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O.
With such a 8 fixed, the inequality (5.15) shows that I q m+ l
_ q’" 12
IIvm + 111 2
= I p m+ l _
p’" 12 --+- 0 as
m-r»,
= lIum + 1-uI12 --+- Oas m --+- oo
(5.16)
The convergence ofu m + 1 to u is thereby proved. Now by (5.14) we see also that the sequence p’" is bounded in L 2 (Q ). We can then extract from p" a subsequence pm’ converging weakly in L 2 (Q) to some element p*. The equation (5.2) gives in the limit v((u, v)) - (p*, div v)
= if,
v), V v EHa(Q),
and by comparison with (5.7), we get (p - p*, div v)
whence
= 0,
V v E Ha(Q),
142
The steady-state Stokes equations
Ch. I, §S
grad (p - p.) = 0, p; = p + const. From any subsequence of p’", we can extract a subsequence converging weakly in L 2(O) to p + c; hence the sequence pm converges as a whole to p for the weak topology of L2 (O)/6t Remark 5.1. Let us define p by imposing the condition
J
n p(x)dx
=O.
Let us suppose that po in L 2 (0) is chosen so that
J
n pO(x)dx
= O.
Then clearly we have
J
n pm (x)dx = 0,
m ;>1,
and the whole sequence p’" converges to p, weakly, in the space L 2 (0).
o
5.2. Arrow-Hurwicz Algorithm.
In this case too, the functions u and p are the limits of two sequences which are recursively defined. We start the algorithm with arbitrary elements u O, pO, uO EHA(O), po EL 2(O). (5.17)
u’", p"
of
I
When pm and u" are known, we defme pm+l and u m+ 1 as the solutions m 1
u + EHA(O) and ((u m+1 - u’", v»
+
pv((u m, v» - p(pm, div v)
= if, v), V v E HA(O), pm+l EL
2(0)
and
cx(pm+l - p’", q)
+ p(div u m+ 1, q) = 0, V q
E L 2(0).
(5.18)
I
(5.19)
Numerical Algorithm,
Ch. I, §5
143
We suppose that p and (X are two strictly positive numbers; conditions on p and (X will appear later. The existence and uniqueness ofu m + 1 EHA(fl.) satisfying (5.18) is easily established with the projection theorem; u m+ 1 is the solution of the Dirichlet problem
_.6.um+ 1 = -.6.u m + pv.6.um - p grad pm + f.
u m + 1 EHb(fl.)•
(5.20)
Then pm+l is explicitly given by (5.19) which is equivalent to
pm+l = pm _
! ..
(X
div u m+ 1 E £2 (fl.).
(5.21)
Convergence of the Algorithm. Theorem 5.2. If the numbers
(X
and p satisfy
2av O o. ForuEWm,P(lRn),m;> 1, 1 ~p 0, luILq(Rn) ~
if
p- ’;i = 0, luILq(eJ) Y bounded set
c(m, p, n)lIu IlWm,p(Rn),
~ c(m, p, n, q, tJ) lIullwm,p(Rn) ~ C
lR n , Y q, 1 ~ q < 00,
(1.1)
Existence and uniqueness theorems
Ch. II, § 1
if
m
1
p - n < 0, IUI~o«(!)
~ ctm, n, p,
V bounded set
159
o ) IIu" wm, P(Rn)
,
o, o C 6ln
If 0 is any open set of 6ln , results similar to (1.1) can usually be obtained if 0 is sufficiently smooth so that: There exists a continuous linear prolongation operator (1.2)
Property (1.2) is satisfied by a locally Lipschitz set O. When (1.2) is satisfied, the properties (1.1) applied to IIu, u E Wm,P(O) give in partic› ular, assuming that u E wm,P(O), m;> 1, 1 < p < 00, and (1.2) holds: if
P- n= q> 0, lulLq(O) ~
if
p- n= 0, lul LQ( l!i) ~
m
1
1
m
1
ctm, p, n, 0) II ull Wm’p(O)’
ctm. p, n, q, o , 0) IIull wm, p(O)’
any q, 1 ~ q < 00, any bounded set () C 0,
if
p- n< 0, lul~o«(!) m
I
~ ctm, p, n, q, 0,
any bounded set o, () C
o
IIullwm,p(o)’
(1.3)
n.
o
When u E Wm,P(O), the function uwhich is equal to u in 0 and to in belongs to wm , p(6ln ), and hence the properties (1.3) are valid without any hypothesis on O. The case of particular interest for us is the case p = 2, m = 1, i.e., the case HACO). Without any regularity property required for 0 we have for u EH5(0)
CO,
n
= 2, luILq(O)~c(q,(),
0) IIuIIH~(O)
V bounded set () CO, V q, 1 ~ q < 00
n« n
3,
lu1L1(O)
~ c(O) IIuIIH~(O)
= 4, lu1 L4(O) ~
n;> 3,
lui
c(O) IIuIIH~(O)
L2n/(n-2) (0) ~ c(O) IIuIIH~(o>,
(1.4)
Ch.U, §1
StudY-Itate Navier-Stokea equattona
160
Compactness Theorems.
Theorem 1.1. Let 0 be any bounded open set of(i{n satisfying (1.2). Then the embedding
(1.5)
is compact for any ql’ 1 E;;;; ql < 00, ifp ’> n, and for any ql’ 1 E;;;; ql (q given by l/p - l/n = l/q) if 1 E;;;;p < n. With the same values of p and ql’ the embedding o
Wl,p(O) C Lql (0)
3):
L2n/(n- 2)(0 ), D j vj E L2 (0 ), wjELn(O), 1 1, we would like to define an approximate solution um of (1.25) by
HA
m
=!
