the solution u of Problem 2.2 belongs to L 2 (0, T*; D(A)). Thus u(t)eD(A)=Vr almost everywhere on (0, T*), and we can find tx arbitrarily small such that u(tt)e V2. The first part of the proof shows that u€%([t l9 T*]; V J O L 2 ^ ; T*; V3). Hence u(f 2 )€ V3 for some t2e [*i, TJ, t2 arbitrarily close to tu and ue<€([t2, T J ; V 3 )nL 2 (f 2 , T*; V4). By induction we arrive at u €<#([*,_!, T*]; V ^ n L 2 ^ , T*; Vr_!) and, since *,_! is arbitrarily close to 0, the result is proved. D
32
PART I.
QUESTIONS RELATED TO SOLUTIONS
4.2. Structure of the singularity set of a weak solution. Let m ^ 1. We say that a solution u of Problem 2.12.2 is VTregular on (tu t2) (0^t1^t2) if ue^dti, t2);Hm(Q)). We say that an H m regularity interval (fl512) is maximal if there does not exist an interval of D0mregularity greater than (tx, t2). The local existence of an H m regular solution is given by Lemma 4.2: if UQG Vm and /eL°°(0, T; Vmi), then there exists an H m regular solution of the NavierStokes equations defined on some interval (0, t0). Also, it follows easily from Lemma 4.2 that if (tu t2) is a maximal interval of H m regularity of a solution u, then (4.12)
lim sup u(r) m = +<». t+t20
We are now able to give some indications on the set of H m regularity of a weak solution. THEOREM 4.1 (n = 3, space periodic case). We assume that u0eH, fe L°°(0, T; Vmi), m ^ l , and that u is a weak solution of the NavierStokes equations (Problem 2.1). Then u is Hmregular on an open set of (0, T) whose complement has Lebesgue measure 0. Moreover, the set of Hmregularity of u is independent of m, i.e. is the same for r = 1 , . . . , m. Proof. Since u is weakly continuous from [0, T] into H, u(t) is well defined for every t and we can define Xr={te[0,Tlu(t)mr(Q)l nr={te[0,Tlu(t)eW(Q)}, + e);Hr(Q))l Ür={te(0,T),3e>0,uec€((tE9t It is clear that Cr is open for every r. For r =. 1, since u e L 2 (0, T; V), Si has Lebesgue measure 0. If t0 belongs to Hi and not to Ot then, according to Theorem 3.2 ii), t0 is the left end of an interval of H ^regularity, i.e. one of the connected components of d. Thus ili\Oi is countable and [0, T]\Oi has Lebesgue measure 0. The theorem is proved for m = 1. We will now complete the proof by showing that Om =01. If (*!, t2) is a connected component of 0% (a maximal interval of (HPregularity), then for every t[ in this interval, u(t[)e V and, according to Lemma 4.2, there exists a unique H m regular solution defined on some interval (t[, t2), t[
New a priori estimates. 4.2 (n = 3, space periodic case). We assume that u0eH, fe L 2 (0, T; Vmi) and that u is a weak solution of the NavierStokes equations THEOREM
NEW A PRIORI ESTIMATES AND APPLICATIONS
33
(Problem 2.1). Then u satisfies (4.13)
ueLM0,T;H;(Q)),
f \u(t)\«
r = l , . . . , m + l, where the constants cr depend on the data v, Q, u 0 ,/, and the ar's are given by (4.14)
. 2rl Proof, i) Let (at, ft), ieN, be the connected components of €u which are also maximal intervals of H m regularity of u. On each interval (ah ft), the inequalities (4.1) are satisfied, r = 1 , . . . , m, and we write them in the slightly stronger form
(4.i5)
ar=
 wm+v I«(OI?+1 *i»(i+iu(t)i?)(i+NWI^^«.
Then we deduce
W*>l»«>g
+
IS^Ëti
Ä r(i + u(t)ß)
By integration in t from at to ßj? we get
~i2r~ " fi+WR o)i;)""" + ( 2 r " J ) (i+«(«,+o)i?)"""
Hta+I^"**^ 1 ''''»* From (4.12), the first term in the lefthand side of this inequality vanishes, since (a t , ft) is a maximal interval of H m regularity. Thus
r ( ui!T(^>* a H*" + """ e> * By summation of these relations for i e N , we find, since u e L 2 ( 0 , T; V),
(416)
tu+iÄ^»*^
r=1
 •m
ii) The proof of (4.13) is now made by induction. The result is true for r = 1. We assume that it is true for 1 , . . . , r, and prove it for r + 1 (r^km). We have
(by Holder's inequality),
34
PART I.
QUESTIONS RELATED TO SOLUTIONS
where 2r ar+l 2 2r— 1 2 2 —a r+1
1 _ ar 2r— 1 2 '
Therefore (4.13) follows for r + 1 , due to (4.16) and the induction assumption. D Remark 4.1. As indicated at the beginning of this section, all the results have been established in the periodic case. We do not know whether they are valid in the bounded case (although it is likely that they are) except for the following special results: (4.1) and (4.2) for r = 1, which coincide with (3.24); in Theorem 4.1 the fact that the complement of Ox in [0, T] has Lebesgue measure 0 and is closed; and (4.13) for r = 2(a r =§) (same proofs)1. The following interesting consequence of Theorem 4.2, which relies only on (4.13) with r = 2, is valid in both the periodic and the bounded cases. THEOREM 4.3. We assume that n = 3 and we consider the periodic or bounded case. We assume that u0eH and feU°(0, T; H). Then any weak solution u of Problem 2.1 belongs to L a (0, T;L°°(0)), Ü = ÜorQ. Proof. Because of (2.26)
iu(oiLwêciiu(oni/2iAu(or2, and by Holder's inequality
Jju(0 L  (o) Sc(jAu(r) 2 ' 3
^
Regularity and Fractional Dimension
If the weak solutions of the NavierStokes equations develop singularities, as Leray has conjectured, then a natural problem is to study the nature of the singularities. B. Mandelbrot conjectured in [l][2] that the singularities are located on sets of Hausdorff dimension < 4 (in space and time), and V. Scheffer has given an estimate of the dimension of the singularities which he successively improved in [l}[4]. A more recent (and improved) estimate is due to L. CaffarelliR. KohnL. Nirenberg [1]. In this section we present some results concerning the Hausdorff dimension of the singular set of weak solutions, following (for one of them) the presentation in C. FoiasR. Temam [5]. 5.1. Hausdorff measure. Time singularities. We recall some basic definitions concerning the Hausdorff dimension (cf. H. Federer [1]). Let X be a metric space and let D > 0 . The Ddimensional Hausdorff measure of a subset Y of X is (5.1)
M Y ) = Hm to,em = s u p t o , e ( Y ) , e^O
e > 0
where (5.2)
ILDÛ(Y) = inf X (diam B,) D , j
the infimum being taken over all the coverings of Y by balls By such that diam B, ( = diameter of B/) ^ e. It is clear that fxDe(Y)^/LLD,e<(Y) for e ^ e ' and fx D (Y)e[0, +oo]. Since jx D , e (Y)^e D ~%i D o e (Y) for D>D0, then if JLLDO(Y)D0. In this case the number inf{D,im D (Y)0} = inf{D,ui D (Y)<œ} is called the Hausdorff dimension of Y If the Hausdorff dimension of a set Y is finite, then Y is homeomorphic to a subset of a finite dimensional Euclidean space. Finally, let us also mention the useful fact that jxD() is countably additive on the Borel subsets of X (see Fédérer [1]). The first result on the Hausdorff dimension of singularities concerns the set of t in [0, T ] on which u(t) is singular, u(t)£ V. Actually this is a restatement by V. Scheffer [1] of a result of J. Leray [3] (cf. also S. KanielM. Shinbrot [1]). THEOREM 5.1. Let n = 3, 0 = Q, or Q, and let u be a weak solution to the NavierStokes equations (Problem 2.1). Then there exists a closed set <£<=[(), T] whose ^dimensional Hausdorff measure vanishes, and such that u is (at least) continuous from [0, T]\€ into V. Proof, i) The proof of this theorem is partly contained in that of Theorem 4.1 (cf. also Remark 4.1). We set # = [0, T]\Ou and we only have to prove that the ^Hausdorff measure of € is 0. 35
36
PART I.
QUESTIONS RELATED TO SOLUTIONS
Let (ah ft), iE7, be the connected components of Ot. A preliminary result, due to J. Leray [3], is
Ità«,) 1 ' 2 «».
(5.3)
Indeed, let (ah ft) be one of these intervals and let t e(ah ft). According to Theorem 3.2 and (3.29),
where K6 depends only on /, v and 0 ( = il or Q). Thus (ft_(*1/2gl
+ u(0l2,
<€(<*, ft).
Then we integrate on (<*;, ft) to obtain
2VF6(ßi*i)mMßi<*i)+ PlMOII2 dt, and we add all these relations for iel to get
2v / ^I(ft« i ) 1/2 ^T+ fTM(r)2dr0 , we can find a finite part Ie of I such that
(54)
X (ft«JSe,
I(fta ( ) 1/2 Se.
it h
Uh
The set [0, T]\Uiei e («» ft) is the union of a finite number of mutually disjoint closed intervals, say Bf, j = 1 , . . . ,N. It is clear that UfU Bj 3 '> and since the intervals (ah ft) are mutually disjoint, each interval (ah ft), i^/ e , is included in one, and only one, interval Bj. We denote by i, the set of f s such that Bj =>(<**, ft). It is clear that Ie, Iu..., IN, is a partition of I and that Bj = ( L U (<*, ft)) U (ft H *) for all j = 1 , . . . , N. Hence diam Bf. « S (ft  a,) ^ e
(by (5.3)),
and N
N
P i À . « ) ^ I (diam B,) 1 ' 2 S £ 1=1
/
\ 1/2
£ (ft  a,) )
1 = 1 ^ielj
Letting e » 0, we find jm1/2(tf ) = 0.
'
S I (ft  a f ) 1 / 2 ë 6. i*I e
D
5.2. Space and time singularities. The second result concerns the Hausdorff dimension in space and time of the set of (possible) singularities of a weak
37
REGULARITY AND FRACTIONAL DIMENSION
solution (cf. V. Scheffer [l][4], C. FoiasR. Temam [5]): THEOREM 5.2. Let n — 3,û = SlorQ; let u be a weak solution to the NavierStokes equations (Problem 2.1) and assume moreover that (5.5)
u0eD(A).
Then there exists a subset 00<^0 such that (5.6)
esssupu(x, t)]<<* for all xeÛ\Û0
and
Te(0, oo)?
te(0,T)
(5.7)
the Hausdorff dimension of Û0 is ^ § .
Before proving Theorem 5.2 we present some preliminary material of intrinsic interest. Lemma 5.1 is borrowed from V. Scheffer [1]; we reproduce it for the convenience of the reader 1 n LEMMA 5.1. For a > 0 and feV(U ), let Aa(f) be the set of those xeUn such that there exists mx with f
(5.8)
'yx^2m
/(y)dy^2m
for all m ^ m x .
n
Then the Hausdorff dimension of IR \A a (/) is ^ a . Proof. By definition of A a (/), for any e > 0 and xeUn\Aa(f) ball B e (x) centered in x such that (5.9)
f
/(y)dy>2" l/(y)l
rt
(diamB e (x)) a
and
there exists a
diam B e (x)i=e.
J
BF(x)
By a Vitali covering lemma (see E. M. Stein [1]), there exists a system {Be(Xj):jeJ}c:{Be(x):xeUn\Aa(f)} such that the B e (x,)'s are mutually disjoint, / is at most countable and (5.10)
U xeR n \A t t (/)
B e (x)c=Uß;(x J ), jeJ
where B'e(Xj) (jeJ) denotes the ball centered in x, and with diameter 5(diamB e (x,)). By virtue of (5.1), (5.9), (5.10), it follows that M R n \ A a ( / ) ) = lim ^5e(Un\Aa(f))^irnmf e0
HmlO*Z f
£ (diam B^x,))" e^O . 6J ,>o 6J
\f\dx^10a[ /dx:<+oo,
and the Hausdorff dimension of Un\ Aa(f) does not exceed a. 1 In Lemma 5.1 and Theorem 5.3 the dimension of space n is any integer =^1 (and is not restricted as everywhere else to 2 or 3).
38
PART I.
QUESTIONS RELATED TO SOLUTIONS
5.3. Let G be an open subset in Un and let Te (0, *>), Kp<<» / € L ( G x ( 0 , T)) be given. Let moreover THEOREM
and
p
vi L~(0, T; L\G)) HL 2 (0, T; H&tf})
(5.11) be such that du
(5.12)
v Av = /
in fhe distribution sense in G X (0, T)
and (5.13)
t>(f)e«(G)
/ora.e. f e ( 0 , T )
and 2
u(0)€«(G).
Then tftere exists a subset G0<^G such that (5.14)
esssupu(x, r)
ifxeG\G0
te(0,T)
and rhe Hausdorff dimension of G 0 is ^max{n + 2  2 p , 0}. Proo/. We infer from (5.11)(5.12) and the regularity theory for the heat equation (cf. O. A. LadyzhenskayaV. A. SolonnikovN. N. Ural'ceva [11]) that: —eLp(Gx(0,T)) dt
and
u e L p ( 0 , T; W2"(G))
(where W 2 p ( G ) is the usual Sobolev space of functions defined on G, which together with all their derivatives up to order 2 belong to LF(G)). Ife CQ(G) and vx = v, then vx will satisfy (5.11)—(5.13), and also dV\
dt
v bkVx = J fx
in the distribution sense in G x (0, T),
where fx is a suitable function which also belongs to L p ( G x ( 0 , T)). Therefore, taking into account that the Hausdorff measures are Borel measures, we can consider only the special case (5.15)
u(f)e«(G),
G=Un,
suppv(t)^K
for all te(0, T)\o>,
0^
where K is a certain compact set czjRn while œ is of Lebesgue measure =0. It is clear that in this case u(x, t) coincides almost everywhere in Rnx(0, T) with the function w — w0+wx, 2 It is well known (cf. for instance J. L. Lions [1] or R. Temam [RT, Chap. Ill, Lemma 1.2]) that if v satisfies (5.11) and (5.12) then v is equal for almost every fe(0, T) to a function in «([0, T]; L2(G)) and therefore u(0) makes sense.
REGULARITY AND FRACTIONAL DIMENSION
39
where w0(x,f) = J
E(xy,t)v(y,0)dy,
Rn
(5.16) Wi(x, t) =
E(x  y, T)/(V, T) dy dr
and JE(x, 0 = — 7 = ^ ne x p  — J . V 4w/ (2V7rW) Therefore we can assume that, for any t e (0, T)\co, u(x, f) = w(x, t) everywhere on IRn. It follows that for such f's and for all x 0 e!R n , we have — I
\v(x, f)w 0 (x, i)\dx JxXol^r
^\ f
IwxUOldx
(5.17)
s J ' f E(x0x,tr)\\
\f(y,r)\dydrdx
Jo J R "
^ f
r
Jy_xsr
E ( x 0  x , r  r ) / * ( x , T) dx dr « wî(x, t)
Jo JR"
(by (4.16) and the change of variables x' = y  x + x 0 , y' = y) where, for all {x, s}eUn x(0, »),
/*(x,s) = su P 4rf r
r
l/(y, s)l dy
JyxSr
From the classical theorem on the maximal functions (see E. M. Stein [1]), we have /* eLp(Qrx(0,T)).
(5.18)
Since for t e [ 0 , T]\
i?(x, f )  ^ w0(x, f) + wî(x, r) for all xeUn
and
f €(0, T)\o>.
But, setting M,={(y,T):yx^2i,T^0,rT^22j}
for / = 1, 2 , . . . ,
40
PART I.
QUESTIONS RELATED TO SOLUTIONS
we have for all xe(R n , te(0, T), and because of standard estimates on E:
f
E(xy9tT)f*(y,T)dydT
Jo J r
ac
"fJ(i>,iv4.(,T)rr(y'T) " » *
«41 T
(5.20)
f*(y,r)dydr T))\M, I*
y
,ti J JM,.\M,.+1 (  x  y  2 + 4i/(fT)) n/
^c;(c^(J T J n
+ 2
f*(y9rrdydryi
>
,l,
/
(öT^)IU/*">""")
Schiff /*(y,T)dydTJ + £ 2"'(22'2n*)(p"1)/p(f
/*(y,T)pdyd
â c ^ f + Ï 2("«"+2)(p1)/p])'( f , =i
\J\yx\s2>
+2
2
= c U + c„,l)l2C'" ^ >'(fX J 1=1
fV(y, rrdTdy) 1 " Jo
/
g(y)dyV" yx^2'
/
where c^, c„,... are suitable constants (with respect to {x, t}) and g()=
fT
/*(,T) p dT€L 1 (R n )
(see (5.18)). Jo Therefore A a (g) (see Lemma 5.1) makes sense, and if a > max {0, n + 2  2p}, obviously sup w*(x, f)
for x e A a ( g ) .
te(0,T)
Consequently (since vv0 is continuous on (R"x(0, T)), by virtue of (5.19) we have esssupu(x, f)
for x e A a (g).
REGULARITY AND FRACTIONAL DIMENSION
41
Therefore the set G 0 :  x e Un : ess sup u(x, f) = °°} (0,T) te(O.T)
I
J
n
is included in (R \A a (g). From Lemma 5.1 we infer that the Hausdorff dimension of G 0 is ^ a . The conclusion (5.14) is now obtained by letting a —* max {0, n + 2  2p}. Remark 5.1. The proof of Theorem 5.2 shows that the result remains valid if G = Q and we replace (5.11) by v G L°°(0, T; L2(G)) H L 2 (0, T; Hj(G)). We now turn to the proof of Theorem 5.2. Proof of Theorem 5.2. Obviously it is sufficient to prove the result for a fixed Te (0, o°). Also it is clear that each component v(x, t) = u,(x, t) (j = 1, 2, 3) of u(t) satisfies (5.11)(5.12). Since u e L°°(0, T; L2(ft)) H L 2 (0, T; L 6 ((l)), we find by interpolation that ueU(0, T; L 6r/(3r " 4) (0)) for every r ^ 2 (see J. L. LionsJ. Peetre [1]), whereupon, for r = ^ , we obtain UGL10/3(0,T;L10/3(Ü)).