Um . 1=1’ ~t mWt’ ~I .. mER
(1.27)
1I«Um, Wk)) + b(Um, Um’ Wk) =
’
k = 1, ... , m.
(1.28)
The equations (1.27) - (1.28) are a system of nonlinear equations for
~ I, m’ .. , ~m. m’ and the existence of a solution of this system is not
obvious, but follows from the next lemma.
Lemma 1.4. Let X be a finite dimensional Hilbert space with scalar product [.,.] and norm [.] and let P be a continuous mapping from X into itself such that [P(~),
~]
> 0 for
Then there exists P(~)
~ E
=k
[~]
X,
[~]
~
> O.
(1.29)
k, such that
= O.
(1.30)
The proof of Lemma 1.4 follows the proof of Theorem 1.2. We apply this lemma for proving the existence of Um, as follows: X = the space spanned by wI, .. , wm ; the scalar product on X is the scalar product « .,.)) induced by V; and P = Pm is defmed by [Pm (U), v] = «Pm (u), v)) = J.’«U, v)) + b(u, u, v)-
if, v), Yu, v EX.
Ch.n, §1
Existence and uniqueness theorems
165
The continuity of the mapping Pm is obvious; let us show (1.29). [Pm (U), U]
= vlluII2 + b(u,
u, U) -
(I, U)
= (by (1.22» = vlluII2 -
(I, U)
~ vlluII2 -lltllprlluII,
(1.31) IIuII (vllull - Iltllv’). It follows that [Pmu, u] > 0 for IIull = k, and k sufficiently large: more precisely, k > ltv IltllV" The hypotheses of Lemma 1.4 are satisfied and [Pm (U), u] ;>
there exists a solution um of (1.27) - (1.28). Passage to the Limit. We multiply (1.28) by k = 1, ... , m; this gives
m
~k,
and add corresponding equalities for
vII"", 11 2 + b("""""" um) = (f, um)
or, because of (1.24), vII"", 11 2
=(I, um) ~
Iltllv,IIu", II.
We obtain then the a priori estimate: I
II"",II ~ ;- IttlIV’.
(1.32)
Since the sequence um remains bounded in V, there exists some u in V and a subsequence m’~oo such that u",,-"u for the weak topology of V.
(1.33)
The compactness theorem 1.2 shows in particular that the injection of V into L 2 (.0) is compact, so we have also um’ ~ u in the norm of L 2 (.0).
(1.34)
Let us admit for a short time the following lemma. Lemma 1.5.
then
n», converges
b(u lJ , ulJ ’
v)~b(u,
U,
to u in V weakly and in L2 (n ) strongly, v), V v E "f’:
(1.35)
Steady-atate Navier-8tokel equatioss
166
Ch. II, § 1
Then we can pass to the limit in (1.28) with the subsequence m’-+oo. From (1.33), (1.34), (1.35) we find that v«u, v»
+ b(u, u, v) = (f, v)
(1.36)
for any v = wI’ ... , wm Equation (1.36) is also true for any v which is a linear combination of wI’ ... , wm ’ Since these combinations are dense in a continuity argument finally shows that (1.36) holds for each v E Vand that u is a solution of (1.25).
v:
Proof of Lemma 1.4. This is an easy consequence of the Brouwer fixed point theorem. Suppose that P has no zero in the ball D of X centered at 0 and with radius k, Then the following application
=-
~-+S(~)
k
P(~)
[P(~)]
maps D into itself and is continuous. The Brouwer theorem implies then that S has a fixed point in D: there exists ~o ED, such that [~o]
k P(~o) [P(~o)]
- I! - ~o•
If we take the norm of both sides of this equation we see that = k, and if we take the scalar product of each side with ~o, we find
[~ ]2
o
= k2 =_ k [P(~o),
[P(~o)]
~o]
This equality contradicts (1.29) and thus P(~)
ofD.
must vanish at some point
Proof of Lemma 1.5. It is easy to show, as for (1.22) - (1.23), that b(up., up.’ v) = - b(up., v, Up.)
But Up.i converges to "t in L 2 (n ) strongly; since Div,E L"" (n), it is easy to check that
J
n u ....,V,ljdx-+
f
n "",D,vj dx.
Existence and uniqueness theorems
Ch. II, § 1
Hence
b(u",~
u"’) converges to b(u~
v~
u)
v~
=-
b(u~
167
u, v).
Uniqueness. FOI uniqueness we only have the following result:
Theorem 1.3. If n ~ 4 and if v is sufficiently large or f "sufficiently small" so that (1.37) then there exists a unique solution u of(1.25).
The constant c(n) in (1.37) is the constant c(n) in (l.18); its estimation is connected with the estimation of the constants in (104) and this is given for instance in Lions [I]. Proof of Theorem 1.3. We can take v n E;;; 4; we obtain with (1.22)
= u in (1.25) since V= V for (1.38)
so that any solution u of (1.25) satisfies 1
lIuli E;;; - Iltllv’ . v
(1.39)
Now let u; and u be two different solutions of (1.25) and let u = u; - u .We subtract the equations (1.25) corresponding to u; and u and we obtain v«u, v) + b (u.~
We take v
vllull
u, v) + btu, u., v)
V v E V.
(lAO)
= u in (lAO) and use again (1.22); hence: 2
=-
b(u, u., u).
With (1.18) and (1.39) this gives (for u V
= 0,
= u.)
lIull2 E;;; c(n) lIull2l1u.1I E;;;
c(n)
c(n)
(v - -
v
2
-lltllv,lIuli , v IIfllv ’) lIuli
2
~
O.
Because of (1.37) this inequality implies lIu’l = 0, which means u.. =u.....
Steady•state Navie,..Stokes equations
168
Ch. II, § 1
Remark 1.1. The solution of (1.25) is probably not unique if (1.37) is not satisfied or at least for v small enough if fixed). A non-uniqueness result for v small will be proved in Section 4 for a problem very similar to (1.25). Remark 1.2. For n > 4, Theorem 1.2 shows the existence of solutions of (1.25) satisfying (1.39): the maiorations (1.32) and (1.33) give indeed:
U
.
I
lIuli 0, such that
vllvll 2 + b(v, v, v) + b(v, t/J, v) + b(t/J, v, v);> 13 IIvll2, V v or because of (1.22) vllvIl 2+b(v,
t/J, v)~l3l1vIl2,
V vE
v:
E
V, (1.71)
Now (1.71) will certainly be satisfied if we can find t/J which satisfies (1.67) and
(1.72) In order to show this, we will prove the following lemma: Lemma 1.8. For any
7> 0, there
exists some t/J = t/J(7) satisfying (1.67)
175
Existence and uniqueness theorems
Ch. II, § 1
and (1.73) Before this we prove two other lemmas. Lemma 1.9. Let p(x) = d(x, f) = the distance from x to f. For any > 0, there exists a function e E ee 2 (0) such that 0e= I in some neighbourhood off (which depends on e). (1.74) 0e= 0
26(e), 6(e)
ifp(x)~
IDkOe(x)I~-ifp(x)~26(e),
= exp
e
p(x)
I (- -) e
(1.75)
k= l, ... ,n.
(1.76)
Proof. Let us consider with E. Hopf [2], the function for X ~O by
A-+~e(A)
defined
I if A < 6(e)2
~e(A)
=
e log
(6~e)
) if 6(ep
< A < 6(e)
(1.77)
o if A > 6(e) and let us denote by Xe the function Xe(x)
= ~e(p(x».
(1.78)
Since the function p belongs to~2(n), the function Xe satisfies (1.74) - (1. 76) and 0e is obtained by regularization of Xe. Lemma 1.10. There exists a positive constant cl depending only on 0 such that I
I pVIL2(n)~clllvIIH~(n)’
1
V vEHo(O).
(1.79)
Proof. By using a partition of unity subordinated to a covering of I’, and local coordinates near the boundary, we reduce the problem to the same problem with 0 = a half-space = {x = (x n, x’), x n> 0, x’= (Xl’ ... , xn-l) E tRn - 1 } . In this case p (x) = Xn, and it is sufficient to check that
Steady-state Nallier-Stokes equations
176
Ch. II, § 1
(1.80) This inequality is obvious if one proves the following one-dimensional inequality:
f:-1 V~)
I’ds 0, we find (the sum is actually finite): +...
1Uh(M)I(n - l)p/n -
P~
c1 (n, p)h j
~
16ihuh(M - (r
+ ~ ) hi) I
{I uh(M - rhj)ln(p - l)/n - P + IUh(M - (r+ l)hi )ln(p-1)/n- p}.
(2.6)
We strengthen inequality (2.6) by replacing the sum on the right-hand side by the sum for r E1t; we can then interpret the sum as an integral and majorize it by
c, (n, p) {
.~,
J__ + ...
(Jl, ~,
I""
16.. u.(II" &)1• +
~ h,)I’(P - 1)/. - d~" p}
, 1Jn) are the coordinates of M and f1t = (1J1’ .,., /oti- b 1J;+ l’ , /otn)• In a similar way we denote by ~ the vector (xl’ ., Xi-1 , Xf+.1’ , xn ) and then write x = (~, xt). For any X E 0h (M), inequality (2.6) gives now
where (/ot1’
1Uh (x)l(n - l)p/n- p= IUh (M)I(n -1)p/n - p
..c, Let us now set
(n, p)
.L~
~
+ ...
__ 16..
,Iu. (,£"
U.(~"
~,+ ~’
(2.7)
&)1
)I•r.-
1)/. -
p} dt,. (2.8)
Discrete inequalities and compactness theorems
Ch.II, §2
Then, IWi(~)ln
-1
183
is majorized by the right-hand side of (2.7), hence
{.~, IU.(~, ~i+ ~i)I-(P-l)/- P}~d~i’ E;;;
0, there exists 11 > 0 such that, IT12 uh - uhILql(/Rn):E;;;
for any uh E it and any translation operator (T12 cP)
(x)
Q = (Ql’
= cP(x + Q).
(2.25)
e, .
,
Q,.,), with IQI :E;;; 11; T12 denotes the (2.26)
Proof of (i). Because of the Sobolev inequalities (2.3) and (2.11), the family 8 is bounded in J!l (n) where q is given by (2.19) if p < n, and q is some fixed number, q > ql> otherwise (p = n). By the Holder inequality, we then get
Discrete inequalities and compactness theorems
Ch, II, § 2
J
IUhlql dx:S;;; c(meas(n-K)l-(qdq),
189
V Uh E C.