It follows that (A = Z " A " 6 L 5/4 (0, T; L5/4(H))
(5.21) since
L
10/3.L2c:L5/4>
We then get du v Aw hgrad p = h (5.22)
in 0 x (0, T),
divw = 0, u=0
o n 3 T x ( 0 , T),
u = u 0 €H 2 ((î)
for f = 0,
where (see (5.21)) fi/*6L5/4((lx(0,T)). By referring to the regularity theory for the Stokes system (5.22) (cf. K. K. GolovkinO. A. Ladyzhenskaya [1], K. K. GolovkinV. A. Solonnikov [1], V. A. Solonnikov [2]), we see that D,p€La(nx(0,T)) 3
This boundary condition must be replaced by the periodicity condition when C = Q; see Remark 5.1.
42
PART I.
QUESTIONS RELATED TO SOLUTIONS
for any K a < i Thus for u(x, t) = u,(x, t) we obtain —  v Au = h , ^  D ^ e l a ( n x ( 0 , T)) on ftx(0, T), so that by virtue of Theorem 5.3, we obtain that the set fto = I * e ^ : e ss sup \v(x, t)\ = oo [ «
te(O.T)
J
has the Hausdorff dimension ^ 5  2 a . The conclusion follows by letting
a^i
D
Successive Regularity and Compatibility 6 Conditions at t = 0 (Bounded Case) In this section we assume that n = 2 or 3 and consider the flow in a bounded domain ftc=Rn. We are interested in the study of higher regularity properties of strong solutions, assuming that the data u 0 , /, ft possess further regularity properties. While this question is essentially solved in § 4 for the space periodic case (through Lemmas 4.2 and 4.1), the situation in the bounded case is more involved. In particular, we derive in this section the socalled compatibility conditions for the NavierStokes equations, i.e., the necessary and sufficient conditions on the data (on dft at t = 0) for the solution u to be smooth up to time f = 0. 6.1* Further properties of the operators A and B. We recall that if ft satisfies (1.7) with r = m + 2, and if m is an integer >rc/2 ( m ^ 2 for n = 3 , 2), then H m (ft) is a multiplicative algebra: (6.1)
If u, u e H m ( f t ) , mi=2, n ^ 3 then u • u € H m ( f t ) and \u • u m
sc^(n)iiUMmFor every integer m we define the space (6.2)
JEm =H m (ft)fïH,
which is a Hubert space for the norm induced by H m (ft) = H m (ft) n . It is clear that Em+1czEm, for all m and that E0 = H: (6.3)
E m + 1 c : E m c : • . . c E 1 c E 0 = H.
The orthogonal projection from L2(ft) onto H being denoted by P as before, we introduce the operator si (6.4)
du = P Aw,
which is linear continuous from Em, m ^ 2, into E0 = H. Actually the operator P is linear continuous from H m (n) into H m ( f l ) n H = £ m (cf. [RT, Chap. I, Remark 1.6]), and therefore (6.5)
si is linear continuous from Em+2 into Em.
Now if u e E m + 2 n V , m ^ 0 , and s£u=feEm, then A u + / = ( I  P ) A w belongs to H^, which amounts to saying (cf. §2.5) that there exists peH1^) such that A u + / = Vp. Hence (6.6)
Au+Vp=/
in ft,
divu=0
in ft,
u=0
on aft, 43
44
PART I.
QUESTIONS RELATED TO SOLUTIONS
and u is the solution to the Stokes problem associated with f. The theorem on the regularity of the solutions of Stokes problem mentioned in § 2.5 implies then that (6.7)
si is an isomorphism from Em+2 H V onto 1 Em,
m ^ 0.
It is clear from (6.6) that 2 (6.8)
Mu=Au
VueD(A).
Operator B. Lemma 2.1 shows us that b(u, v, w) is a trilinear continuous form on H m ' ( ( l ) x H m 2 + 1 ( n ) x H if m1 + m2>n/2 (or m 1  f m 2 ^ n / 2 if m ^ O for all i). This amounts to saying that the operator B defined in (2.41) is bilinear continuous from H m, (fl)x[Hl m2+1 (n) into H, under the same conditions on m1, m 2 . Clearly (w • V)v belongs in this case to L2(H) and (6.9)
B(u,u) = P[(uV)u].
We have LEMMA 6.1.
(6.10)
ß(£m+1xEm+1)e£m
formal.
Proof. If m = 1, u, v G H 2 ( ( Î ) , then obviously (u • V)v el 2 (ft) and A ( ( u • V)ü) = (DiU) • (Vu) + u • (AVu) belongs to L2(fl), since the first derivatives of u and v are in L6(H) (at least) and u and D are continuous on Ù. The operator P mapping H r (fl) into H r ( n ) n H,B(u,v) = P((wV)v) is in H 1 ( f l ) n H = £ 1 . If m ^ 2 , then i^ and D^ are in H 2 (H), and their product is too, due to (6.1). Hence (u • V)ue(H]m(n) and P((u • V)v) as well3. 6.2. (6.11)
Regularity results. If u0 and / are given, satisfying u0eV,
/€L2(0, T;H),
then Theorem 3.2 asserts the existence of a (strong) solution u of Problem 2.2 defined on some interval [0, T"], T" = T if n = 2, T ' ^ T if n = 3. Changing our notation for convenience, we write T instead of T', and (6.12) (6.13)
1
u e L 2 (0, T; D(A)) n «([0, T]; V), U'GL2(0,
T;H).
If ^ is the inverse of ^  E m + 2 n v , ^ u = u, for all u € E m + 2 n V. but in general ^dufix for an arbitrary u e H m + 2 , m ^ 0 . 2 Because of (6.7) and (6.8), Vm is a closed subspace of E m , for all meN, the norm induced by Em being equivalent to that of Vm. In the space periodic case, the analogue of Em =H™(Q)nH is equal to Vm. 3 The proof shows also that B ( £ m x E m + 1 ) c E m for m ^ 2 .
SUCCESSIVE REGULARITY AND COMPATIBILITY CONDITIONS AT t = 0
45
Our goal is to establish further regularity properties on u, assuming more regularity on u 0 , /. We introduce the space (6.14)
Wm = ( i ) e ^ ( [ 0 , T];Em),
U('> = ^ G « ( [ 0 ,
T]; Em_2j),j
= l,...,/},
where m is an integer and / = [m/2] is the integer part of m/2, i.e. m = 21 or m = 2I + l. We begin by assuming that for some m ^ 2 (6.15)
u0eEmfW,
feWm_2,
and we make the following observation: LEMMA 6.2. If u is a strong solution to the NavierStokes equations and u0 and f satisfy (6.15) for some m $ 2 , then the derivatives at t = 0 of u, u 0) (0) = (d j u/dr')(0), / = 1 , . . . , / , can be determined "explicitly" in terms of u0 and /(0), and (6.16)
u ( <>(0)eE m  2] ,
/ = l,...,I^[—J.
Proof By successive differentiation of (2.43), we find du(i) ^—] + ^ u ( j ) + ß(i) = f(i\ dt
(6.17)
where we set 4>{i) = d]ldt\ ß(t) = Bu(t), so that ß 0 ) = I ( 7 ) ß ( u °  ° , u ( 0 ).
(6.18)
i=o M>
Now by (6.5), (6.10), (6.15), ß(0) = Bu(0) eEm_u iAstfu(0)B(u(0)) + /(0)eE m _ 2 . Similarly, if m ^ 4 , u"(0) = vdu'(0)B(u'(0),
and
u'(0) =
u(0))B(u(0), u'(0)) + f (0)eE,'m—4*
The proof continues by induction on j using (6.5), (6.10), (6.15), and shows that u(i\0) = vsiu(j1\0);t
( / T 1 )B(u°' 1  i ) (0),
MW(0))+/°"1)(0)
^ ^m2j?
for / = 1 , . . . , I . D It is known that for an initial and boundary value problem, the solutions may not be smooth near t = 0, even if the data are c€°°. For NavierStokes equations, the following theorem gives the necessary and sufficient conditions on the data for regularity up to t = 0, the compatibility conditions (cf. (6.21)).
SUCCESSIVE REGULARITY AND COMPATIBILITY CONDITIONS AT t = 0
47
For that purpose we write (6.17), (6.18) in the form vMu(i) = ~uii+l)
(6.27)
£ ( { W i i «  0 , u (0 ) + / 0 ) ; i=0
(,)
W
(,+1)
/ and u are in «([0, T ] ; H ) (at least). Because of Lemma 2.1 and (2.36), B is bilinear continuous from V x V into V_1/2. Hence by (6.24), for i = 0 , . . . , / , and i = 0 , . . . , 1  1 , B(u ( i °, U ( 0 )G «([0, T ] ; V_ 1/2 ).
(6.28)
Since A is an isomorphism from V 3/2 onto V_1/2, (6.27) implies then that u(i) €<€([(), T ] ; V 3/2 ), j = 0 , . . . , J  1 . Using again Lemma 2.1, B is a bilinear continuous operator form V3/2 x V3/2 into H. Thus, for the same values of i and ; as in (6.28),
and (6.26) follows, iv) Finally we show that u e Wm, i.e., (6.29)
u ( i ) e « ( L 0 , T ] ; E m _ 2 i ),
/ = 0,..., I
For j = I, this is included in (6.25). For ; = I  1 , using (6.10) and (6.26) we find that £
(f)B(u°,\ii
i=0
V'
and then (6.5), (6.27) and (6.25) show that K^VfcVŒO, T ] ; E m _ 2 I + 2 ). The proof continues by induction for j = I — 2 , . . . , 0. Theorem 6.1 is proved. D 6.3.
Other results. We particularize Theorem 6.1 to the case m = 3. 6.2. We assume that n = 2 or 3, H satisfies (1.7) with r = 5,u0e H 3 (ft)nV,/e<e([0, T I J H ^ H H ) and dfldteL2(0,T\H). Then the solution u of Problem 2.2 defined by Theorem 3.2 on some interval [0, T*], 0 < T * ^ T , belongs to «([0, T*];H 3 (n)n V), if and only if THEOREM
(6.30)
yPAuo + P I
u 0 i —5) + /(0) = 0
V*
dx
<;
onT.
The only condition left in (6.21) is (6.30). Since u ' ( 0 ) € E 1 = H 1 ( f t ) n H , u'(0) • v = 0 on T, and (6.30) is a restrictive assumption for the tangential value on T of (6.31)
dU0
¥ 0 = i> Au 0 + £ u 0 i —2 + i=l
dXt' °Xi
f(0)eH\to).
Another way to formulate (6.30) is this: * 0 = PV0 + Vq0, where (cf. [RT, Chap.
48
PART I.
QUESTIONS RELATED TO SOLUTIONS
I, Thm. 1.5]) q0 is a solution of the Neumann problem:
6 32
(  )
Aq0 = d i v ^ 0
in«,
dq0 . — = ^0 ' v dv
_ on T;
^ o must be such that the tangential components on T of ^ 0 and Vq0 coincide: (6.33)
V^oW.
onT;
(6.32), (6.33) constitute an overdetermined boundary value problem (Cauchy problem on T for the Laplacian) which does possess a solution. If the compatibility conditions (6.21) are not satisfied, the other hypotheses of Theorem 6.1 being verified, we get a similar result of regularity on ftx(0,T]: THEOREM 6.3. We make the same hypotheses as in Theorem 6.1, but the conditions in (6.21) are not satisfied5. Then (6.34)
^ G «((0, T]; E m _ 2 j ),
/ = 1 , . . . , I.
Proof. Thanks to (3.39), there exists 0 < t0< T%, t0 arbitrarily close to 0, such that u(t0)eD(A). Equation (6.19) for t0 and j = \ shows that u'(t0)eH. We conclude as in Theorem 6.1 or 3.2 that u'eL2(t0, T*; V)n<ë([f0, T*];H). We choose tu t0
It suffices also to assume that u 0 € V instead of u 0 e Em O V.
SUCCESSIVE REGULARITY AND COMPATIBILITY CONDITIONS AT t = 0
49
belong to V. If we introduce the analogue if/(t) of (6.31)
rtö=i'AM(d+£i*(ö^+/(d, i= l
oXj
and write that the overdetermined boundary value problem similar to (6.32), (6.33) possesses a solution, we conclude that u(t) belongs, for t in €u to a complicated "manifold" of V: i> depends on u(t) (and / ) ; q(t) = ^T(div i>(f), ijj(t) • v), where Jf is the "Green's function" of the Neumann problem (6.32): the condition (6.33) is the "equation" of the manifold. A similar remark holds for every f > 0 for a strong solution; similar remarks follow from the other conditions (6.21) if m ^ 5 . Remark 6.4. If we introduce pressure, then we clearly get regularity results for pressure: » Vp =
du dt
' + i^Au(u • V)u+/.
In the situation of Theorem 6.1,
(6.35)
pMe <ê([0, W T Tl f r H—*
^ € «([0, T]; H "  1 " 2 ' « ! ) ) ,
dt
/ = !,... ,11, and in the situation of Theorem 6.3,
(6.36)
p € «((0, T ] ; Hm\Ü)),
M^n**™.
^ e «((0, T ] ; H "  * " " « » ) ,
„
/= 1
11.
7 Analyticity in Time In this section we prove that the strong solutions are analytic in time as D(A)valued functions. We assume for simplicity that feH is independent of t: we could as well assume that / is an Hvalued analytic function in a neighborhood in C of the positive real axis. The interest of the proof given below is that it is quite simple and relies on the same type of method as that used for existence. The method applies also to more general nonlinear evolution equations with an analytic nonlinearity. 7.1.
The analyticity result. The main result is the following one: 7.1. Let there be given u0 and f, u0eV,feH independent of t. If n — 2, the (strong) solution u of Problem 2.2 given by Theorem 3.2 is analytic in time, in a neighborhood of the positive real axis, as a D(A)valued function. If n = 3, the (strong) solution u of Problem 2.2 given by Theorem 3.2 is analytic in time, in a neighborhood in C of the interval (0, T*), as a D(A)valued function. Proof, i) Let C denote the complex plane and H c the complexified space of H, whose elements are denoted u + iv, u,veH,i= VT. Similarly Vc, V'c are the complexified V, V, and X c is the complexified space of a real space X. By linearity, A (resp. Pm, resp. B) extends to a selfadjoint operator in Hc (resp. to the orthogonal projection in H c , V c , V^, onto the space Cwx+ • • • +Cw m , resp. a bilinear operator from Vc x V c into Vc). Consider now the complexified form of the Galerkin approximation of NavierStokes equations, i.e., (compare with (3.41)(3.45)) the complex differential system in PmHc: THEOREM
(7.1) (7.2)
^(C)+vAum(C) dC
+ PmB(um(C)9um(C)) = Pmf, um(0) = Pmuo,
where f e C , and um maps C (or an open subset of C) into PmHc = CvVi+ • • • +Cw m . The complex differential system (7.1), (7.2) possesses a unique solution u^ defined in C in some neighborhood of the origin. It is clear that the restriction of um(£) to some interval (0, Tm) of the real axis coincides with the Galerkin approximation um(t) defined in the real field by (3.41H3.45). ii) As in the real case, we now prove some a priori estimates on um. We take the scalar product in Hc of (7.1) with Aum(Ç) ( ( ^ J p . «.(£))) + v Au m (0 2 +
AiUfl).
52
PART I.
QUESTIONS RELATED TO SOLUTIONS
We multiply this relation by e' e , with £ = seie, and for fixed 0, 0 < 7r/2, we get:
M(^)) Therefore ~  k , ( s e w )   2 + vcos 9 Au m (se i9 ) 2 2 as < 7  3)
=  R e e'6(B(um(seie),
lUse")), A i U « " »
,
+ Ree" (/,Au m (se«»)>. The absolute value of the term involving / is less than or equal to l/l •  A « m ( 5 e * )  S ^ ^  A u m ( S e « ' )  2 + ^ — 1 /  2 . 4 i> cos 0 For the term involving B, we use the inequality (2.32), which clearly extends to the complex case 1 , (7.4)
(7.5) \b(u,v,w)\^ct\\u\\\\v\\1,2\Av\m\w\
V ue
D(Ac),veVc,weHc.
This allows us to write \(B(um(seie\um(seie)\Aum(seie))\ ^c?um(Seiö)irAum(seie)3/2 1/COS0,
t
à
Ä \4x
c^
•tAM m (sé») 2 +: Ti3l!um(seiS)6 cos COS < 0 (by Young's inequality, see (3.10)),
where c' depends on the data v, u0, H. If we take into account this majorization and (7.4), then (7.3) becomes (7.6) ~ u m ( Se i9 ) 2 + v cos 0 \A*4*r)ß as
â^—IfP+r^ji v cos 0 cos 0
\K(se«)\\*.
The inequality (3.25) is still valid in the complex case
and thus (7.7)
^um(Seie)2+^1cosoum(Sei9)2â^/2+r^3um($C'a)ir. as v cos 0 cos 0
1 This relation, which is not so good as the first relation (2.31), is valid for both dimensions, n = 2 and 3.
ANALYTICITY IN TIME
53
This differential inequality is of the same type as (3.27), setting y(s) = \\um(seie)\\2,c'1 = v\l cos 6,C2 = 2/(v cos 6) / 2 , C3 = c'/cos 03. We conclude, as in Lemma 3.2, that there exists K'6 which depends only on /, v and O (or Q), such that (7.8)
i2(l +   u 0 J 2 ) ^ 2 ( l + u02),
l+IMse*
for (7.9)
S^COSÖ
3
T;(U0),
f; UT[(\\U0J)= ^ A (HUo ) V (l+Kmll ) / This shows that the solution um of (7.1)(7.2), which was defined and analytic in a neighborhood of £ = 0, actually extends to an analytic solution of this equation in an open set of C containing (see Fig. 7.1): (7.10)
(7.11)
T;(11U01D
A(uo) = {£ = s e i e , O < s <  c o s 0  3 T ; (   u o   ) ,  ö  <  } .