(2.27)
a-K
The right-hand side of (2.27) (and hence the left-hand side) can be made less that e, by choosing the compact K sufficiently large; (i) is proved. . Proof of (ii), First, we show that (2.25) may be replaced by a similar condition on 1T2 uh - uh ILP (condition (2.30) below). Case (a): q1:S;;; p, For any f E H (n) we have f E Lql (n) as well as fE lJ’ (n) since n is bounded and q1 :s;;; p < q. Also, O:S;;; l/q1 - l/p < 1. By the Holder inequality,
ltlLql (a):S;;; (meas
1
n) /ql-l/P.
ltlLP(a) = Const. ltlLP(a)"
Case (b): q1 > p. For any function fE Lq (n) we can write, using the Holder inequality,
J
IfI q, dx
= :s;;;
f
lfI,q’IfI(1-8)q• dx
(J ) (f ltl Ql 6P dx
l/p
ltlql (l - 6)p’ dx
)
lip’
where 8 E (0, 1), p > 1, and as usuaII/p’+ I/p = 1. We can choose 8 and p S9 that q1 8p = p, q1 (1 - 8)p’= q;
this defines p and 8 uniquely, and these numbers belong to the specified intervals, (8ql(q-ql) (q-p) =p(q - ql)’ and p(q - ql) = q - p). Then
(2.28) In particular, for any 2 and
l12 u h
- Uh ILq1(Rn)
uh’
:s;;; IT2Uh - uhliq~R~
IT2Uh - uhl~(Rn)
Steady-state Navier-Stokes equations
190
~
HTl/UhILq(61n)
+ 1UhILq(61n)} 1-
1- 8 :s;: 2 1- 6 1Uh 1Lq(61~
-.
Since the family
Ch. II, §2 6
11Q Uh -Uh l;;’(61 n)
8 ITl/Uh - Uh1lJ’(61n)’
is bounded in Lq (6ln ),
~
I1Q Uh - uh ILql (61 n)
~
(2.29)
cl1Q uh - uh 1;;’(61 n)’
Inequality (2.29) shows us that it suffices to prove condition (ii) with ql replaced by p: V e > 0, 3 11, such that ITl/Uh - uh IL P(61 n) ~
(2.30)
e
for IQI ~ 11 and uh Eg’. The proof of (2.30) follows easily from (2.23) and the next two lemmas. Lemma 2.2.
n
ITl/Uh - uh IL P(61 n)
where
~
Qi
~ ~
denotes the vector ~ht
(2.31)
l’IQ[Uh - uh IL P(61 n)’ ~
Proof. Denoting by I the identity operator, one can check easily the identity n
Tl/-
I=
L i=1
(2.32)
Tt ... Tt (Tt -l). i i-I i
This identity allows us to majorize the norm
ITl/uh- uh I p n L (61 )
by
n
L i=1
ITt .. T2’= (T2’= Uh - uh)l P n’ i 1-1 I L (61 )
We obtain (2.31) recalling that IT(l/lLP(61n)
= lfl LP(61 n)’
for any ex. E 6ln and any functionfE IP(6l n ).
(2.33)
Ch. II, § 2
Discrete inequalities and compactness
191
theorems
Lemma 2.3. ’T2iUh - uh I E;;; LP(6t n)
c(I~1
+ 1~ll/p)
lc5ihUhILP(6tn),
I E;;;iE;;;n. (2.34)
Proof. Since IT2iUh - Uh’LP(6tn)
= IT_2iUh -
uh ILP(6t n
>’
we can suppose that Qi~ 0 and we then set
~ = (~+
Pi) h;, where ai is an integer
and 0 E;;; Pi < I, I E;;; i E;;; n,
We write 011
Tt Uh - Uh 1
~
o,}
(2.35)
-1
= j=O L
T/.t. (Tt. 1
1
u" -
Uh)
+ TOI1 h 1 (Tp1 t.1 Uh -
(2.36)
uh)’
From (2.36) and (2.33), we get the majoration 011 -
IT~
Uh - UhILP(6tn) E;;;
But
"h1 -
I
~
1 ITt. Uh - UhILP(6tn)
+ ITPs~
Uh - UhILP(6tn)"
(2.37)
= hI "It/2 c5 l h 1
and the sum on the right-hand side of (2.37) is equal to alhllc5lhUhILP(6tn);
since al hI E;;; Ql’ we obtain
IT~Uh
- uhILP(6tn) E;;; QlI6lhUhILP(6tn)
..
+ 1’Tp. h. uh
Let us now majorize the norm of Tn" h Uh - Uh. For x E ah(M), x = (xl’ ... , x n ) , and ME Rh , M we have
o if (ml -
- Uh’LP(6t n)’
= (mlh l, ... , mnh n),
I 2")hl <xl «ml -
Jt
hI c5 l h uh (M +~)
2
(2.38)
if (ml -
I
f3t + 2")h l I
f3t + -
I xl 0,
provided
(3
lIuhllh =k,andk>-.
ao
Lemma 1.4 gives the existence of at least one uh such that P(Uh)
=0
or «P(Uh), Ph »h
= 0, V Ph
(3.12)
E Vh,
which is exactly equation (3.9). Let us suppose that (3.8) and (3.10) hold and let us show that uh is unique. Iful and ul* are two solutions of (3.9) and ifuh = ul - ul*, then ah (Uh. Ph)
+ bh (ul, ul, Ph) - bh (ul*, ul*, Ph)
= 0, VPh
E Vh ;
taking Ph = uh and using (3.6) we find ah (uh. Uh)
= bh (uh , ul, uh ).