The estimate (7.8)(7.10) shows that (7.12)
sup u m (£) 2 ^const2(l + u02). £eA(u 0 )
iii) The analyticity of um and Cauchy's formula allow us to deduce from (7.12) a priori estimates on the derivatives of um (with respect to £) on compact subsets of A(u0). Indeed, for £eA(u 0 ) and keN, fc^l,
(7.13)
^(0~{
m bCr
where d = d(£, dA(u0)) is the distance of £ to the boundary dA(u0) of A(u0).
y*
FIG. 7.1. The region A(u0) in C.
54
PART I.
QUESTIONS RELATED TO SOLUTIONS
Therefore dku
II 2k/c»
dC
' d
 ^ ( O l N ^ f  sup IM*)«, zeA(u 0 )
and by (7.12), for any compact set K<=A(u0) (7.14) 2
sup C«KD
[d(K,dA(u0))J
dC
(l+ll«ol22\l/2 )
Considering (7.1), we find that v\AumU)\Z\f\
+
+B(« m (#,Mm(4))l
dC
am*
+ c3\\um(C)\\3,2\Aum(0\V2 dt (0 dum dÇ
U/H
(by (2.32))
3 (0 +^\Au m(C)\+f\\um(0\\ . 2 2v
Thus Aum(^)â/+%um(dl3 + 2
(7.15)
dUn
(()
and it follows from (7.14) that for every compact subset K of A(u0) (7.16)
suplAu^l^c;,
where C4<°°, depends on K and the data, (7.17)
7v2r2
1
c'<{K)=\f\ +
C
5(l +  k f ) 3 ' 2 +
yfT,[d(K,dA(u0))]
(l+ll«ol2)1/2
Using again Cauchy's formula (7.13) and (7.16), we obtain also for every (eK and IceN: d£ (7.18)
k
"m(£)
sup
2fcfc' —r^suplAu^)!, [d(K,dA(u0))f^
um(0\
2 k fc!c 4 (K')
where K' (containing K) is the set (7.19)
{z e A(i40), d(z, aA(u 0 ))i^d(K, dA(wo))}.
iv) We now pass to the limit, m—»oo. Since the set {DG V C , u^p} is compact in Hc for any 0 < p < oc? we can apply to the sequence um the vector 'd(X, Y) = inf x e X y e Y d(x,y).
ANALYTICITY IN TIME
55
version of the classical Vitali's theorem. Thus we can extract a subsequence um9 which converges in H c , uniformly on every compact subset of A(u0) to an H c valued function u*(£) which is analytic in A(u0) and satisfies obviously: sup   M ( a i 2 S 2 ( l + u02).
(7.20)
£eA(u 0 )
Since the restriction of um to the real axis coincides with the Galerkin approximation in R+ of the NavierStokes equation, it is clear that the restriction of u*(£) to some interval (0, T") of the real axis coincides with the unique (strong) solution u of the NavierStokes equations given by Theorem 3.2. Hence u* is nothing else than the analytic continuation to A(u0) (at least) of u, and we will denote the limit u(£) instead of w*(£). Secondly the whole sequence um(*) converges to u() in the above sense (i.e. uniformly on compact subsets of A(u0), for the norm of H c ). Since the injection of D(A) in V is compact, it also follows from (7.16) and Vitali's theorem that the sequence um converges to u in V uniformly on every compact subset of A(u0), and that (7.21)
supAu(f)^ci(K),
with the same constant c'A(K) as in (7.17). Finally the majorizations (7.14), (7.18) imply that dkujd£k converges to dku/d£k in V uniformly on every compact subset K of A(u0), and
<722)

(7.23)
d^l a w£X>)r (1 + IMF)1 " sup
Adku(0\^
dC
2kk\c'4{K')
r[d(K,aA(u0))]k
K' defined in (7.19). To conclude, we observe that the reasoning made at t = 0 can be made at any other point f 0 €(0, °°) s u c n t n a t M('o)e V. We obtain that u is a D(A)valued analytic function at least in the region of C: (7.24)
U{'o + A(u(t0))}, to
where the union is for those r0's for which (f0€ (0, ») and u(t0)e V). Finally we observe that, actually, A(u0) = A(u0) depends only on the norm in V of u 0 and decreases as u0 increases. Therefore if u is bounded in V on some interval [a, ß],sup te[a , ß] w(r)^JR, then (7.25)
U t 0 e[a,ß]
{fo + A( M (t 0 ))}^
U
{to + A(R)},
toe[a,ß]
and this guarantees that the domain of analyticity of u contains the region mentioned in the statement of Theorem 7.1 (See Fig. 7.2.) The proof of Theorem 7.1 is complete.
56
PART I.
QUESTIONS RELATED TO SOLUTIONS
RG.7.2.U (toe[ «.B]^o + A(R)}.
7.2. Remarks. Remark 7.1. Of course the result of analyticity given by Theorem 7.1 applies as well to a weak solution of the NavierStokes equations (Problem 2.1, n = 3) near a point t0 belonging to the set of H ^regularity (cf. § 4). If fe H is independent of f, and (a, ß) is a maximal interval of H ^regularity of a weak solution u, then u is a D(A)valued analytic function in U
it+A(u(t))}.
te(<x,ß)
Remark 7.2. Theorem 7.1 and Remark 7.1 extend easily to the case where / is an H c valued analytic function of time in a neighborhood of the positive real axis: it suffices to replace everywhere in the proof A(w0) ( ° r t0 + A(u(t0))) by its intersection with the domain of analyticity of /. We conclude this section with the following: PROPOSITION 7.1. Under the same hypotheses as in Theorem 7.1, for 0
,(fc)
\Au(k\t)\
l$0 =
dt>•it)
'fc!
d+lkll22\l/2 )
.c's(k).ck(k)
fcsi,
where k\ c^(k) =  — (1 +  k   2 ) , / 2 , c'6(k) = 2kk!i(2WVA,
^3/2^2
?(l+ll«ofP )•
Proof It suffices to apply (7.22), (7.23) with K = {t}. The distance d(t, dA(u0)) appears in the righthand side of these inequalities, and an elementary calculation shows that for 0^r^Ti(w o ), this distance is exactly t. D Remark 7.3. When the compatibility conditions (6.21) are not satisfied, u is not smooth near f = 0 and the Hnorm of the derivatives u(k)(t) tends to infinity as t—»0. The relations (7.21), (7.22) give some indications on the way in which they tend to infinity. A similar result has been obtained by totally different methods by G. Iooss [1] and by D. Brézis [1].
8 Lagrangian Representation of the Flow We assume that the fluid fills a bounded region H of IR3. The Lagrangian representation of the flow of the fluid, mentioned in § 1, is determined by a function <^:{a, r } e u x ( 0 , T)—>4>(a, t)efl, where 4>(a, t) represents the position at time t of the particle of fluid which was at point a at time t = 0 (flow studied for time f, O ^ f ^ T ) . This also amounts to saying that {t»»4>(a, t)} is the parametric representation of the trajectory of this particle. The Lagrangian representation of the flow is not used too often because the NavierStokes equations in Lagrangian coordinates are highly nonlinear. It plays an important role, however, in two cases at least: it is used in the numerical computation of a flow with a free boundary, and, in the mathematical theory of the NavierStokes equations, it is the starting point of the geometrical approach developed by V. I. Arnold [1], D. EbinJ. Marsden [1], among others. If we are given u, a strong solution of the NavierStokes equations, then it is easy to determine the trajectories of the particles of fluid for the corresponding flow: for every aeCl, the function f»>4>(a, t), also denoted £ or £a, is a solution of the ordinary differential equation (8.1)
^
= u(£(0,0
with initial (or Cauchy) data (8.2)
m
= a.
Our goal in this section is to show that, using one of the new a priori estimates mentioned in § 4, it is possible to determine the trajectories of the fluid (in some weak sense), even if u is a weak solution of the NavierStokes equations. 8.1. (8.3)
The main result. We assume for convenience that U0GV,
/e«([0,T];V),
and that we are given u, a weak solution of the NavierStokes equations (Problem 2.1) associated with u0 and /, and that Cl satisfies (1.7) with r = 2. We have first to give meaning to (8.1), (8.2) since u is not regular. LEMMA 8.1. If £ is a continuous function from [0, T] into Ù and u is a weak solution of Problem 2.1, then u(Ç(t), t) is defined for almost every te[0, T], the function f i»u((f), r) belongs to L^O, T;IR3) and (8.1) and (8.2) make sense. Proof. It follows from Theorem 4.1, Remark 4.1 and Theorem 5.1, that u is continuous from [0, T ] \ £ into D(A), where € has Lebesgue measure 0. Due to (1.7), H2_(fi) is included in «(Ö) (cf. §§2.3 and 2.5, dimension n = 3), and D ( A ) c z ^ ( n ) 3 . Hence u((f), f) is well defined for every t e [ 0 , T]\<£ and is 57
58
PART I.
QUESTIONS RELATED TO SOLUTIONS
continuous from [0, T ] \ t into R 3 . The function t »» u(£(t), i) is measurable. If we show that this function is integrable, then (8.1) will make sense in the distribution sense and (8.2) obviously makes sense. But, (8.4)
\u(t(t),t)\^\u(t)\L~m
Vfe[0,T]\tf,
and according to Theorem 4.3, t *+\u(t)\L°°(a) is L 1 . D The main result can be stated: 3 THEOREM 8.1. Let ft be an open bounded set of U which satisfies (1.7) with r = 2, and let u be a weak solution of the NavierStokes equations corresponding to the data u0, f, which satisfy (8.3). Then, for every aefl, the equations (8.1), (8.2) possess (at least) one continuous solution from [0, T] into Ô. Moreover, we can choose this solution in such a way that the mapping <$>:{a, t}**£a(t), belongs to L°°(Ox(0, T)) 3 l and d(f>/dteL\ax(0,T))3. 8.2. Proof of Theorem 8.1. i) Let wj9 j e N, denote the eigenfunctions of the operator A (cf. (2.17)) and, as in (3.44) we denote by Pm the orthogonal projector in H on the space spanned by wlf..., vvm. The function u being continuous from [0, T] into V with u'eL4/3(0, T; V), u^ is, in particular, continuous from [0, T ] into D(A), with u^eL4/3(0, T; D(A)). Due to Agmon's inequality (2.21) (cf. § 2.5), there exists a constant c[ depending only on ft, and such that
(8.5)
K. (n) sc;inriAvr
Pm being a projector in V and (8.6)
D(A),
Mm(0Lcn)^cI
for every te[0, T]\€,
V» 6 D(A) ;
IKitW^lAu^itr^Oit),
where o(r) = c;w(r) 1/2 A W (0 1/2 ,
(8.7)
6eV(0, T) by Theorem 4.2 and Remark 4.1. Now as m ^ oo (8.8)
um + u
in L 2 (0, T; V) and L\0,
T; L°°(n)).
Indeed the convergence in L 2 (0, T; V) is a straightforward consequence of the properties of Pm and the Lebesgue dominated convergence theorem. The convergence in Ll(0, T; L°°(ft)) is proved in a similar manner, using the following consequence of (8.5): *Uf)~«
and even ^eU°(Ü; «([0, T];IR3)).
Vf^,
59
LAGRANGIAN REPRESENTATION OF THE FLOW
and by Holder's inequality2 fT
\um(t)u(t)\L~(n)dt *o
(8.9)
/ [T
\l/4,çT
\3/4
Sei:1 ( T  M m (r) M (f) 2 d t ) 1 M (  T  A u m ( 0  Au(t) 2 / 3 dt)~ ii) For every m, we define in a trivial manner an approximation £m of £, where £m is solution of (6.1), (6.2) with u replaced by u^. The usual theorem on the Cauchy problem for ordinary differential equations (O.D.E.'s) asserts that _£m is defined on some interval of time [0, T(m, a)], 0 < T ( m , a)^T. But um :ft—>(R3 vanishes on dft, and if we extend um by 0 outside Ù, we see that a trajectory £m(t) starting at time t = 0 from a e Ö (condition (8.2)) cannot leave H a t a time
m(a,0 = £nato
Urn = Ïma)
is in « H f t x p ) , T ])3 divum=0, the Jacobian of 4>m, Since det (Dm(, t)/Da) is equal to one for every t and therefore 4>m(, t) : fî »> n is locally invertible for every t. Furthermore, if aedfl, Hm(t) = a is a trivial solution of (8.1), (8.2) (with u replaced by I O , and hence O m (a, t) = a for all a G dfl and for all t. The classical theorems on the global invertibility of *€1 mappings (cf. F. Browder [1]) imply that, for every f, 4>m(, t) is a <£* diffeomorphism from Ö onto itself. Finally, as div u^ = 0, m(\ f) preserves the areas, iii) We now pass to the limit m —» », /or a ßxed a e Ü . Due to (8.6),
dUO
1^(^(0,01^0(0 Vre£,
and since 0 e 1/(0, T), the functions £m  £ma are equicontinuous. Thus, there exist a subsequence m' (depending on a) and a continuous function £ = £a from [0, T] into Ö, such that fm > £ as m '  ^ œ ? uniformly on [0, T]. We will conclude that  is a solution of (8.1), (8.2), provided we establish that (8.11) But, for ti €,
Mfin<0,0^Mtf()!0
inL 1 (0,T).
IM^W,0w(«0,0l ^ IMê»<(*>,0  uUm'it), 0I+"(4n<(0,0  ««(0,0I ^  M 0  " ( 0 l L  ( n ) + w(Ên'(0, 0  w ( « 0 , Ol2
Since the smooth functions are not dense in L°°, and L 2/3 (0, T; D(A)) is not a normed space, there is not much flexibility in the construction of a sequence of smooth functions u m approximating u in the norm of L\0, T;L°°(ft)).
60
PART I.
QUESTIONS RELATED TO SOLUTIONS
Because of (8.8), (8.9), it remains, to prove (8.11), to show that (8.12)
u(£ m '(), •)+"(£(•),•)
inLHO.n
For every t<£ #, u(,t)eD(A) is a continuous function on Ù; hence M(6n'(0»Ö*M.({(Ö,0f and u(£m(a, *) = £(*), and clearly, O possesses the desired properties. D Remark 8.1. It would be interesting to prove further properties of O: for instance that 4> preserves the areas, that <ï>(a, •) is unique at least for almost every a e f i , etc. 8.3. Appendix. For the convenience of the reader, we recall here the measurable selection theorem (cf. C. Castaing [1]): THEOREM 8.A. Let X and Y be two separable Banach spaces and A a multiplevalued mapping from X to the set of nonempty closed subsets of Y, the graph of A being closed. Then A admits a universally Radon measurable section, i.e., there exists a mapping L from X into Y, such that L(x)eA(x)
VxeX,
and L is measurable for any Radon measure defined on the Borel sets of X.
PART II Questions Related to Stationary Solutions and Functional Invariant Sets (Attractors) Orientation. The question studied in Part II pertains to the study of the behavior of the solutions of the NavierStokes equations when t —> », and to the understanding of turbulence. Of course, before raising such questions, we have to be sure that a welldefined solution exists for arbitrarily large time; since this question is not solved if n = 3, we will often restrict ourselves to the case n = 2 or, for n = 3, we will make the (maybe restrictive) assumption that a strong solution exists for all time for the problem considered. In § 9 we start by describing the classical CouetteTaylor experiment of hydrodynamics, in which there has recently been a resurgence of interest. We present here our motivations for the problems studied in Part II. In § 10 we study the time independent solutions of the NavierStokes equations. In § 11, which is somewhat technical, we present a squeezing property of the trajectories of the flow (in the function space) which seems important. In § 12 we show that, if the dimension of space is n = 2, then every trajectory converges (in the function space) to a bounded functional invariant set (which may or may not be an attracting set), and that the Hausdorff dimension of any bounded functional invariant set is finite. This supports the physical idea that, in a turbulent flow, all but a finite number of modes are damped. Throughout §§10 to 12 we are considering the flow in a bounded domain (A or Q) with indifferently either 0 or periodic boundary conditions. We will not recall this point in the statements of the theorems.
W The CouetteTaylor Experiment We recall what the Couette flow between rotating cylinders is: the fluid fills a domain ft of U3 which is limited by two vertical axisymmetric cylinders with the same axis (radii r1? r2, 0^ 0 in (1.9) (but <£ • v = 0). This case was not studied in Part I to avoid purely technical difficulties, but its mathematical treatment is very similar to that considered in Part I (cf., e.g., C. FoiasR. Temam [3], [4], [5]; J. C. SautR. Temam [2]). When a)x(t) is constant, co^f) = <öu a Reynolds number can be defined for this flow (cf. (1.6)), Re = öj1r1d/v, d = r2ru which corresponds for instance to the choice of L* = d as a characteristic length for the flow, and to U* = <ö1r1 = the velocity on the inner cylinder as a characteristic velocity. The Taylor experiment on the Couette flow is as follows. We start with the fluid at rest and we increase the angular velocity of the inner cylinder from 0 to some value (Oil i) If cox (more precisely Re, which has no dimension) is sufficiently small, then after a very short transient period we observe a steady axisymmetric flow. The trajectories of the particles of fluid are essentially circles with the same axis as the cylinders. ii) If cô^Re) is larger than some threshold À', but not too large, then, after the transient period, another steady state appears, different from that in i). A cellular mode now appears: the trajectory of the particles of fluid is now the superposition of the motion around the axis and a rolllike motion in the azimuthal plane. The steady flow (see Fig. 9.1) remains axisymmetric, and in adjacent cells the fluid particles move in counterrotating spiral paths. With changing (i.e., increasing) (öu a cellular structure may lose its stability and another cellular steady motion may appear, with more cells. iii) When cô^Re) is larger than a higher threshold value, the flow observed after the transient period becomes unsteady with its cellular structure perturbed by circumferential travelling waves. At this point the flow is periodic in time. As Re increases, the temporal variations of the flow become more complex, quasiperiodic perhaps, with two or more incommensurate frequencies. Finally the flow becomes totally turbulent with no apparent structure at all. This is a somewhat simplified description of the phenomena, the flow observed depending in a sensitive way on the ratio d/l and/or on how the 1
The gravity forces are of the form grad (pgx 3 ), where 0x 3 is a vertical axis pointing upward, and we can incorporate them in the pressure term. 63
64
PART II.
QUESTIONS RELATED TO STATIONARY SOLUTIONS
z
o" 5cells
3cells FIG. 9.1
constant angular velocity œl is attained; different situations may also appear if the flow does not start from rest. We refer the reader to T. B. Benjamin [1], T. B. BenjaminT. Mullin [1], [2], P. R. FenstermacherH. L. SwinneyJ. P. Gollub [1], H. L. Swinney [1], J. P. Gollub [1]. Let us also mention the remarkable fact reported in T. B. Benjamin [1] that different stable steady motions have been observed for the same geometry and the same boundary velocity (same values of v, ru r2, K <*>i). Although the above description of the experiment is oversimplified, it is quite typical, and similar phenomena are observed for other experiments, such as Benard flow and flow past a sphere. Let us interpret, for the flows studied in Part I, what the corresponding mathematical problems are. We start fromrest (u0 = 0), and we can imagine that f(t) is "increased" from 0 to some value f e H through some complicated path, or through a linear one: f(t) = k(t) f0,f0eH, k(t)eU, increasing from 0 to A. The parameter 2 (9.1)
R =
v2zTL3n/2
l/l,
can play the role of the Reynolds number, L being a characteristic length of the domain (Q, or Q), n the dimension of space and  /  the Hnorm of JeH. Then the conjectures on bifurcation and onset of turbulence are as follows (cf. D. Ruelle [1], D. RuelleF. Takens [1]): a) If R is sufficiently small, the solution u(t) of Problem 2.2 tends, for t —> °°, to ü a time independent solution of the NavierStokes equations. b) For larger values of R, ü may lose its stability and u(t) converges for 2 \L =
R has no dimension if p = 1; if p ^ 1, the number without dimension is R =pjx V.
2
 /  L3
n/2
,
THE COUETTETAYLOR EXPERIMENT
65
t—»oo to another stationary solution w'; this can be repeated for higher values of R, u{t)*ü'\.... c) After a higher threshold, u(t) converges, for £>o°, to a time periodic solution of the NavierStokes equations un(t), or to a quasiperiodic solution. d) For higher values of R, u(t) as t —» oo tends to lie on a "strange attractor," such as the product of a Cantor set with an interval. This would explain the chaotic behavior of turbulence. The small contributions presented below to this list of outstanding problems are : • The proof of a), which is elementary and has been known for a long time; • Some properties of the set of stationary solutions of the NavierStokes equations in connection with b) (§ 10); • Existence and a property of the limit set if n = 2 in the case d) (§ 12).
10
Stationary Solutions of the NavierStokes Equations
Assuming that the forces are independent of time, we are looking for time independent solutions of the NavierStokes equations, i.e., a function u = u(x) (and a function p = p(x)) which satisfies (1.4), (1.5), and either (1.9) with = 0, or (1.10): (10.1)
i/Au + (u • V)u + V p = /
(10.2)
(io.3)
divw=0
mO(=Q,or
Q),
in©,
w = o onr = an if o = n,
or (10.4)
u(x + Lei) = w(x),
i = 1 , . . . , n, xeQ
if€ = Q.
In the functional setting of § 2, the problem is (10.5) (10.6)
Given f in H (or V ) , to find u e V which satisfies *««, !>)) + &(«, * o )  ( f , t > )
VueV,
or (10.7)
*>Au+J3u=/
inH(orV').
10.1. Behavior for t —• oo. The trivial case. We start by recalling briefly the results of existence and uniqueness of solutions for (10.5)(10.7). THEOREM 10.1. We consider the flow in a bounded domain with periodic or zero boundary conditions (0 = 0, or Q), and n =2 or 3. Then: i) For every f given in V and y > 0 , there exists at least one solution of (10.5H10.7). ii) / / / belongs to H, all the solutions belong to D(A). iii) Finally, if (10.8)
v2>Cl\\f\\v;
where cx is a constant depending only on O1, then the solution of (10.5)(10.7) is unique. Proof. We give only the principle of the proof and refer the reader to the literature for further details. For existence we implement a Galerkin method (cf. (3.41)~(3.45)) and look, l
Ci is the constant in (2.30) when ml = m3= 1, m 2 = 0. Since / = VXjf v . if /eff, a sufficient condition for (10.8) is v2>cï\fk~l\f\i which can be compared to (9.1). 67
68
PART II.
QUESTIONS RELATED TO STATIONARY SOLUTIONS
for every m e N , for an approximate solution um, m
(10.9) such that (10.10)
« m ' i t t
êimeR,
^((wm, u)) + b(u m , um, u) =
for every v in Wm = the space spanned by wu . . ., wm. Equation (10.10) is also equivalent to (10.11)
vAum + PmBum = Pmf.
The existence of a solution um of (lO.lO)(lO.ll) follows from the Brouwer fixed point theorem (cf. [RT, Chap. II, § 1] for the details). Taking v = um in (10.10) and taking into account (2.34), we get
(io.i2)
HI"J 2 =
and therefore (10.13)
\\um\\^v We extract from um a sequence u m , which converges weakly in V to some limit u, and since the injection of V in H is compact, this convergence holds also in the norm of H : (10.14) um > u weakly in V, strongly in H. Passing to the limit in (10.10) with the sequence m', we find that u is a solution of (10.6). To prove ii), we note that if u e V, then Bu e V_1/2 (instead of V ) because of (2.36) (applied with ml = 1, m2 = 0, m3 = ^). Hence u = v~lA~l(fBu) is in V 3/2 . Applying again Lemma 2.1 (with m1 =§, m2 = h m 3 = 0), we conclude now that BueH, and thus u is in D(A). We can provide useful a priori estimates for the norm of u in V and in D{A). Setting v = u in (10.6) we obtain (compare to (10.12)(10.13)) (10.15)
(10.16)
v\\u\\2 =
(f,umf\\v'\\ul
(ë^=/if/eH).
M=4MV.
For the norm in D(A) we infer from (10.7) that v\Au\^\f\
+ \Bu\^\f\ + c2\\u\\3/2\Au\m c21
v = 1/1+:; IAu +  w3 2 2^ 2 il/l + ^ l A , u K 4 °^ l/ 2/ 1If l3 •2."" '2V A?
(by the first inequality (2.32)) (by the Schwarz inequality) (with (10.6)).
STATIONARY SOLUTIONS OF THE NAVIERSTOKES EQUATIONS
69
Finally
2,„,
c\
 A « f S  l / 1 + ^ Ä l/l13
(10.17)
For the uniqueness result hi), let us assume that ux and u2 are two solutions of (10.6): v((uu v)) + b(uu Mi, t)) = . Setting v = Ui — u2, one:
we
obtain by subtracting the second relation from the first
v MI  w22 =  6 ( " i , Mi, Mi ~ "2) + b(u2, u2, ux  Ma) = fc("i  u2, u2, uY  u2) (with (2.34)) S Ci M! — M2II2 HMJ
(using (2.30) with ml = m 3 = 1, m 2 = 0)
^—11/11 vII"1W2II2
(with (10.16) applied to u 2 ).
Therefore (^^V')K«#SÛ and Ui  u 2 = 0 if (10.8) holds. D Concerning the behavior for r * 00 of the solutions of the timedependent NavierStokes equations, the easy case (which corresponds to point a) in § 9) is the following: THEOREM 10.2. We consider the flow in a bounded domain with periodic or zero boundary conditions (0 = (l or Q) and n = 2 or 3. We are given f G H, v>0 and we assume that
(10 18)

V
\7V > ^ l / i + 7 I F i
where c 2 , c2 depend only on Û. Then the solution of (10.7) (denoted u«,) is unique. If u() is any weak solution2 of Problem 2.1 with u0eH arbitrary and f(t) = f for all r, then (10.19)
u(t)>u„
in H
as f> 00.
Proof. Let w(t) = u(t)uO0. We have, by differences, . dw(t) —L1^vAw(t) + Bu(t)Buoo = 0, dt 2
If n = 2, u() is unique and is a strong solution (at least for f >0). If n = 3, u() is not necessarily unique, and we must assume that u ( ) satisfies the energy inequality (see Remark 3.2).
70
PART II.
QUESTIONS RELATED TO STATIONARY SOLUTIONS
and, taking the scalar product with w(f), (10.20)
~  w ( f )  2 + v\\w(t)\\2 + b{u(t), u(t\ w($))b(u^ 2 at
u., w(r)) = 0 3 .
Hence, with (2.34), i 4 Iw(t) 2 +^ »w(r)2 = b(w(t), 2 at
u„, w(t)).
Using (2.30) with ml = \, m2 = 1, m3 = 0, we can majorize the righthand side of this equality by ciw(r) 1/2 w(r)Au 00 , which, because of (2.20), is less than or equal to c;w(0 3/2 w(t) 1/2 AuJ; with (3.10) this is bounded by 2 ^lw(t) +^3w(r) 2 Au a ^IMOIP+^sl*
where c 2 = 3( Cl /2) 4/3 . Therefore, (10.21)
4 H O I 2 + v lw(r
i w(t) 2 \AuJf»,
(10.22)
^IwW^^A.^lAuJ^lwW^O.
If (10.23)
v = ü A ^  % AuJ4/3>0,
then (10.22) shows that w(r) decays exponentially towards 0 when t—»°°:
M0Sw(0)e*, w(0) = u o u o o . Using the estimation (10.17) for u«,, we obtain a sufficient condition for (10.23), which is exactly (10.18). If we replace u(t) by another stationary solution u* of (10.7) in the computations leading to (10.22), we obtain instead of (10.22) H"£"oo 2 ^o. Thus (1023) and (10.18) ensure that u£= u«,; i.e., they are sufficient conditions for uniqueness of a stationary solution, like (10.8). The proof is complete. D 3
The proof that <w'(0, w(f)) = £(d/dt) w(f)2 is not totally easy when n = 2, and relies on [RT, Chap. Ill, Lemma 1.2]. If n = 3 we do not even have an equality in (10.20), but an inequality ^, which is sufficient for our purposes: a technically similar situation arises in [RT, Chap. Ill, § 3.6].
STATIONARY SOLUTIONS OF THE NAVIERSTOKES EQUATIONS
71
Remark 10.1. Under the assumptions of Theorem 10.2, consider the linear operator sd from D(A) into H defined by (Si, * ) = y(A , U., i/r) 4 b( Moo , t£, $
V
whose adjoint ^/* from D(A) into H is given by (st*, i/r) = v(A4>, il*) +fc(«fcu«,, <£) +fe(Woo,$ 4)
V <^>, i/f G D(AJ.
Then with the same notation as in the proof of Theorem 10.2, we have dw(t) —— + sdw(t) + Bw(t) = 0, at and (10.20) is the same as 1 w
x^w(r)2+y(^w(f),w(r)) = 0 2 at or 2 ^ Iw(r)2 + ^ —  — w(0, w(i) J = 0. The operator ( ( ^ + ^ * ) / 2 ) _ 1 is self adjoint and compact from H into itself, and the conclusions of Theorem 10.2 will still hold if we replace (10.18) by the conditions that the eigenvalues of (s/i + si*)/2 are >0. 10.2. An abstract theorem on stationary solutions. In this section we derive an abstract theorem on the structure of the set of solutions of a general equation (10.24)
N(u) = f;
this theorem will then be applied to (10.7). Nonlinear Fredholm operators. If X and Y are two real Banach spaces, a linear continuous operator L from X into Y is called a Fredholm operator if i) dimkerL<°o, ii) range L is closed, iii) coker L = Y/rangeL has finite dimension. In such a case the index of L is the integer (10.25)
i(L) « dim ker L  d i m coker L.
For instance, if L = L1 + L2 where L x is compact from X into Y and L 2 is an isomorphism (resp. is surjective and d i m k e r L 2 = q), then L is Fredholm of index 0 (resp. of index q). For the properties of Fredholm operators, see for instance R. Palais [1], S. Smale [1]. Now let co be a connected open set of X, and N a nonlinear operator from
72
PART II.
QUESTIONS RELATED TO STATIONARY SOLUTIONS
Let N b e a ^ 1 mapping from an open set o> of X into Y, X, Y being again two real Banach spaces. We recall that u e X i s called a regular point of N if N'(u) is onto, and a singular point of N otherwise. The image given by N of the set of singular points of N constitutes the set of singular values of N. Its complement in Y constitutes the set of regular values of N. Thus a regular value of N is a point fe Y which does not belong to the image N(max (index N, 0), then the set of regular values of N is a dense open set of Y We deduce easily from this theorem: THEOREM 10.3. Let X and Y be two real Banach spaces and co a connected open set of X, and let N:(o> Y be a proper c€k Fredholm map, fcäl, of index 0. Then there exists a dense open set co1 in Y and, for every fe 0 and k^q, then there exists a dense open set o)l in Y and, for every feù)u N~l(f) is empty or is a manifold in œ of class c€k and dimension qProof. We just take CJ1 = the set of regular values of N which is dense and open by Smale's theorem. For every fe col5 the set N _ 1 ( / ) is compact since N is proper. If index (N) = 0, then for every fe o)1 and u e N~x(f), N'(u) is onto (by definition of CÜX) and is onetoone since dim ker N'(u) = dim coker N'(u) = 0. Thus N'(u) is an isomorphism, and by the implicit function theorem, u is an isolated solution of N(v) = f. We conclude that N _ 1 ( / ) is compact and made of isolated points: this set is discrete. If index (N) = q^k, for every fe
X = D(A),
Y = H,
N(u) = vAu + Bu.
It follows from (2.36) that B(, •) is continuous from D(A)xD(A) and even ^3/2 x ^3/2 into H and N makes sense as a mapping from D(A) into H.
STATIONARY SOLUTIONS OF THE NAVIERSTOKES EQUATIONS
73
10.1. N:D(A)> H is proper. Proof. Let K denote a compact set of H. Since K is bounded in H, it follows from the a priori estimation (10.17) that N~\K) is bounded in D(A) and thus compact in V ^ . As observed before, B(«, •) is continuous from V 3 / 2 x V3/2 into H, and thus J B ( N  1 ( 1 0 ) is compact in H. We conclude that the set N _ 1 ( K ) is included in LEMMA
v'A'iKBiN'iK))), which is relatively compact in D(A), and the result follows. D We have the following generic properties of the set of stationary solutions to the NavierStokes equations. THEOREM 10.4. We consider the stationary NavierStokes equations in a bounded domain (fl or Q) with periodic or zero boundary conditions, and n = 2 or 3. Then, for every v>0, there exists a dense open set CV<^H such that for every feOv, the set of solution of (10.5)(10.7) is finite and odd in number. On every connected component of Ov, the number of solutions is constant, and each solution is a c#°° function of f. Proof, i) We apply Theorem 10.3 with the choice of X, Y, N indicated in (10.26) (and w = X). It is clear that N is a ^°° mapping from D(A) into H and that (10.27)
N'(u)'V = vAv + B(u,v) + B(v,u)
It follows from (3.20) that for every ueD(A),
v^B(u,v),
Vu,veD(A). the linear mappings
v^B(v,u)
are continuous from V into H and they are therefore compact from D(A) into H. Since A is an isomorphism from D(A) onto H, it follows from the properties of Fredholm operators (recalled in § 10.2) that N'(u) is a Fredholm operator of index 0. We have shown in Lemma 10.1 that N is proper: all the assumptions of Theorem 10.3 are satisfied. Setting Ov = the set of regular values of N = Nv, we conclude that ûv is open and dense in H and, for every feOv, N~1(f), which is the set of solutions of (10.7), is finite. ii) Let (Oi)ieI be the connected components of €v (which are open), and let /o> A t>e t w o points of Oi for some i. Let u0eN~1(f0). There exists a continuous curve S
E[0,lM(s)e^,
/(0) = /o,
/(D=/i,
and the implicit function theorem shows the existence and uniqueness of a continuous curve s*+u(s) with N(u(s)) = f(s),
u(0) = uo.
Since f(s) is a regular value of N, for all s e[0, 1], u(s) is defined for every s, O ^ s ^ l , and therefore u(l)eN~ 1 (/ 1 ). Such a curve {S»»M(S)} can be con
74
PART II.
QUESTIONS RELATED TO STATIONARY SOLUTIONS
structed, starting from any uk e N _ 1 (/ 0 ). Two different curves cannot reach the same point u^eN"1^^ and cannot intersect at all, since this would not be consistent with the implicit function theorem around u* or around the intersection point. Hence there are at least as many points in N  1 ^ ) as in N _ 1 (/ 0 ). By symmetry the number of points is the same. It is clear that each solution uk = uk(f) is a c#°° function of f on every O. iii) It remains to show that the number of solutions is odd. This is an easy application of the LeraySchauder degree theory. For fixed v>0 and feOv, we rewrite (1.7) in the form (10.28)
%(u) = vu + AxBu
= A~lf = g,
u, g e D ( A ) . By (10.17), every solution uK of Tv(uK) = Ag, O^A ^ 1, satisfies \Auk\
R = 1 + ^  /  + ^ l/l3.