Because of (3.2) and (3.8), we have aolluhll~ "’c"ul"h lIuhll~. If we set J)h = u~ in the equation (3.9) satisfied by
uZ, we find
(3.13)
ah(ul, ul) = (Qh. ut>,
and with (3.2) and (3.3),
ao lIuI I~
’" (3l1ul Ilh ’
I!ulllh’ ’"
1..
(3.14)
ao
Using this majoration and (3.13) we obtain,
(ao- :) lIuhl ~
’" 0;
(3.15)
if (3.10) holds, this shows that uh = O. Theorem 3.1. We assume that conditions (3.2) to (3.7) are satisfied; uh
Ch. 11, §3
Approximation of the stationary Navier-Stokes equations
203
is some solution of (3.9). Ifn , (3.16)
0:0 IIuh IIh .
(3.18)
As h’ ~ 0, according to (3.4), (3.5), (3.7), ah,(uh’. rh’v) ~ v«u, v)), bh,(uh’, uh’, ’h’v) ~ b(u,
u, v),
($lh" rh’v> ~ if, v).
Hence U belongs to V and satisfies v«u. v))
+ btu, u,
v) = if, v), Vv
Er.
(3.19)
If n c
"!lI v’,
(3.77)
inequality (3.76) shows that the error lIuh - U II has the same order as IIrh U - U II. Since c is the constant c(n), n = 2, in (1.18), the inequality (3.77) is exactly inequality (1.37) which ensures the uniqueness of the solution U of the exact problem. Approximation (APX5). If we consider the approximation (APX5) of
V, we can set
(3.78) (3.79) (3.80)
(3.81)
b h" (uh’ vh’ wh)
=-
~
-2I L..
i.I"l
f
n
uih vlh (D ih wlh) dx.
(3.82)
It is clear that b", bl: and bh are trilinear forms on Vh and since Vh has a finite dimension, these forms are continuous. We have to check (3.6) and (3.7); (3.6) is obvious with our choice of the form bh , and (3.7) is the purpose of next lemma. lemma 3.3. Assume that n then
< 3.
Ifphuh converges
weakly to
WU,
(3.83)
Approximation of the stationary Navier-Stokes equations
Ch. II, § 3
215
Proof. By definition, we are assuming that uh -+ U in L 2 (n) weakly,
and
D ih uh -+ Diu in L2 (n ) weakly, 1 ~ i ~
(3.84)
n.
(3.85)
The Compactness Theorem 2.4 shows that uh -+ U in L 2 ( n ) strongly.
or.
(3.86)
We know that if v E Ph ’h v converges to wv in F strongly; but the proofs of Propositions 1.4.12 and 1.4.15 show that furthermore ’h v -+ v in the norm of Loa (n), (3.87) Dih’h
v -+ Div in the norm of Loa (n).
(3.88)
The proof of (3.7) will be complete if we prove that b,,(Uh’ uh’ ’hv)-+b’(u, U, v),
(3.89)
b;;(Uh’ uh’ ’h v) -+ b"(u, u, v) .
(3.90)
For (3.89) we write
n
2:
i.]» 1
n
2:
i.]» 1
J
n U, ’jCDih Ujh - D, Uj) dx .
All the preceding integrals converge to 0 and (3.89) follows. The proof of (3.90) is similar. 0 The convergence result given by Theorem 3.1 states that uh ’ -+
u in L2(n) strongly,
Dih, uh ’ -+ Diu in L 2 (n) strongly, 1 E>;; i ~ n
(we recall that n = 2 or 3 only).
(3.91) (3.92)
216
Steady-state
Navier-Stoke« equotions
Ch.n, §3
Exactly as in the linear case it can be shown that there exists some step function 1fh constant on each g, g E 9j, , and vanishing outside n(h) such that v«Uh’ vh ))h + bh (uh’ uh’ vh) - (1fh’ divh vh)
= (f, vh)’
VVh E Wh .
(3.93) When condition (3.8) and some condition similar to (3.10) are satis› fied, uh and U are unique and the error between U and uh can be esti› mated as in the linear case, if moreover U EW 3 (n ) and p E~2(n). Lemma 3.4. Let U and p denote the exact solution of (1.8)-(1.1 0) and let us suppose that U Ei’3(n), p Ei’2(f2), and that 0. = 0. (h). Then v«u, vh ))h + b h (u, U, vh) - (P, div vh) = (f, vh) + Qh (vh)’ (3.94) where (3.95)
Proof. We take the scalar product in L 2 (n ) ofvh E Wh with the equa› tion (1.8) written in the form 1
- v/).u
Since 0.
~
n
+ 2" ~1
= n(h),
(ui Diu - Di(uiu))
+ grad p = f ,
we find
( 1 tv(/).u, Vh)9’ + 2" (Ui Diu - Di(uiu), Vh)9’
+ (grad p ’ Vh)9’ - (f, Vh)9’
t= o.
The Green formula applied several times in each simplex f/ shows that the left-hand side of this relation is equal to {v«u, vh ))h
+ S(u,
u, vh) - (P, divh vh) - (f, vh)
- Qh(vh)} = 0
where
Approximation of the stationary Navier-Stokes
Ch.U, §3
equations
217
The estimation (3.95) of Qh (vh) follows easily from Proposition 1.4.16. 0 We now proceed as for (3.71). We take vh (3.94) and subtract these relations. We get v«u - uh’ uh - ’hu»h
+ b h (u,
=uh
- ’hu in (3.93) and
U, uh - ’hu)
- b h (Uh’ uh’ uh - ’h u) - (p - 1rh’ divh (’h u - uh»
= Qh (vh)
(3.96)
.