Therefore the LeraySchauder degree d(Tv, Ag, BR) is well defined, with BR the ball of D(A) of radius JR. Also, when Ag is a regular value of Tv, i.e., A/ is a regular value of N, the set T~*(Ag) is discrete = {uu..., u k }, and d(Tv, Ag, B R ) = Z?=i i(Uj) where i(u,) = index u. It follows from Theorem 10.1 that there exists A ^ G [ 0 , 1], and for O ^ A ^ A*, N~l(\f) contains only one point ux. By arguments similar to that used in the proof of Theorem 10.1, one can show that N'(uk) is an isomorphism, and hence for these values of A, d(Tv, Ag, B R ) = ± 1 . By the homotopy invariance property of degree, d(T v , g, BR) = ±1 and consequently k must be an odd number. The proof of Theorem 10.4 is complete. D Remark 10.2. i) The set 0 is actually unbounded in H ; cf. C. FoiasR. Temam [13]. ii) Similar generic results have been proved for the flow in a bounded domain with a nonhomogeneous boundary condition (i.e.,^0 in (1.9)): generic finiteness with respect to / for fixed,with respect to <£> for / fixed and with respect to the pair /,; see C. FoiasR. Temam [3], [4], J. C. SautR. Temam [2], and for the case of time periodic solutions, J. C. SautR. Temam [2], R. Temam [7]. iii) When ft is unbounded, we lack a compactness theorem for the Sobolev spaces H m (ft) (and lack the Fredholm property). We do not know whether results similar to that in Theorem 10.4 are valid in that case; cf. D. Serre [1] where a line of stationary solutions of NavierStokes equations is constructed for an unbounded domain ft. We now present another application of Theorem 10.3, with an operator of index 1, leading to a generic result in bifurcation. We denote by S(f, i>)<= D(A) the set of solutions of (1.5)(1.7) and (10.29)
S(f)=
U S(f,v). v>0
STATIONARY SOLUTIONS OF THE NAVIERSTOKES EQUATIONS
FIG.
75
10.1
THEOREM 10.5. Under the same hypotheses as in Theorem 10.4, there exists a Gôdense set C<^H, such that for every feû, the set S(f) defined in (10.29) is a 9?°° manifold of dimension 1. Proof. We apply Theorem 10.3 with X = D(A) xR, Y = H, co = ù)m = D(A) x(1/m, <*>)<= X, meN, N(u, v) = vAu+Bu, for all (u, v)eX. It is clear that N is %™ from
N'(u, v)  (v, LI) = vAv + Biu, v) + B(v, u) + /u,Aw. For (u, v)eB(u, u) + B(t;, U ) + JULAU, 4
which is compact , and the operator (u,
/UL)
>
MD,
which is onto and has a kernel of dimension 1. Thus N'(u, v) is a Fredholm operator of index 1 and N is a nonlinear Fredholm mapping of index 1. The proof of Lemma 10.1 and (10.17) shows that N is proper on D(A)x (i^o, °°), for all v0>0 and in particular v0=l/m. Hence Theorem 10.3 shows that there exists an open dense set €m<^H, and for every feÛ^N^if) is a manifold of dimension 1, where N m is the restriction of N to com. We set 0 = Ç)m^i@m, which is a dense G8 set in H, and, for every feO,S(f) = Umäi Nml(f) is a manifold of dimension 1. Remark 10.3. i) By the uniqueness result in Theorem 10.1, S(f) contains an infinite branch corresponding to the large values of v. ii) Since S(f) is a ^°° manifold of dimension 1, it is made of the union of curves which cannot intersect. Hence the usual bifurcation picture (Fig. 10.1) is nongeneric and is a schematization (perfectly legitimate of course!) of generic situations of the type shown in Fig. 10.2. Remark 10.4. Other properties of the set S(f, v) are given in C. FoiasR. Temam [3], [4] and J. C. SautR. Temam [2]. In particular, for every v*f, S if, v) is a real compact analytic set of finite dimension. 4
Same proof essentially as in Theorem 10.4.
76
PART II.
QUESTIONS RELATED TO STATIONARY SOLUTIONS
10.4. Counterexamples. A natural question concerning Theorem 10.4 is whether ûv is the whole space H or just a subset. Since Theorem 10.4 is a straightforward consequence of Theorem 10.3, this question has to be raised at the level of Theorem 10.3. Unfortunately, the following examples show that we cannot answer this question at the level of generality of the abstract Theorem 10.3, since for one of the two examples presented ûv = H, while for the second one Ov±H. Example 1. The first example is the onedimensional Burgers equation which has been sometimes considered in the past as a model for the NavierStokes equations: consider a given v>0 and a function / : [ 0 , 1 ]  * R which satisfies d2u du v — +u — = f dx dx
(10.30) (10.31)
on (0,1),
u(0) = u(l) = 0.
For the functional setting we take H = L 2 (0,1), V = Hl(0,1), Hl(0, l ) n H 2 ( 0 , 1 ) , Au = ~d2uldx2, for all ueD(A), dv B(u,v) = u—, dx
Vu, veV9
Bu=B(u,u).
Then, given f in H (or V ) , the problem is to find ueD(A) satisfies (10.32)
D(A) =
(or V) which
vAu + Bu=f.
We can apply Theorem 10.3 with X = D(A), Y = H,N(u) = vAu + Bu. The mapping N is obviously ^°° and N'(w) • v = vAv + B(u, u) + B(u, u); N'(u), as the sum of an isomorphism and a compact operator, is a Fredholm operator of index 0. Now we claim that every ueD(A) is a regular point, so that (ol = H. In order to prove that u is a regular point, we have to show that the kernel of N'(u) is 0 (which is equivalent to proving that N'(u) is onto, as index N'(u) = 0). Let v belong to N'(u); v satisfies vC^+u^ dx
dx
+ v^ = 0 on (0,1), dx
t>(0) = u(l) = 0.
By integration in;'4 uv = constant = a, and by a second integration taking into account the boundary conditions we find that v = 0. Since the solution is unique if / = 0 (u = 0 ) , we conclude that (10.33)
(10.30M10.31) possesses a unique solution
V v, V /.
Example 2. The second example is due to Gh. Minea [1] and corresponds to a space H of finite dimension; actually H = U3.
STATIONARY SOLUTIONS OF THE NAVIERSTOKES EQUATIONS
77
Theorem 10.3 is applied with X= Y = U3, N(u) = vAu+Bu for all u = (ul9 u 2 , U 3 ) G R 3 . The linear operator A is just the identity, and the nonlinear (quadratic) operator B is defined by Bu = (ô(ul\u3l),—ÔUiU2,—ôu1u3) and possesses the orthogonality property Bu • u = 0. The equation N(u) = f reads
vu2Sulu2 = f2, vu3 — 8u1u3 = f3. It is elementary to solve this equation explicitly. There are one or three solutions if / 2  +1/ 3  7^ 0. In the "nongeneric case", f2 = f3 = 0, we get either one solution, or one solution and a whole circle of solutions: U^ ^ Y in this case.
The Squeezing Property In this section we establish a squeezing property of the semigroup of operators associated with the timedependent NavierStokes equations. This property, which will be useful for the next section, is also interesting by itself and has found other applications (see in particular FoiasTemam [7]): the squeezing property shows that, up to some error which can be made arbitrarily small, the flow is essentially characterized by a finite number of parameters. 11.1. An a priori estimate on strong solutions. The following a priori estimate is valid for all time ; it is a particular case of the more general results stated in § 12.3 (Lemma 12.2); we refer the reader to that section for more details. LEMMA 11.1. Assume that u0eH, and f satisfies f is continuous and bounded from [0, ») into H, /' is continuous and bounded from [0, ») into V , and let u be the strong solution to the NavierStokes equations given by Theorem 3.2, defined on [0, °o) //
n
= 2, on [0, T^UQ)] if n = 3.
Then u is bounded in V on [0, &)ifn = 2, on [0, T^UQ)] ifn = 3, and for every a > 0 , Au is bounded on [a, o°) if n = 2, on [a, T^UQ)] if n = 3,
(11.2)
supu(r)^c'(iio,/,v,n),
(11.3) 1
supAu(t)ëc"(a,u 0 ,/ ( v,fi)
where c' depends on u0, /> v* ß (or Q), and c" depends on the same data and furthermore a. Remark 11.1. i) If n = 3, (11.2) follows from (3.28)(3.29), but (11.2) must be proved for large time if n = 2. ii) If u is defined on some interval (T 0 , TX) and sup To3StsiT Ju(f)^ci0 , sup T o + a < t ^ T l Au(f)^cï, where c\ depends on c[ and the data u 0 , /, v, Ü (or Q). 11.2. The squeezing property. We assume that n = 2 or 3. Let u 0 and v0 be given in V, HuoH^i*, u0 = ^ and let / be given satisfying f is continuous and bounded from [0, o°) into H, / ' is continuous and bounded from [0, ») into V. 1 0
80
PART II.
QUESTIONS RELATED TO STATIONARY SOLUTIONS
We denote by u and v the strong solutions of NavierStokes equations corresponding respectively to (u 0 , / ) and to (v0, / ) , which are given by Theorem 3.2, and are defined on (0, o°) if n = 2 and on [0, TxCR)] if n = 3, where Tl(R)
(11.5)
=
(l + R2)2
(see (3.29)). We recall that vvm, Am denote the eigenfunctions and eigenvalues of A and that Pm is the projector in H (or V, V, D(A)) on the space Wm spanned by w 4 , . . . , wm. THEOREM 11.1. Under the above hypotheses, for every a > 0 , there exist two constants c5, c6, which depend on a, R, f, v, ft (or Q) and such that, for every t^a, for every m sufficiently large, i.e. satisfying (11.6)
A m + 1 ^c 6 ,
we have either (11.7)
u(r)i;(t)^V2P m (u(r)i;(0)l
ur
u(t)i>(t)ëec'x.
(11.8)
Proo/. We assume in this proof that t £ a/2 if n = 2, a/2 S t S T^J?) if n = 3. i) We set U
vv(t) = u ( 0  u ( 0 , p(t) = Pmw(t),
'
q(t) = (/F m )w(().
We have, by taking the difference of the equations (2.43) for u and v, dw — + Aw + B(u, w) + B(w,u) = 0. at
(11.10)
Taking the scalar product, in H, of (11.10) with Pmw(t), we obtain
i£p 2 +Hp m w 2 2 at (11.11)
=b(u,w,P m w)b(w,t;,P m w) = b(u, Pmw, (IPm)w)b(Pmw,
v, Pmw)
b((IPm)w, v, Pmw) (with (2.33M2.34)). We apply Lemma 2.1 (i.e. (2.29)) respectively with m1( m2, m3 = 2, 0,0; \, 1, \; 0 , 1 , 1 , and we use (11.3) for u and v, with a replaced by a/2: b(u, Pmw, (IPJw) (11 12)
_fc P
is  C l c " Pmw q,
W
< m . V, P m w)âC lC "P m wf /4 âcip 3 / 2 P m w 1 / 2  b ( ( I  P m ) w , v, PmW)^Clc"q
(with (2.20)),
\\Pmw\\.
THE SQUEEZING PROPERTY
81
With these lower bounds, (11.11) becomes ^p2^^Pmw2c^Pmwc;p3/2Pmwr2.
(11.13)
Since AxSAm+i and PmwSA^lPmw^AjaiR we have p ^ ^  p ( v A m + 1 p + c;A^ 1 Ar 1 / 4 p + ^A^ / 2 r l q). at In a similar manner, taking the scalar product in H of (11.10) with we obtain (11.14)
(IPm)w,
iq2+v(7Pm)w2S^p(/Pm)w + c;(JPm)wrq3'2. But (11.15)
(/Pm)wâA^+1(/Pm)wâAj/2)
and thus q^\\(IPm)wHv\^q + c^ + c^l'4q). at We set A = A m+1 , p = max( C ;A7 1/4 , c'2, c'it c^m). Thus (11.14)(11.16) take the form: (11.16)
(n.17)
p ^ S  p ( v A p + pp + pA1/2q), ™ qftmiPM\(^mq+PP
+ Pq),
and if (11.18)
 O A ^  p ^ + pp^O, 2
then for almost every t
(11.19)
at
f s A ^  O ^  p f o + pp]. ii) We now want to prove the alternative (11.7)—(11.8). Let us consider a 2 The first relation of (11.19) follows easily from the first relation of (11.17) at a point f0 where p(r o )>0. If p(r0) = 0 then either p is not differentiable at r0 (which can only occur on a set of points f0 of measure 0) or dp(to)/dt^0. The differentiability almost everywhere of p follows trivially from the differentiability almost everywhere in V" of u and v, and thus w = uv.
82
PART II.
QUESTIONS RELATED TO STATIONARY SOLUTIONS
specific point t0, t0 = a if n = 2, a^t0^Tx(R) if n = 3. If q(t0)^p(t0) then (11.7) is satisfied at f0 and we have nothing to prove. Therefore we assume that (11.20)
q(t0)>p(to\
and on the other hand we assume that m is sufficiently large so that v\mp>2p.
(11.21) This implies that
(v\mp)q(t)>2pp(t),
(11.22)
in a neighborhood of (0At this point two possibilities can occur: either (11.22) is valid on the whole interval [t0a/2, t 0 ], or (11.22) is valid for te(tu t0), with tx^t0 —a/2 and (vkmp)q(t1)
(11.23)
= 2pp(t1).
The second possibility is discussed in point iii) of the proof; in the first case, since (11.18) and (11.19) are satisfied on [t0a/2, t0], we have
y*—
(^p)q(r),
r.[r 0 ,r 0 J,
and consequently (11.24)
q(to)âexp (   A 1 / 2 ( v A 1 / 2  p ) « ) q ( t 0   ) .
Due to (11.2),
«('°i)4('«DI^ and (11.8) follows at t0 with appropriate constants c 5 , c 6 . iii) The last step consists in proving (11.7) at t = t0, a ^ ^ ( ^ T ^ J R ) if n = 3), assuming that (11.22) is satisfied on an interval (tu f0), and (11.23) is satisfied at tx (r 0 <*/2 ^ *!<*())• We denote by (11.19)' the differential system obtained by replacing the inequalities in {11.19) by equality signs. It is natural to formally associate with (11.19)' the differential system dp_(v\1/2 + p)p + pq dq (v\l,2p)q~pp' We prove by elementary calculations that 4>(p(r), q(t)) remains constant when p(), q(') are solutions of (11.19)' and (11.26)
/ v\ll2q \
83
THE SQUEEZING PROPERTY
By elementary calculations we also check that at when p(), q(') are solutions of (11.19), in particular on [fto t0]. Thus
By (11.23), <Wp(h), qUi)) =
H 2p
q(h) • exp
( ^ ^ J ,
and on the other hand, with (11.20), *(p(lo),
(11.27)
q(fOSwfri)Sc7^ we
and p(t0)
ll(f)^ci
SUp
ü(r)^cl
so that sup
Au(r)^cï
sup
Au(r)^c'i<«>.
ii) Of course we can replace the coefficient V2 in the righthand side of (11.6) by any constant ß > 1. It is clear that in such case c5 and c 6 depend also on ß. Remark 11.3. It is reasonable to consider as physically "undistinguishable" two strong solutions u, v, which satisfy (11.8) for some m sufficiently large. Mathematically, we can consider an equivalence relation among the strong solutions which are defined on (T 0 , TX) (and bounded in V on this interval): two solutions u, v are said to be equivalent if there exist two constants c5, c 6 such that (11.6)(11.8) hold on (T 0 , T^. It is easy to check that this relation is indeed an equivalence one. The squeezing property implies that Pmu characterizes the equivalence class of u, i.e., characterizes u up to an error exp (ch^i). Remark 11.4. Let us mention an alternate form of the squeezing property which is useful for later purposes.
84
PART II.
QUESTIONS RELATED TO STATIONARY SOLUTIONS
Let u and v be two strong solutions of the NavierStokes equations, defined and bounded in V on some finite interval T 0 ^ t ^ rx < <». For every a > 0, there exist two constants c*, c* which depend on r 0 , ru a, £ , /, v, H 3 , such that, on [T 0 + CX, T J , we have either (11.7) or (11.28)
\u(t)v(t)\^\u(t0)v(t0)\exV(cnXlù
for every m satisfying (11.29)
Xm+1^ct
The proof is exactly the same, except that in (11.25) and (11.27) we replace the majorizations
»4*fM*f)
;^M
N(t^ti(fi)<4
VA, VÀ! by the estimate ( ^ C "  W ( T 0 )  ) which follows from the next lemma. LEMMA 11.2. If u and v are two strong solutions defined and bounded in V on a finite interval T o ^ t ^ T i , n = 2 or 3, then there exists a constant c" which depends on T 0 , r l 5 /, v, d (or Q), and R, with (11.30)
R^
sup u(0,
R^
te[T ( ) ,T,]
sup t>(0ll, te[T 0 ,T,]
such that (11.31) \u(t)v(t)\^cff'\u(T0)v(r0)\ V r e ^ r j . Proof. It follows easily from the energy inequality (3.24), valid for strong solutions, and from the assumptions on u and v, that Au and Av belong to ^2(TO> TI ; H) with their norm in this space bounded by a constant depending on T0, Ti, /, v, H ( o r Q ) and JR. We take the scalar product in H of (11.10) with w(f), and using (2.29), (2.33), (2.34) and (11.30) and Remark 11.1 ii), we get, as for (11.13): ^w2+HHI2"Mw,U,w)^cjAt;wriwr ^\\w\\2
(11.32)
+
c'5\W\2\Avr,
ft\W\2të2c'5\AvrW\
and anc\ since since \Av\ \Av4/3e L\0, T), (11.31) follows from Gronwall's lemma.