Since divh (’hu - uh) vanishes, the corresponding term disappears. We then estimate the difference bh(uh, uh, uh - ’h u) - b h (u, U, uh - ’h u)
= bh(uh - ’h u, uh’ uh - ’hu)+bh(’h U, Uh’"h - ’h U) - bh(u, U, uh - ’hu)
=
= b h (uh - ’h u, "h’"h -’h") + bh(’h u, ’h" - U, uh - ’hu)
+ bh(’h u
- U, U, uh - ’h U),
The absolute value of this sum can be majorized, because of (l.18) and (3.65), by
+ c{ lI’hullh + lIull} "’h u
cIIuh IIl1uh - ’huI12
- ullh•
(3.97)
lIuh - ’hullh .
We recall that vlluh I ~
= (f,
and therefore
Iluh IIh E;;;
I
-
v
uh>
IfI .
Hence the sum (3.97) is less or equal to
=-v IfI lIuh - ’hulI~
+ c{
lI’hulih
+ lIulI} "’h u
- ullh•
lI’hU - Uh II h
(3.98)
With this majoration and (3.95) we get from (3.96)
( v -;- IfI ) lu, -
r,uft~
.. vllu, - r,u!, lIu - r,ull,
+ c{ Il’hulih + lIulI} lIuh - ’hul1h lIu - ’hullh + C(U, p) p(h) lIuh - ’hullh .
(3.99)
Steady-state Navier-Stokes equations
218
Ch.U, §3
Finally
(
- ;- Ifl ) lIu. - ,.ull. ".lIu - ,.uR. + c{
IIrhullh
+ lIull} lIu -
rhullh
+ c(u,
p) p(h) .
(3.100)
With the assumption
v2
> clfl ,
(3.101)
the inequality (3.100) gives a majoration of the error between uh and ’h u and hence between u and uh .
3.3. Numerical Algorithms. The following analysis is restricted to the dimensions n a must satisfy. The existence of um + 1 satisfying (3.107) is not obvious, but can be proved using the Galerkin method, exactly as in Theorem 1.2. There› fore we will skip the proof. It is not difficult to see that tim + 1 is the solution of the following nonlinear Dirichlet problem:
um + 1 EHA(n) n
2:
-vilu m + 1 +
=-
i= 1
rrz + 1 D.um + 1
u,
’
+ ~ (div um+ 1 )u m+1 (3.109) 2
grad pm + fEH-l (n).
The solution of (3.107)-(3.109) is not, in general, unique. When u,m +1 is known, pm + 1 is explicitly given by (3.108) which is equivalent to pm+l =pm _ p divum+1 EL2(n). (3.110) To investigate convergence we will assume that c(n)
v- -
v
Ilfllv’
_
= v > 0;
(3.111)
with (3.102), (3.103) and Theorem 1.3, the condition (3.111) implies the uniqueness of the solution of (1.8)-(1.11); p is unique up to an additive constant; we fix this constant by req uiring that
f
p(x)dx
n
= O.
(3.112)
Ch. II, §3
Steady-state Navier-Stokes equations
220
Proposition 3.2. We assume that n ~ 4 and that condition (3.00) holds.
We suppose also that the number p satisfies 0 0 v v
(3.126)
and that 001*
(3.127)
O r1 > 0), having the same vertical axis. The inner cylinder is rotating with an angular velocity e, while the other is at rest. Since we are looking for axi-symmetrical solutions, we will use cylindrical coordinates in ~3, say r, e, z, where the Oz axis is the axis of the cylinders. The fluid thus fills the domain 0:
r1 = t/>* = Tt/> - At/> = t/>’ - t/>", t/> E V . We have f}f* £v*
= -M(f, v)
=-
(4.22)
N(f, v)
Since TO = 0, we must prove that IW"IIv -+ 0 as 11.1.11
’
IIt/>IIv
’#’
v
-+ 0 .
(4.23)
With (4.22) and the methods of Lemma 4.3 one easily sees that IM(f, V)IL 2( (!JL )
~collt/>II}
,
IN(f, V)IL 2((QL ) ~ cl IIt/>II} ,
(Ci
= constants).
Then, by Lemma 4.2,
Ilf*IIH 2((!J L ) ~
c21M(f, V)IL 2( (!J L )
~ c3 IIt/>II} ,
IIv*IIH 2 ((!J L ) ~
c4 1N(f, V)IL 2((!J L )
~
Cs
IIt/>II} .
In particular
IIt/>*IIv ~c611t/>II} and (4.23) follows.
0
,
4.1.5. A uniqueness result. Before starting the proof of the non-uniqueness of solutions, we es› tablish a simple uniqueness result (for A "small"), which is exactly the adaptation of Theorem 1.6. to the Taylor problem. Proposition 4.2. If A is sufficiently small O~A = {f, v}be a solution of (4.12)-(4.14). We multiply (4.12) by t, (4.13) by v, and integrate these equations in (9L with respect to the measure r dr dz. We have
f
[M(f, v)’f - N(f, v)•vJ r dr dz
=0 .
If)L
Using then (4.16) we get
Integrating by parts, we see that the right-hand side of this equation is equal to (a
+ b)
at r dr dz
v -
az
.
This is bounded by
Ar2 Suple + bl Iv IL 2(If)L )
l.c v = 0 when (4.24) is satisfied.
dz 0 and ’Y < O. Then
+ 1] ,
The Green’s function G(r, s) of M under the boundary condi› tions G(–I, s) = 0, is negative for r, s E (-1, +1) (i)
(ii) The Green’s function H(r, s) of M2 under the boundary condi› tions H(–I, s) = (3H)/3z (–I, s) = 0, is positive for r..s E (-1, + 1).
We infer from this the Lemma 4.8. The Green’s functions H(k; r, s) and G (k; r, s) of (Jt- k 2)2 and - GJt- k 2) on (r 1, r2), under the boundary conditions (4.29) are positive on (r1’ r2) X (r 1, r2)’
With these kernels, we can convert (4.28), (4.29), into integral equations f; (r)
J’2
=n a
G(n a; r, s) a(s) vn (s) ds,
(4.30)
H(na;r,s)bfn(s)ds.
(4.31)
’1
’2
J
v~(r)=na
’1
An eigenvector of En is a pair of functions lfn, vn }, such that
En lfn, vn}
= Xlfn, vn}
for some X E lR. Therefore the relations (4.28) (4.29) hold with = Xv n . These are equivalent to the following ones deduced from (4.30), (4.31):
t; = Xfn , v~
f n (r)
= An
’2
a
J
’1
G(n a; r, s) a(s)
Vn (s)
ds ,
(4.32)
Bifurcation theory and non-uniqueness results
Ch. II, §4 Vn (r)
f
237
’2
= "An
a
H(n a; r, s)b f n (s) ds .
(4.33)
’1
Eliminating Vn ’ we also get,
where Jl
K(n a; r, s) f n (s) ds ,
(4.34)
J
(4.35)
= A2 and
Kik;r, s)
= k2
’2
G(k; r, t)H(k; t, s) a(t)b dt .
’1
We shall now give some properties of the eigenvalues of the operators En. They are based on the following result whose proof can be found for example in Witting [I ] . (1) Lemma 4.9. Let K(r, s) denote a real continuous function defined on the square [rl’ r2] X [rl’ r2]’ which is strictly positive on the interior of this square. The eigenvalue problem
f(r)
’2
J
=A
K(r, s) f(s) ds, rl
< r < r2 ’
(4.36)
’1
possesses a solution Al > 0 which corresponds to an eigenfunction f etO([rl’ r2]),f(r) > Ofo rr 1 AI’ (1)This is a particular case of a general result of Krein-Rutman [1) concerning linear compact operators leaving invariant a cone of a Banach space. These results are infinite dimensional extensions of the Perron-Frobenius theorem for positive matrices, well-known in linear algebra (see for instance R. S. Varga [1]).
Ch. II, §4
Steady-state Navier-Stokes equations
238
Lemma 4.10. The operator En possesses an eigenvalue X~ > 0, which corresponds to an eigenvector {fJ, v~ } with fJ (r) > 0, v~ (r) > 0 for rl Xl ~ n
n
n
+ bl
max]s
0
Proof. Lemma 4.10 gives IXn I > X~. in (4.37). We have (Jt-
(no)2)2 f~(r)
-(Jt-
(no))
v~
= X~ (r) = X~
(4.37)
.
Let us prove the second inequality
(4.38)
a no v~(r) b no f~ (r) .
We multiply the first relation (4.38) by r f~(r), the second by r v~(r), then we integrate and integrate by parts. We obtain:
J(j~,
v~) = X~
where J(j, v) =
no (
’z
J
(a + b)f~
’1
(4.39)
dr,
’1
r [(I’n’
+ (no)4J2 +
v~r
+ 2(na)’ (
(~
)’
+
T~
(dV)2 1 ] dr + r 2 v2 + (no)2 v2
f’ ) + r dr.
Ch.n, §4
Bifurcation theory and non-uniqueness results
239
The right-hand side of (4.39) is bounded by r2
A~ n a maxls
+ bl
J
If~
Ir dr
v~
I Jr
rl
la + bl no max -2-
E;;; A~
2
no
If~
2r
1
1 dr + no
Jr
rl
2
Iv11 2 , dr
rl
I
la+bl
~
E;;; - - max - - 1(f1, vI). (no)2 2 n n
We can divide by l(f~,
v~)
which is non-zero, and (4.37) follows.
0
4.2.3. Spectral properties of B. We first observe that if If, v} is an eigenvector of B with eigenvalue A, then the ifn , v n } corresponding to the expansion (4.27) of If, v} are respectively eigenvectors of the operators B n , with eigenvalue A. We will conversely deduce from Section 4.2.2 a spectral property of B. We consider all the A~, n » 1, given by Lemma 4.10. Due to (4.37), Aln -+ 00 as n -+ 00, and therefore Inf A~
n;;.l
is fmite and strictly positive. We denote by m the largest integer n such that A~ = inf A~. It may happen that A~ = A~ for some other values p;;’l
of n, n < m. We would like to avoid this situation and actually we have: Lemma 4.12. One can choose the period L, so that Al In this case Al
=
> A~,
Vn
> 1.
Inf A~ is denoted AI; Al is a simple eigenvalue of B
n> 1
in iT and any other eigenvalue A of B satisfies IAI > AI. The eigenvector ¢ 1 = if1, vI} corresponding to A1 admits a Fourier expansion of type (4.27) with fJ = v~ = 0, V n > 1.
Proof. Let L be arbitrarily chosen and let a = 21T ILand m be defined as We set 0* = rna, above (m = the largest n such that A~ = Inf A~). n>l
L = LIm. For the corresponding operator B, Al is an eigenvalue of B l’ and Al < A~, V n > 1. The other properties stated in the Lemma
240
Steady-state Navier-Stokel equation«
are now obvious.