»Here Ä£sup r « [TfcTl] u(t), RÊsup t e [ ^ T l ] Mt).
D
J^ J Hausdorff Dimension of an Attractor In this section we derive an interesting property of a functional invariant set which is bounded in V, and in particular of an attractor bounded in V (see the definitions below). This holds in space dimension n = 2 or 3, but for n = 2, we also prove that a solution w() of the NavierStokes equations tends in H, as f—>», to a functional invariant set bounded in V (distance in H of the solution to the set » 0). When it applies, this result shows that the flow depends, for large t, on a finite number of parameters. Another property of a functional invariant set is established in § 13. 12.1. Functional invariant sets and attractors. Throughout this section, we assume that f(t)=f is independent of t and belongs to H. Let S(t) be the semigroup associated with the strong solution of the NavierStokes equation du „ —+ vAu+Bu=f J dt i.e., the mapping u0 = u(0)**u(t), which is defined on V for all r ^ O if n =2, and for te[0, T^UQ)] (at least), if n =3. DEFINITION 12.1. A functional invariant set for the NavierStokes equations is a subset X of V which satisfies the following properties: i) For every u0e X,S(t)u0 is defined for every f > 0 . ii) S(t)X=X for all t>0. We recall that an attractor (in V or H) is a functional invariant set X which satisfies furthermore the condition: iii) X possesses an open neighborhood o) (in V or H), and for every u0ea), S(t)u0 tends to X, in V or H, as t —> oo. If n=2, we have the following: LEMMA 12.1. Let us assume that n = 2, and that u0 and f are given in H. Then there exists a functional invariant setX<^ V, compact in H, such that the distance in Hofu(t) to X tends to 0 as t —» o°, where w() is the solution of Problem 2.1. Proof We set (12.1)
X=riW(s),s>r} TâO
where the closures are taken in H. An element e H is in X if and only if (12.2)
Ve>0,
VT>0,
3S>T
suchthat«(s)4>^e.
It is easy to see that X is also equal to (12.3)
D {u(s),s>r} 85
(t0^0).
86
PART II.
QUESTIONS RELATED TO STATIONARY SOLUTIONS
Since for t 0 > 0 , the set (12.4)
{u(s\s^t0}
is bounded in V (cf. Lemma 11.1), so is its closure in V and so is X. By Theorem 3.2, since n = 2, S(t)is defined for all t > 0, for every eX, and the first condition i) in Definition 12.1 is satisfied. For condition ii) we observe that S ( ( ) X c X follows easily from the fact that S(t)u(s) = u(s + f), for all s, f^O. In order to show that S(t)X = X, we note that S(t) is injective because of the time analyticity property (see § 7) and therefore since the set (12.4) is relatively compact in H(f o >0), the sequence u(Sj),jeN, converges in H whenever the sequence S(t)u(Sj) converges. Indeed, if S(f)w(s,) converges in H to some limit , as /—>oo (and ty —• «>), then by (12.2), <£eX. The sequence u(s J ), being bounded in V, is relatively compact in H and possesses cluster points in H ; if i/f and (// are two cluster points, S(f)i/f = S(f)i// = so that «// = i//, the cluster point is unique and the whole sequence w(s,) converges to it in H as /—•«>. Finally we show that the distance in H of u(r) to X tends to 0 as t —> *. We argue by contradiction. If this were not true, we could find e0>0, and a sequence sy —» o°? such that V
inf  u ( s , )  < H ^ e o > 0
/
4>eX
Since u(sy) is bounded in V we can, by extracting a sequence, assume that u(s ; ) converges in H to some limit i/f. By (12.2), i^eX, in contradiction with M(Sj)0 The lemma is proved.
V/.
D
12.2.
Hausdorff dimension of functional invariant sets. 12.1. Let n = 2 or 3 and letfeH. We assume that X is a functional invariant set bounded in V. Then the Hausdorff dimension of X is finite and this dimension is bounded by a number c, which depends on CI (or Q), v, \f\ and1 ^=sup^eX^. Proof, i) We are going to make use of the squeezing property, in the conditions indicated in Remark 11.4: we fix a>0 and choose T 0 = 0, r1 = 2a, and te[a, 2a] 2 . We then choose m sufficiently large so that (11.29) is verified and (see (11.28)) THEOREM
(12.5)
cxp(cWJili)^V,
and the value (or one value) of r\ which is appropriate below is (12.6)
= 71
1
2V2 8
The bound given in the proof is, in fact, an increasing function of v \ / and R. Since, by the definition of a functional invariant set, S(t)is defined for every r^O, for every <j>eX we are free to choose T 0 , TX in Remark 11.4, O ^ T O ^ T ^ + Q O . 2
HAUSDORFF DIMENSION OF AN ATTRACTOR
87
We set S(r) = S, then Theorem 11.1 and Remark 11.4 show us that for all <£, i/f G X, we have either (12.7)
S<ÊSi/^V2P m (S<ÊSi/r)
or
\S4>S^\^7\
*f.
Given r > 0, since X is relatively compact in H, it can be covered by a finite number of balls of H of radius
(12.8)
X c U Jc =
S(J3 k nX) l
and (12.7) implies that for all k = 1 , . . . , M, and for all u,ve S(Bk OX) we have either (12.9)
\uv\^\f2\Pm(uv)\,
or
MD^T
diam Bk.
Letbe an arbitrary point of S(Bk OX) and let Yk be the largest subset of S(Bk flX), containing /2Pm(*0)
ViA,ÖGY k ;
3
of course Yk may be reduced to <£ . ii) We now define a reiterated covering of X. Let E f t , . . . , BSvfk be a covering of P m S(B k H X ) by balls of PmH of radius ^ r k = 5 d i a m ß k . For those ;'s such that PmYk CiB™,^ 0 , we choose arbitrarily an ukJ G Yk, such that PmukJ ePmYkDBknj, and we denote by Bkj the ball of H centered at ukj and of radius r k = y/2 r'k + TJ diam B k = ( — I TJ 1 diam B k . Setting B kj = 0 if B53nP m Y k  0 , we have (12.10)
S(BkOX)c=UBkj,
Indeed, if uGS(B k OX), then there exists u'e Yk (u' = u if UG Yk) such that, by (12.9) and the definition of Yk, u — u'l^T) diamB k . Then  u  u k j .  ^  w  M l + w'u k j ^TidiamB k +V2P m ( M 'u k j ) ^TjdiamBk+\/2rk. Before proceeding to the next step, let us observe that we can derive an upper bound for M k . 3
If Yk * S(Bk O X) then sup„ e S ( B k n x ) X Y k dist (ifc Yk) ^ TJ diam Bk.
88
PART II.
QUESTIONS RELATED TO STATIONARY SOLUTIONS
By Lemma 11.2, \SSil/\^c"\ il/\
V
hence \uv\^c"diam (12.11)
Bk
V u,ve S(Bk HX),
diam S(Bk H X ) ^ c " d i a m Bk, diam P m S(B k H X) ^ c" diam B k .
Since PmS(Bk H X ) is included in a ball of P m H of radius c" diam B k , we have (12.12)
Mk
^\4cw/'
where /m(o\) is, in (Rm, the minimum number of balls of radius ^a necessary to cover a ball of radius 1, lm (a) ^ 2~m/2cr~m. iii) We have
which is
diam B kj ^ ( — + 2TJ J diam B k ^ediamBk
(with the choice (12.6) of TJ)
(e = (2V2)/4); with (12.10)(12.12), for any
7
>0,
£ (diam B k / ) ^ M k £ n d i a m BkV ^ ^ ( ^ ^ ( d i a m
Bk)\
Since the B kj , k = 1 , . . . , M, j = 1 , . . . , M k , for all k, constitute a covering of X by balls of radius ^er, we infer from the definition of JLLXS(X) in (5.2) that ^, e r (X)=g I
£ (diamBwrs^rWeL — K
k=l j=l
^
c
/
£ (diamB k )\ k= l
This last inequality, valid for any covering of X by balls of radius ^r, implies in its turn (12.13)
^ > e r (X)^Afx^ r (X),
where A = lm(l/4c")ey. (12.14)
By reiteration, ^,eJr(X)^AKr(X).
Since s < 1, if y is sufficiently large, then A < 1 and AJ » 0 as / —» o°, so that liyiX) = sup iLyJX) = lim fxxeV(X)  0. p>0
HAUSDORFF DIMENSION OF AN ATTRACTOR
89
Therefore, provided (12.15)
7S
_logUl/4c") log((2V2))/4
the Hausdorff dimension of X is ä=y. The proof is complete. D Remark 12.1. We do not know, even if n = 2, whether every functional invariant set is bounded in V. But Lemma 12.2 shows us that this is the case for those associated with the limit of u(t) as f —»oo. The question of the boundedness of X in V is totally open if n = 3. 12.3. Other properties of functional invariant sets. We conclude this section by indicating some regularity results and a priori estimates which are valid for long times, and we infer from these estimates some properties of functional invariant se'ts; in particular, if / is sufficiently regular and n = 2, a functional invariant set is necessarily carried by the space of ^°° functions. LEMMA 12.2. Let n = 2 or 3, and consider the timedependent NavierStokes equations with the data u0, f satisfying (12.16) (12.17)
u0eH, d'f —7 is continuous and bounded from [0, ») into (HT dt' i L , ;
2j 2
(û) n H,
/ = 0,...,/l, (12.18)
dlf —j is continuous and bounded from [0, ») into Vm_2l_1,
where m is an integer ^1 and l = [m/2]4. We assume that the solution u to Problem 2.2 satisfies (12.19)
u e L°°(t0, oo; V)
/brsometo^O.
Then u is continuous and bounded from [TJ, oo) into Mm(Û), d]u\dt] is continuous and bounded from [TJ, OO) into Hm~2](€), j = 1,. . . , /, V TJ > 0 , and the corresponding norms of u and d]u/dtJ are bounded by constants which depend only on the data, t0, rj, and uL~(t0>oo; v ) . We do not give the proof of these a priori estimates, which is rather long; we refer the reader to C. Guillopé [1]. We recall that, if n = 2, then (12.19) is automatically satisfied for every t 0 > 0 and the result holds for TJ > 0 arbitrarily small. When n = 3, we do not know whether (12.19) is true; however, the boundedness of u(f) for t —> oo is closely related to Leray's conjecture on turbulence. Let us recall that Leray's hypothesis, and his motivation in introducing the concept of weak solution was that singularities may develop spontaneously
90
PART II.
QUESTIONS RELATED TO STATIONARY SOLUTIONS
over a finite interval. We know that \u(t)\ remains bounded even for weak solutions but the enstrophy u(t) may perhaps become infinite. We formulate Leray's assumption in a more precise way: (12.20)
There exist n<=[R3, T > 0 , v>D, u0e V, feLT(0, T; H) such that M(0 becomes infinite at some time te(0, T), u being the weak solution of the NavierStokes equations.
Let us assume now that fe L°°(0, <»; H) and is not "too chaotic" at infinity, i.e., (12.21)
There exists a > 0 such that for any sequence (j —» +oo5 the sequence of functions fj(t) = f(t + ti),0
This property is trivially satisfied if / is independent of time, or more generally if (12.22)
/eLJL(0,oo; V), f € 1 ^ ( 0 , « ; V )
with l/lL2(s,S+a;V) + /lL2(s,S+a;V') =
C
(/)^
for all s > 0 , a > 0 , where c(/) is independent of s. In such a case we have the following result proved elsewhere (cf. C. FoiasR. Temam [13]): 3 THEOREM 12.2. Given CteU , T, v and /eL°°(0,oo ; H) satisfying (12.21), then either (12.19) or (12.20) holds; i.e., there exists u0eV for which \\u(t)\\ becomes infinite on the interval [0, a ] , a > 0 given in (12.21). Finally we infer from Lemma 12.2 the following regularity result for a functional invariant set (or attractor): n 5 THEOREM 12.3. Lef us assume that f is independent of t, and f e %~({f) f) H . Then any functional invariant set X, bounded in V, for the corresponding NavierStokes equations, is contained in ^°°(ü) n . Proof. We consider the semigroup operator S(t) defined at the beginning of § 12.1. Since S(t)X = X, for all f > 0 , and S(t) is onetoone by the analyticity property, everyeX is of the form = S(t)u0, with u0eX, and therefore by Lemma 12.2, is in Hm(Ö), for all m. Remark 12.2. i) This result is comparable to regularity results for the solutions of stationary NavierStokes equations, but it applies also to periodic, and quasi periodic solutions, among others. ii) The conclusion of Theorem 12.2 is valid if we assume only that X is bounded in V1/2+e for some e > 0 . See C Guillopé [1] for the details. 5
If we consider the flow in ft(0 = ft), we also need (1.7) with r = °o.
PART III Questions Related to the Numerical Approximation
Orientation. In Part III, containing §§13 and 14, we develop some results related to the numerical approximation of the NavierStokes equations. This is far from being an exhaustive treatment of a subject which is developing rapidly, due to the needs of modern sophisticated technologies and the appearance of more and more powerful computers. Part III does not discuss any practical developments concerning implementation of computational fluid dynamics algorithms1 (though these are important). It is limited to the presentation of two questions related to the numerical analysis of the NavierStokes equations. The first is the question of stability andj convergence of a particular nonlinear~scheme; this is developed in §13. The particular scheme was chosen almost arbitrarily among several schemes presently used in large scale computations and for which convergence has been proved. The second question, considered in § 14, is that of the approximation of the NavierStokes equations for large time: in the context of the questions studied in § 12 (see also § 9), the problem of numerical computation of turbulent flows is connected with the computation of u(t) for t large (while the force, or more generally the excitation, is independent of time). Section 14 contains some useful remarks pertaining to this question. 1
See the comments following § 14.
2) Finite Time Approximation In this section we study the convergence of a space and time discretization scheme for the evolution NavierStokes equations. This scheme combines a discretization in time by an alternating direction (or decomposition) method with a discretization in space by finite elements. It belongs to a series of efficient schemes which have been introduced and studied from a theoretical point of view in the past (cf. A. J. Chorin [2], [3], [4], R. Temam [1], [2], [5], [6]), and which have recently regained interest and have been implemented in large scale computations for industrial applications (cf. e.g., H. O. BristeauR. GlowinskiB. MantelJ. PeriauxP. PerrierO. Pironneau [1], R. GlowinskiB. MantelJ. Periaux [1]). The proof of convergence is the same as that appearing in [RT, Chap. Ill, §§5, 6, 7] and in R. Temam [1], [2], [4], [5] for closely related schemes. However, for the proof of the strong convergence result which is necessary for the passage to the limit, we use a compactness argument different and slightly simpler than that considered in those references. This compactness result is given in § 13.3. 13.1. An example of spacetime discretization. We consider the flow in a bounded domain with periodic or zero boundary condition (i.e., 0 =ft or Q), and n = 2 or 3. We are given u0 and / as in Theorem 3.2 and we want to approximate the strong solution u to Problem 2.2 defined in Theorem 3.2, on [0, T]ii n = 2, on [0, T J if n = 3. We are given a family of finite dimensional subspaces Vh, hedf£, of V, such that (13.1)
(J Vh
is dense in V.
For instance we can take X = N, V h =the space spanned by the eigenvectors w b . . . , wh, and the discretization would correspond to a Galerkin method based on the spectral functions. Also Vh may be a more general subspace corresponding to a more general Galerkin method, but the most interesting case we have in mind is that in which Vh is a space of finite element functions with the set 9C properly defined: we refer to [RT, Chap. I, § 4] for the precise definition of a possible finite element space (Approximation APX4). Actually, in order to be able to treat more finite element schemes we will consider the following more general situation: (\T> 9Ï *}
U n
,
^ h ' ^ € ^ ' *s a family of finite dimensional subspaces oi W = HJ(Û) or Hl(Q\ such that \Jh^ Wk is dense in W. For every h, Vh is a subspace of Wh, such that the family Vh, heffl, constitutes an external approximation 1 of V.
1 For the precise definition of an external approximation cf. [RT, Chap. I, § 3.1]. The reader who so wishes may concentrate on the first, simpler case: Vh<=V with (13.1).
93
94
PART III. QUESTIONS RELATED TO THE NUMERICAL APPROXIMATION
The cases covered by (13.2), (13.3) contain the finite element approximations of V called APX2, APX3, APX3' in [RT, Chap. I, §4] and others published in the literature (cf. M. BercovierM. Engelman [1], V. GiraultP. A. Raviart [1], among others). For every h, let uoh be the projection of u0 in W, on Vh, i.e., uoh e Vu,
(13.4)
(("OK, vh))
= ((u0, vh))
Vvhe
Vh.
2
Let N be an integer , k = T'/N. For every h and k we recursively define a family u™+1/2 of elements of Vh, m = 0 , . . . , N1, i = 1, 2. We start with "H =M oh
(13.5)
Assuming that u£ (or more generally u™, m ^ O ) is known, we define u™+1/2 m
m+ 1/2
(13.6)
Uh
w
G Vh,
J("r 1/2 u:^K)+2"(w +1/2 ^K))=(/ m ,^) VüH6vH,
where i
(i3.7)
r=7
k Jmm k
and (13.8)
r ( m + l)k
1
1
«d*
v
ll
l(ur uv \vh)+{{ur\vh))^b{ur\ur\vh)=() Here 3 b(w, u, w) is the modification of the nonlinear term which was introduced in R. Temam [1], [2] and which guarantees the existence of a solution to (13.8), and the stability of the approximation for (13.8) or similar schemes: (13.9)
b(u,v,w)=
J
\ I {ui[{Divi)wivi{Dlwi)'\}
dx.
The existence and uniqueness of a solution u^ + 1 / 2 for (13.6) follow from the Riesz representation theorem (or LaxMilgram theorem). The existence of a solution u™+1 e Wh of (13.8) (a nonlinear finite dimensional problem) follows from the Brouwer fixed point theorem, using [RT, Chap. II, (2.34) and Lemma 1.4]. Problem (13.6) is just a variant of the linear stationary Stokes problem, and (13.8) is a relatively standard nonlinear Dirichlet problem. We refer to Glowinski, Mantel and Periaux [1] and to F. Thomasset [1], for the practical 2
With the notation of Theorem 3.2, V = T if n = 2, V = T* = min (T, Tx{uQ)) if n = 3. b(u,u, w) may be ^ 0 , while 6(u, u, w) = 0, for u, veHlp(fl) (or Hlp(Q)), by lack of the free divergence property. 3
FINITE TIME APPROXIMATION
95
resolution of these problems. We note that, as usual (cf. R. Temam [3], [RT, Chap. Ill, § 7]), the alternating direction or splittingup method allows us to decompose, and treat separately, the difficulties related to nonlinearity and to the incompressibility constraint div u = 0 (contained of course in (13.6)). We associate with this family of elements u™+i/2 of Wh the following functions defined on [0, T'] (cf. R. Temam [3], [RT]): • ukl) is the piecewise constant function which is equal to u™+l/2 on [mfc, (m + l)fc), i = 1, 2, m = 0 , . . . , N1. • uk° is the continuous function from [0, T"] into Wh, which is linear on (mfc, (m 4 l)fc) and equal to u%+i/2 at mfc, i = 1, 2, m = 0 , . . . , N1. By adding (13.6) and (13.8) we obtain a relation which can be reinterpreted in terms of these functions as (13.10)
law } \ v ( ^ ( t  f c ) , t ; H ) +  ( ( u ^ r ) + u(k2)(r),i;H)) + b(u(k2\t), uk2\t\ vh) = (fk(t), vh)
V vh e Vh
where fk(t) = fm
(13.11)
for f€[mfc,(m + l)fc).
Similarly, by adding (13.8) to the relation (13.6) for m + 1, we arrive at an equation which is equivalent to (13.12)
( ^ ( 0 , t ; K ) +  ( ( 4 1 ) a +fc)+ u k 2) (0^H)) * b(uk2\t\
uk2\t\ vh) = (fk(t + fc), VH) V t* e Vh.
13.2. The convergence theorem. We now discuss the behavior of these functions u(kl), ük\ as h and fc —> 0. THEOREM 13.1. Under the above assumptions, the functions uk\ ük\ i = 1, 2, be/ong to a bounded set of L 2 (0, T';W)n L°°(0, T'; G), G = L2(tf). As fc and h —» 0, u(kl) and ükl) converge to the solution u of Problem 2.2 in 2 L (0, V; W) and L q (0, T\ G) for all l^q«*>. Proof, i) We first derive the a priori estimates on the functions. We set tyi = u™+1/2 in (13.6) and, observing that (ab,a)
= ^(\a\2\b\2 + \ab\2)
Va,beH,
we get m +1/
uh (13.13)
l2 + wr1,2uz\2 +vij«r+lT Tl«r m
= 2k(f , <
+1/2
)S2k/m
_,m + l/2 M
l h
1
^ — If 1ll"r +1/2 ll 7ÄT 2fc a y UT + 1/2112
!
^K
\r
12
96
PART III. QUESTIONS RELATED TO THE NUMERICAL APPROXIMATION
whence (13.14)
Mm
+ 1/22_um2+
 M m + l/2_Mm2
+
2
VAx
Similarly, taking vh = u™+1 in (13.8) and taking into account (2.34), we obtain
(i3.i5)
iurTi"r+1T+kr+1wr+1/T+^ii"r+1ii2=o.
By adding all the relations (13.14), (13.15) for m = 0 , . . . , N  l , we find
KI2+Y
dur^ur^T+iur 172 ^! 2 )
m=0 m=C kv'. N  l
rT I diur+1/2i!2+2iiur+1ii2) Z
m=0
^IUOHI2^^1 Z i n
(13.16)
âuohr +
2
v
*\
m=0 N  l I /»(m + Dk
^ 1
m = 0 Un
/(s) ds
^l"oh 2 +
s) 2 ds
(by the Schwarz inequality).
Due to (13.4), UOHI = MO, and therefore the sequence u0h is bounded in H. We then conclude that (13.17)
(wr'ur^+wr1'2"^2)^^
Z m=0 Nl
(13.18)
k Z (ur +1/2  2 + 2   u r + 1   2 ) ^
2Ln
where (13.19)
2 L1=f«ol +1>A4 f W ds. Ax 1 J0
By adding the relations (13.14) and (13.16) for m = 0 , . . . , p, we obtain, after simplification, (13.20)
urT^i,
P =0,...,
N1.
Adding the relations (13.14) for m = 0 , . . . , p and (13.16) for m = 0 , . . . , p  1 , we find, after dropping unnecessary terms, (13.21)
\uph+1/2\2^Lu
p= 0,...,Nl.
The relations (13.18)(13.21) amount to saying that the functions u(k°, û(k°, i = 1, 2, belong to a bounded set of L°°(0, T'; G), and that u(k1}, uk2), belong to a bounded set of L 2 (0, T'; W). In order to show that u(k° also belongs to a bounded set of L 2 (0, T'; W), we observe by direct calculation (cf. [RT, Chap.
FINITE TIME APPROXIMATION
97
III, Lemma 4.8]) that k ff  I i (0_ I :(i)2
— V „m+i/2 •^ m = l
_.ml+i/22
(13.22) g^i
(by (13.17)).
ii) We want to pass to the limit as fc —*• 0, h * 0. Due to the previous a priori estimates, there exist a subsequence (denoted fc, h) and u (1) , u<2> in L 2 (0, T'; W)nL°°(0, T'; G) such that (13.23)
_^ ^„{weakly in L 2 (0, T; W), tweakstar in L°°(0, T'; H),
u
i = l,2.
The relations (13.17) and (13.22) imply that the same is true for ü(£\ i = 1,2. Now we infer from (13.17) that (13.24)
K^uSc'W.T^ïifcL,
and therefore u£)uikl)*0
inL 2 (0, T ' ; H ) ,
so that u(2) = u(1). Also we infer from the properties of the Vh's (which constitute an external approximation 4 of V) that un) belongs in fact to L 2 (0, T ; V)nL°°(0, T';H): u (1) = u(2) = u*G L 2 (0, T; V) n L°°(0, T; H).
(13.25)
The last (and main) step in the proof is to show that u* is a solution to Problem 2.2. We need for that purpose a result of strong convergence which will be proved with the help of the compactness theorem in § 13.3. 13.3.
A compactness theorem. 13.2. Let X and Y be two (not necessarily reflexive) Banach spaces
THEOREM
with (13.26)
Y c X,
the injection being compact.
Let ^ be a set of functions in L*(R; Y ) n L p ( R ; X), p> 1, with « is bounded in LP(U; X)
(13.27) (13.28)
and
g(t + s)g(s)&
L\R;
Y);
ast^O,
J—oo
uniformly for (\ \ oo\ U3 Z9j *
ge^;
tne su
PPort °f me functions ge<ê is included in a fixed compact of R, say [U+LI
Then the set % is relatively compact in LP(U; X). 1
For the details see [RT].
98
PART III. QUESTIONS RELATED TO THE NUMERICAL APPROXIMATION
Proof, i) For every a > 0 , and for every geLp(U;X) Jag : RI* X, by setting (JaZ)(s) = ^ \S+ag(t)dt
(13.30)
we define the function
= ^ rag(s + t)dt.
La Js_a
La Ja
It follows from the first relation in (13.30) that the function s**(Jag)(s) is continuous from R into X; on the other hand J a g e L P (R; X), and furthermore 1 / C+a
(Jflg)(s)x^— (J
\1/p
Iflj + Ôfc*)
(2a) w
(by the Holder inequality), ç +oo
J
ç +oo
1
(J«g)(s)fc
i» + a
— j
ë
g( s + t ) & A d s
lg(t)Rd«, J—oo
(13.31)
VgeL p ([R;X).
/agUR;X)^gUR;x)
The same reasoning shows that JageLx(R; (13.32)
/rfL^Y)S.IL^;ri
Y) and
VgeL^jY). P
ii) It is easy to see that Jag > g in L 0R; X) (resp. I^flR; Y)) as a > 0 for every g e L p ( R ; X ) (resp. L\U; Y)). Due to (13.31), it suffices to prove this point for functions g which are c#QO with values in X (resp. Y) and have a compact support, and this is obvious. We then observe that, due to (13.28), (13.33)
Jag > g
in LP(R ; X)
as a * 0
uniformly for g e <§.
In order to prove (13.33), we write 1 Jag(s)
~ g(s)
(+a
= —
(g(s
+ 0  g ( s ) ) dt,
2a J_a (2a) 1/p ' /f +a  / a g ( s )  g ( 5 ) 
Joo
X
^
i
/«g(s)g(s)&
 ^  (J
a
\1/p (s + 0  g ( s ) & A )
,
— g(s + 0  g ( s )  i d t d s joo La J_a
Ü5
\g(s +
t)g(s)\*xdsdt
La J_ a J_oo /•+00
s sup
g(s + 0g(s)l&
 a S t S a J—c»
and (13.28) implies that this last quantity converges to 0, uniformly for g G ^ as a —>0.
99
FINITE TIME APPROXIMATION
iii) We now show that (<%
For every a > 0 fixed, the set ^ a = {/ag, g e <S} is uniformly equicontinuous in ^ ( R ; X ) .
Indeed, for every s, teU \Jag(s)  J«g(r) x = Ya 
0 ( g ( 5 + Z)
"
g ( + Z)) d 2
'
=^f S+a g(2)dzf%(z),dz 2a IJs_a
=
Jf_a +a ^ l f \^)dz[Js+ag(z)dz 2a IJs_a 2lsfl1/p'
2a
• I g l L P (R;X)
^csr1/p'
Vge<
Hence we have (13.34). We then apply Ascoli's theorem (cf. Bourbaki [1]) to show that the set <Sa is relatively compact in ^ ( R ; X ) . All the functions Jag have their support included in a fixed compact, say [—L — 1, L + 1 ] (assuming that a(si 1). Due to (13.34), the only assumption of Ascoli's theorem which remains to be checked is that (13.35)
For every seR,
the set {Jag(s), g e ^ } is relatively compact in X
But according to (13.26), it suffices to show that this set is bounded in Y, and we have with (13.27) \Jag(s)\y=—\ g(s + t)dt\ ^—\ \g(s + t)\Ydt 2a U_a IY 2a J.« 1 S—lgLt^Y)^const 2a
Vge<£
The set ^a is therefore relatively compact in ^(R; X), and due to set is, for every fixed a, relatively compact in LP(U; X). iv) Finally we prove that the set <S itself is relatively compact For instance, we have to prove (cf. Bourbaki [1]) that for every exist a finite number of points g 1 ? . . . , g^, in L P (R; X), such that $ in the union of the balls centered at gt and of radius e. According to (13.33), for every e > 0 there exists a such that (13.36)
JaggL'(R;X)^3
(13.29) this in L P (R;X). e > 0 there is included
VgG1
Since ^ a is compact, it can be covered by a finite number of balls of LP((R; X), centered at Jagu . . . , JagN, and of radius e/3; finally it is clear with
100
PART III.
QUESTIONS RELATED TO THE NUMERICAL APPROXIMATION
(13.36) that the balls of LP(IR; X) centered at g l 5 . . . , gN, and of radius e cover 8. The proof of Theorem 13.2 is complete. D We have a similar result for functions g defined on a bounded interval, say [0, T]. THEOREM 13.3. We assume that X and Y are two Banach spaces which satisfy (13.26). Let % be a set bounded in L 1 «), T; Y) and L p (0, T; X), T > 0, p > 1, such that rra
(13.37)
\g(a + s)g(s)\xds+0
Jo
as a —» 0, uniformly for ge%
Then ^ is relatively compact in L q (0, T; X) /or any q,l^q
a
\g(a + s)g(s)\«zds+\
Jrt
\g(s)\x ds + J \ o
\g(s)\xds
, p/q
g(a + S)g(S)^ds)P
ëTi
+ 2a'<"p(j g(s)&d s y' \ q/p
g T .^/pQ
a
 g(û + s)g(s)^dS)"
and this goes to 0 uniformly with respect to g e 8 as a —> 0. Remark 13.1. Of course, under the assumptions of Theorem 13.2 we can obtain that ^ is relativelv compact in L p (0, T; X) itself if we assume, instead of (13.37), that f (13.38)
j()
a
\g(a + s)g(s)\pxds+
f h a ~
g(s)&ds+fag(s)&ds>0 Jo
as a —» 0, uniformly for g e % 13.4. Proof of Theorem 13.1 (conclusion), i) By application of Theorem 13.3, we show that the functions w(kl), u(k° converge to u strongly in L 2 (0, T'; G) or even L q (0, T'; G), l^q<<*>. By integration of (13.12), (ßi»(»+a)ßi»(t), «h) =  } ' " { ^ ( « ( s +fc)+ «(k2)(s), «*.)) + 6(u
FINITE TIME APPROXIMATION
101
We majorize the absolute value of the righthand side of this equality as follows:
IF**
k»(s+ * ) , ! » , ) ) * \^a^\\vh\\(\'*
1/2
a
\K\s
+ k)fdï)
^a^\\vh\\[\\\^\S
+
k)fdS)m
^c[a"2\\vh\\, where c[ depends only on the data, u0, /, v, ft (or Q), T". Similarly,
icia w k,
\^'+\(ui2\s),vh))ds\ +a
{fk(s + k),Vh)ds\: cW12 Ik«.
For the term involving b, we use Lemma 2.1 with mi —\, m2 = m 3 = 1: ,+°&(u(s), u(fc2)(s), c ) d s  ^ ï 
+a
ulc2)(s)1/2 K2)(s) Ik« ds
âc;kf' + a ulc 2 ) (s) , / 2 «l 2 ) (s)irds /p+a
^cJka1/4(J
\l/2
u(k2)(s)2ds)
(by Holder's inequality and since ul2» is bounded in L~(0, T'; G)) S
1/4
KU \fvhe Vh.
5
Since ü^ is a Vhvalued function , we can take vh = u ^ f + a)û(k1)(f), and we find after integration with respect to f: "\üil\t + a)  ü
5
a
Jo
a
ffik»(t + a )  Ûk»(r)P dt) ,/2 ,
ü£\t + a)  ßg?(0P * S c^a1/4.
This is not the case for fijj* (a W^valued function).
102
PART III.
QUESTIONS RELATED TO THE NUMERICAL APPROXIMATION
We apply Theorem 13.3, as follows. ^ is the family of functions ü^ which is bounded in L 2 (0, T; W)nL°°(0, T; G), ( Y = W, X = G), p = 2, so that (13.37) follows from (13.40). We conclude that ßjjp is relatively compact in L q (0, T; G), l ^ q < 2 , and since üjp converges weakly in L q (0, T ; G) to u* (see (13.23)(13.25)), this means that û(k1} converges to u*, strongly in L q (0, T ; G), l ^ q < 2 . The sequence u£> being bounded in L°°(0, T'; G), we conclude from the Lebesgue dominated convergence theorem that we actually have (13.41)
strongly in L q ( 0 , T ' ; G )
ü^^u
V q,
l^q«*.
As for (13.22), it is easy to check by direct calculation that k fi lfj<*>— f i ( i > l 2 ,
C\%AO\
— V ..m+i/2_ « m= l
ml+i/212
and with (13.17) (13.43)
luVuVlUo.r.o^cï,
i = l,2,
This, together with (13.24) and (13.41), shows that (13.44) u(k° and u(k° converge to u strongly in L q (0, T'; G)
V q, 1 ^ q
ii) Using (13.44) and the weak convergences obtained in § 13.2 (in particular (13.23)), the passage to the limit in (13.12) is standard. We will just give the main lines of the proof. Let i/r be any c€1 scalar function on [0, T'] which vanishes near T'; we multiply (13.11) by t/KO, integrate in t and integrate by parts the first term to get T
{(û ( k 1 ) (r), U K^(0)+^((u ( k 1) (r +fc)+ M(k2)(r), tvKO)) + b(u(k2)(r), u(k2)(r), ivMO)} at
(13.45)
(&((.+k>,
Let u be an arbitrary element of V. We choose for vh an approximation of u, and passing to the limit in (13.45) we get
(13.46)
r
{(u*(0, v^(t))+v((u^t), «K0))+K«*(0, MO, «MO» *
Jo
=
Jo
(f(t), v4>(t))dt + (uo,v)4>(0).
FINITE TIME APPROXIMATION
103
(The passage to the limit in the nonlinear term necessitates the strong convergence (13.41); cf. [RT].) We deduce from (13.46) that u* is a solution to Problems 2.12.2, and since this solution is unique in the present situation, we conclude that u*= u. By uniqueness, the entire sequences u(k°, u(k° converge to u as k and h tent to 0. The last step of the proof is to obtain strong convergence in L 2 (0, T"; W).
14
Long Time Approximation of the NavierStokes Equations
The study of the long time behavior of the solutions to the NavierStokes equations is directly related to an understanding of turbulent flow, as can be seen from Part II. In this section we present two results which tend to indicate, as do those in §§11 and 12, that the flow has, for time large, a finite dimensional structure. The first result is related to the flow itself (§ 14.1); the second to its Galerkin approximation (§ 14.2). 14.1. Long time finite dimensional approximation. Let / and g be two continuous bounded functions from [0, ») into H, let UQ and v0 be given in V, and let u and v denote the corresponding solutions to Problem 2.2: (14.1)
^+vAu+Bu=f, at
(14.2)
dv — + vAv+Bv at
u(0) = uo, =&
v(0) = vo.
We assume that u and v are defined and bounded for all time in V: (14.3)
sup u(r) ^ c',
sup u(f) ^ c\
We recall that this property is automatically satisfied if n =2 (cf. Lemma 12.2). We recall also that for every a > 0 (14.4)
sup \Au(t)\ =i c",
sup \Av(t)\ =i c",
where c" depends on a, c' and the data (Lemma 12.2). The space E. Let us now consider a finite dimensional subspace E of V. We denote by P(E) the orthogonal projector in H onto E, and Q(E) = IP(E). Since P(E) is not a projector in V, it may happen that ((<£, i/0) ^ 0, ifeE,il/eV and P(E)il/ = 0. However (see Lemma 14.1 below), there exists p ( E ) , 0 ^ p ( E ) < l , such that (14.5)
((*+))l*fK£)1+H*l
V4>eE,
VIAGV,
We also associate with E the two numbers k(E),
P(E)* = 0.
IJL(E),
HE) = inf{U\\2,eV,P(E) = 0,\ \ = ll n(E) = sup{U\\2,il,eE,\i!j\ 105
= l},
106
PART III.
QUESTIONS RELATED TO THE NUMERICAL APPROXIMATION
so that ^A(E) 1 / 2 W
(14.6)
1
(14.7)
MSpCB) "!*!
V4>eV,
P(E)= 0,
V^€E
When it is not necessary to mention the dépendance on E, we will write simply P, O, p, A, p, instead of P(E), and so forth. We will now prove (14.5). LEMMA 14.1. Under the above assumptions, there exists p = p(E) such that (14.5) holds. Proof. If this were not true, we could find two sequences <£,, i/fJ? j ^ l , <£, G E, tfc G V, P ^ = 0 such that
^ II^H ^ ((^, ^))l ^ ( l ~ y ) ll^ll ll^li. Setting * ; = <MI<M. *} = «MW, we find (14.8)
ia((^,^))äii.
We can extract a subsequence, still denoted /, such that 4>] converges to some limit , <£ = 1,eE (E has finite dimension), and i/fj converges weakly in V to i/f, i/f G V, MI = 1> I V = 0. At the limit, (14.8) gives
\mm±%
u\\=h mm,
so that ifr = 1, i/f = k<^^ 0, in contradiction with Pij/ = 0. D An inequality. We consider the two solutions u, u of (14.1)(14.2), and we set
w = uv, (P = P(E\Q
= Q(E) =
p = Pw, q = Qw,
e=fg,
IP(E)).
We apply the operator Q to the difference between (14.1) and (14.2). We obtain da Tdt+ i / Q A w + QB(v, w) + QB(w, u) = Qe. We then take the scalar product in H with q,
™kl2+Htaf = (B(v, p), q )  (B(p, u), q)  (B(q, u), q) + (Oe, q)  v((p, q)). Using Lemma 2.1 (or (2.36)) and (14.5)(14.7), we find that the righthand side of this inequality is less than or equal to A, Oe q+ vpn1'2 \p\ q + C l (Au + A») p q + C l \Au\ \q\ \\q\\.
LONG TIME APPROXIMATION OF THE NAVIERSTOKES EQUATIONS
If t^a, a>0
107
arbitrary, then this expression is in turn less than or equal to
2 2 2 2 2 flkll +^Oe +^W f^pVlp +fN 3 4ev 3 e 3
+  ^ (  A u  + A U ) 2 p 2 + c 1 c"A 1 ' 2 q 2 ; £ V  U I  2! .+
%0c2+3/(£içl+ 4ev
e \
2
A(
v
2+
Ai«y
1
where s > 0 is arbitrary. If
(14.9)
k(E)>(^J,
then we set (14.10)
C C k{E)m
v' = vc,c"\{EY">0,
e=\
' " 2v
,
2 and we have established: LEMMA 14.2. / / (14.3)(14.5)(14.7) and (14.9) hold then, (14.11)

q 2 + v> I k l f s g l Oe 2 +  ( ^ f +
fort^aX),
, p 2 p ) p 2 ,
with v , e as in (14.10). We now prove THEOREM 14.1. We assume that the space dimension is n = 2 or 3 and that u and v are solutions of (14.1)(14.2) uniformly bounded in V. Let E be a finite dimensional subspace of V such that (14.9) is satisfied. Then, if (14.12) (14.13)
\P(E)(u(t)v(t))\+0, (/P(£))(/(0g(())hO
ast^™,
we have also (14.14)
\(IP(E))(u(t)
 v(t))\ ^ 0
as t * oo,
i.e., (14.15)
u(f) v(t)\  » 0
as r » oo.
Remark 14.1. Theorem 14.1 shows that if the condition (14.9) is satisfied the behavior for f—>oo 0 f u(t) is completely determined by that of P(E)u(t) (although we do not know how to recover u(t), knowing P(E)u(t)). Lemma 14.3 below shows that (14.9) is satisfied if E is "sufficiently large".
108
PART III.
QUESTIONS RELATED TO THE NUMERICAL APPROXIMATION
Proof of Theorem 14.1. We infer from (14.6) and (14.11) that
^\q\^v'\\q\^c[\Qe\^c'2\p\^ at where m,
c\
3A2
6/( C l c") 2 e \ v
LEV
\ )
2
whence for f =^ f0 = <*> (14.16)
k ( 0  2 S k ( ^  2 « ^ X ( t ~ t o > + P [c[ \e(r)\2+c'2 p(r) 2 ]e^ ( t  T ) dr.
Given 8 > 0, there exists M (which we can assume ^ a ) such that for t ^ M P(u(0f(0)l2âô,
(7PXf(0g(0)l2â& Therefore, for t^t0+M, \q(t)\2^c(u,v)ev'K(''»)
(14.11) implies K( T)
+ Ô(c'1 + c^\'
e"'
'
dr
JtM rtM
+ [ c i c ( / , g ) + c£c(ll,ü)]
é T ^ ^ d r , J
to
where
c(w,i>) = supu(f)t>(OI, As r—>«>, limsupq(r) 2 ^ô(c; + c y t*»
c(/,g) = sup/(r)g(r). lle~vkM\ \
v A
/ v'KM
+ [cic(f f g) + c i c ( f f g ) l — r r  . 1/A We let Ô » 0 and then M ^ oo and we obtain (14.14). D As mentioned before, the following lemma shows that condition (14.9) is always satisfied if E is sufficiently large. LEMMA 14.3. If Ef, j ^ 1, is an increasing sequence of subspaces of V such that {JjEj is dense in V, then \(Ej)^> +œ as /*«>. Proof In fact, we will prove that (14.17)
For every integer m there exists jm and, A(E im )^A m ,
where Am is the mth eigenvalue of A.
for / ^ / m , we have
LONG TIME APPROXIMATION OF THE NAVIERSTOKES EQUATIONS
109
By assumption, for every k, Inf wk  iA —» 0
as / * oo.
Hence, for given m and 8 > 0, there exists jm such that Inf wfc  i/f ^ ô
for k = 1 , . . . , m and every / ^ /m.
Thus, for every / ^ / m , there exist tö^;..., wm in Ej9 with w, —w,^& Therefore ife V, ( J  P ( Ê ;  ) ) ^ = 0, we have W 2 = pm^2+(/pm)^2 ^ À m + 1  ( /  P m ) ^ >  2 + A 1 P m ^ 2 m
= A m + 1 <£ 2 (A m + 1 A 1 ) X (
D
14.2. Galerkin approximation. We are going to establish a result similar to that in Theorem 14.1, for the Galerkin approximation of NavierStokes equations when the space dimension n = 2. For simplicity we restrict ourself to the Galerkin approximation using the eigenf unctions vvm of A (as in § 3.3), and we will show that if m is sufficiently large, the behavior as f—>oo of the Galerkin approximation um is completely determined by the behavior as t —» « of a certain number m* of its modes, i.e. the behavior of P ^ i ^ , m*<m. We start with some complementary remarks on Galerkin approximations. For fixed m, the Galerkin approximation i v of the solution u of (14.1) is defined (see § 3.3) by
(14.18)
^+vAurn+PmB(um) at
= Pmf,
f>0,
u m (0) = P m u o , where P m is the projector in H, V, V , D(A),..., onto the space spanned by the first m eigenfunctions of A, wx,..., wm. We have established in § 3.3 some a priori estimates which are valid on a finite interval of time [0, T]; we can obtain a priori estimates valid for all time as follows.
110
PART III.
QUESTIONS RELATED TO THE NUMERICAL APPROXIMATION
We infer from (3.46) that
~ M t ) l 2 + H  i U t ) l l 2 = tf(t), «U0)3É/(Ï) M O I â ^ = 1/(01 I K (r) ^;IK(0ll2+^=l/(0l2, (14.19)
 «U«P+» I M O I I 2 ^ /(0 2 s4"
N (
^
where N(/) = sup t s 0 1/(01 Thus for every f ^ s ^ O (14.20)
lUnittf +
SllUniaWda^luM2*
4(ts)N(f)2
and Um(01 is bounded for all time:  i U r )  2 5  i i o  2 e  ^ + H r27 — W J 2 . v \x
(14.21)
The following a priori estimate is also verified by u^. LEMMA 14.4. 11^(011 is bounded independently of m and t. Proof. For T = a > 0, this is proved on [0, a] in (3.59). In order to obtain the result for r^a, we consider the analogue of (3.11) for u^ (obtained by taking the scalar product of (14.18) with Aum; see § 3): ^   u m ( 0   2 + v  A u m ( 0  2 S   / ( 0  2 + ^u m (0 2 u m (0 4 . at v Therefore with (14.21) ^u m (0 2 + l 'A 1 u m (0 2 SN(/) 2 + ^u m (0 4 (14.22)
dt
sc^i+lMi 2 ) 2 ,
and for 0 ^ s ^ t, we can show by integration that (^««„.(«^^(^«^(s^exp^I'd + IKMRAr). If t ^ a > 0, we integrate in s from t  a to t and we find c'2a{\ + u m (0 2 )S [exp (c^J'
(1 +um(o)2) d o )  1 ] .
Using (14.20), we see that the righthand side of this inequality is bounded by a constant depending on a and the data, but independent of t and m. The lemma follows. D
LONG TIME APPROXIMATION OF THE NAVIERSTOKES EQUATIONS
111
We can now state the result announced. Let um and vm be the Galerkin approximations to the solution u and v of (14.1M14.2), (14.23)
% + vAUn + PnßiuJ at
 Pmf,
1^(0) = Pmu0,
(14.24)
^ + M u m + P m ß ( ü m ) = Pmg,
vm(0) = Pmv0.
THEOREM 14.2. We assume that n = 2 and that m ^ m*, m* sufficiently large so that the condition (14.29) below is verified. Then, if
(14.25)
PnJiU0tVn(0)>0
as r ^ o c ,
(14.26)
l(/PO(fWg(0)>0
asf*«>,
we have (14.27)
l(/PrJ(Wm(0t^(0)>0
as r^ oo,
i.e., (14.28)
w m (0i> m (0>0
asf^oo.
Proo/. We have to obtain an analogue of (14.11); we then proceed as for Theorem 14.1. We set for m*^m (m* to be determined): wm = umvm,
pnH = PnHwm,
q m# =Q mjc w m ,
e=fg,
enH=
Q^m
(Q m =IPm). Then by subtracting (14.21) from (14.23) and applying Q ^ to the equality which we obtain, we arrive at ' + i / A q ^ + Q^P m £(i; m , wm) + Q rr .P m B(w m , 1 0 = 0 ^ , » * . Consequently,, i_o;
^ 3 7 l<W 2 + V Ikm^lP = (B(Vm,
2dr
Wm), Pmq„J
 (B(wm,
U^), P ^ ^ ) + ( O ^ e , PmClm*).
The righthand side is equal to (Q^e, Pmq„J  (B(vm, p„J9 Pmq„J (B(vm,
(IPm)qmJ,
PmqmJ
Because of (2.31) and Lemma 14.4, this quantity is bounded by ICM k j + c \vm\m \\vm\r IPm,l1/2 IIPJI1'2 h j
+ci«uii/2ii«i»iri*j,/2iw3/2 +c i P j i/2 nPmjr I M ikJI+c iqj1« i k j r K I I
112
PART III.
QUESTIONS RELATED TO THE NUMERICAL APPROXIMATION
if (14.29)
W I > ( T )
4
.
we set >U, v' = 2(i/c"A^ 1 )>0,
(14.30)
1 c"KV4 e = ^  " m * + 1 >0, 2 2»>
and we bound the last quantity by 2
LEV + <~"A
2 1/4
LEV
2
11/7 II
Finally we find (14.31)
^  q qmJ ,2 2++vv'' lUS*k J I 2 â ^  lOOmm, e» Pe +2 ^+ ^ ^  f t j . 4 at
ei/
ei>
This inequality is totally similar to (14.11), and starting from (14.31) the proof of Theorem 14.2 is the same as that of Theorem 14.1. D Remark 14.2. We did not establish above any connection between the behavior for t—»o° of u(t) and of its Galerkin approximation Umit). This remains an open problem. See however P. ConstantineC. FoiasR. Temam [i].
Comments and Bibliography
Most of the results in §§ 2 and 3 are classical and, as mentioned, can be found in the books of O. A. Ladyzhenskaya [1], J. L. Lions [1] and R. Temam [6], and in J. Serrin [1]. The only difference is that we have emphasized a little more the space periodic case which is not treated in detail in the literature; also, at the end of § 3 (§ 3.4) we give a simple but interesting result of generic solvability in the large, due to A. V. Fursikov [1]. The new a priori estimates in § 4 are borrowed from C. FoiasC. GuillopéR. Temam [1]. The results on the Hausdorff dimension of the singular set of a weak solution to the NavierStokes equations in § 5 are two typical results, if not the most recent. They rely on the ideas of B. Mandelbrot [1], and on the work of V. Scheffer [1], [4] and C. FoiasR. Temam [5]. The result in § 5.1 (time singularities) is just a restatement by V. Scheffer [1] of an old result of J. Leray [3], and also generalizes a result of S. Kaniel and M. Shinbrot [1]. The result in § 5.2, leading to a Hausdorff dimension of singularity ^ § in space and time (n = 3), is an extension due to C. FoiasR. Temam [5] of a result by V. Scheffer [3], [4]. The best result presently available along those lines is that of L. CaffarelliR. KohnL. Nirenberg [1] (the onedimensional Hausdorff measure of the singular set vanishes). The set of compatibility conditions given in § 6 was derived in R. Temam [8]. The first compatibility condition is mentioned in J. G. Heywood [2]; for the compatibility conditions, cf. also O. A. LadyzhenskayaV. A. Solonnikov [3]. In § 7 we give a simple proof of time analyticity of the solutions to the NavierStokes equations (with values in D(A)), which is a "complexification" pf the standard Galerkin approximation procedure; this presentation is slightly simpler than the original one given in C. FoiasR. Temam [2], [5]. For other results on time analyticity, cf. also C. FoiasC. GuillopéR. Temam [1], and for a proof based on semigroup operators which apparently applies to N.S.E., cf. F. J. Massey III [1]. Finally, in the last section of Part I (§ 8), we use an a priori estimate of § 4, to determine the Lagrangian representation of the flow associated with a weak solution of the N.S.E.; for further developments on this new result, cf. C. FoiasC. GuillopéR. Temam [2]. The properties of the set of stationary solutions to the N.S.E. based on the SardSmale theorem, and derived in § 10, are among the results proved in C. FoiasR. Temam [3], [4]. Results of genericity with respect to the boundary conditions are not presented here; they use somewhat more sophisticated tools in analysis and topology, and can be found in J. C. SautR. Temam [1], [2]. For other recent results on the set of stationary solutions, cf. also C. FoiasJ. C. Saut [3] and Gh. Minéa [1]. The results in §§11 and 12 are taken from C. FoiasR. Temam [5]. The squeezing property of the trajectories is used in § 12, and has played a role in 113
114
COMMENTS AND BIBLIOGRAPHY
the proof of other results on the structure of a turbulent flow: cf. in particular C. FoiasR. Temam [7], [8]. The results mentioned in § 12.3 are proved in C. Guillopé [1]; a similar result is also announced in J. G. HeywoodR. Rannacher [1]. For the numerical approximation of the N.S.E. on a finite time interval (and for stationary solutions), see the references mentioned in § 13; see also R. Temam [6], and M. FortinR. PeyretR. Temam [1], R. Peyret [1], R. Temam [1], [2], [4], [5], F. Thomasset [1], and the references therein. The compactness theorem used in the proof of convergence is new. The results in § 14 are part of a work in progress; cf. C. FoiasR. Temam [9], C. FoiasO. ManleyR. TemamY. Trêve [1] and the references there.
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Branching In The Presence Of Symmetry by D. H. Sattinger Professor Sattinger has been actively involved in the development of bifurcation theory for several years. In these CBMS lectures he discusses some of the latest developments in the field, with emphasis on symmetry breaking and its interrelationship with singularity theory. The notions of universal solutions, symmetry breaking, and unfolding of singularities are discussed in detail. The book not only reviews recent mathematical developments but should provide a stimulus for further research in the field. Contents: Critical Points ofNonconvex Functional • Minimax theorems • Periodic solutions of a nonlinear wave equation • Periodic solutions of Hamiltonian systems • Variational methods for free boundary value problems • Perturbation methods in critical point theory. • Spontaneous Symmetry Breaking • Equivariant equations • Equivariant bifurcation equations • Modules of equivariant mappings • The rotation group • SU(3) model for hadrons • Hopf bifurcation in the presence of spatial symmetries • Stability of bifurcating waves. • Equivariant Singularity Theory • Modules of equivariant mappings • Unfoldings • Universal unfolding of Z2 singularity • Universal unfolding of the Dn singularity • Bifurcation analysis for D3. • Critical Orbits of Linear Group Actions • Orbits and strata • Theorems of L. Michel • A trace criterion for critical orbits • Critical orbits for representations of SO(3). • References 1983, viii + 73 pp., $12.00/list, $9.60/SIAM member
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