Ch. II, §4
0
Until the end of this Section we assume that L is chosen so that Lemma 4.12 holds.
Our last result concerns the degree of Al as an eigenvalue of B. The degree of Al is the dimension of Ker(I - Al B)p , which is independent of p, for large p. Lemma 4.13. Under the conditions of Lemma 4.12, Al is an eigenvalue of B of degree I. Proof. We will show that if (I - A1B)Pif> = 0 for p ~ 2, then (I - AlB) = 0, so that Ker(I - Al B)P is equal for each p to Ker(I - Al B), and its dimension is one, because of Lemma 4.12. We proceed by induction on p and actually we just have to show that (I - Al B)2 if> = 0 implies (I - Al B)if> = O. Let us consider some function if>0 such that (I - A1B)2if>0
= O.
We argue by contradiction and assume that (I - Al B)if>0 is not equal to O. This vector is then equal, within a multiplicative constant, to the previous eigenfunction if>l:
= (I -
if>l
Al
ew.
if>l
= Al Bif>l
(4.40)
We have,
= A1B(if>0 + if> 1),
if>0
which amounts to saying that
to = Al a aza
£2
£vo
a
= Alb az
We recall that fl (r, z)
(vo + vi
(fO +
),
r»
(4.41)
=fl (r) sin(az), vi (r, z) = vI (r) cos(az).
Let us consider the Fourier series of
to and vO ;
Ch. II, §4
Bifurcation theory and non-uniqueness results
2: 12 sin(naz),
n=l
241
00
vO
=
2:
n=l
v~ cos(naz).
The relations (4.41) imply (A’- a 2 ) 2
-i.r-
!? = Al a a (v? = Al
a 2 )v?
ba
+ vi),
ifl + 11),
(4.42)
and for n ~ 2: C/(- a 2 ) 2 ~
= Al a a n v~
-(1- a 2 ) v~
= Al
ban ~.
Since Al is not an eigenvalue of B n for n ~ 2, we see that
~ = v~ = 0 for n ~ 2. We now convert (4.42) into integral equations, as in (4.30), (4.31). Since {f"f, vi} is an eigenvector of B 1 , we obtain
v~ (r)
= Al
J(2
Gto; r, s) a(s)a v~ (s) ds
+ 11 (r),
H(a; r, s) b a!?, (s) ds +
vi (r).
rl
By elimination of v?, and using the kernel K introduced in (4.35), we get !?(r) -
A~
r2
(
i,
K(a; r, s)!?(s)
ds
= 2/1 (r).
(4.43)
The equation (4.43) satisfies the Fredholm alternative. Thus/l is orthogonal to the eigenfunction gl of the adjoint equation: gl (r) -
By Lemma
A~
f
J
r2
K(a; s, r) gl (s) ds
=o.
rl
4.9,/1 andg 1 are positive on (r1’
r2) and this contradicts
Steady-state Navier-Stokea equationa
242
Ch.II,§4
the orthogonality condition
C fl 2
Jr.
(s) gl (s) ds
= 0.
Thus (I - Al B)4Jo = 0, and the proof is complete.
0
4.3. Elements of Topological Degree Theory We recall a few definitions and properties of topological degree theory. For the proofs and further results, the reader is referred to the basic work of J. Leray and J. Schauder [1], or M. A. Krasnoselskii [I], L. Nirenberg [I], P. Rabinowitz [4]. 4.3.1. The topological degree. Let T be a compact operator in a normed space V, and let S = I - T (I = the identity in V). We denote by w, wi’ bounded domains of V; wand aw denote the closure and the boundary of w. If w is a bounded domain of V, if v E V and
v fF. S(aw), one can define an integer d(S, w, v) which is called the topological degree of S, in t», at the point v. The main properties of the degree are the following ones:
= wI
U w2’ and wI n w2 :;: v f$.S(aW2)’ then v f$.S(aw) and
(i)
If w
d(S, w, v)
= d(S,
(ii) If d(S, w, v) the equation (I - T) (u)
rp, if v f$. S(awl ), and
wI’ v) + d(S, w2’ v).
* 0, then v
E S(w), which amounts to saying that
=v
has at least one solution in w. (iii) d(S, w, v) remains constant if S, t», v, varies continuously, in such a way that v never belongs to S(aw). (1) (1) A continuous variation of Sis defined as follows:
S = S(’A.) = I - T(’A.), ’A. E tR (or any topological space), and ’A. ..... T(’A.) I/> is a mapping uni› formly continuous with respect to 1/>(1/> E V).
243
Bifurcation theory and non-uniqueness results
Ch. II, §4
4.3.2. The index. Let Uo be a point of V, v = Suo’ and let us assume that the equation Su = v admits only the solution uo, in some neighbourhood of uo’ In this case, one can define for e small enough, the degree deS, we (uo), v), where we (uo) is the open ball of radius f centred at uo’ According to the property (iii) of the degree, this number is inde› l
pendent of
f,
as
~
O.
We define then, the index of Sat Uo as this degree, for small: i(S, uo)
= deS,
we(uo), v),
f
f
sufficiently
< fO’
Some fundamental properties of this index are listed below: (i) If the equation Su = v posseses a finite number of solutions uk in a bounded domain w, and has no solutions on aw, then
.us, w, v)= L:k
;(S, uk)’
(ii) The index of the identity (T = 0) at any point Uo is one: i(l, uo) = 1.
(iii) Let us assume that T admits at the point uo, a Frechet differ› ential A. Then A is compact like T If 1 - A is one to one (i.e. 1 is not an eigenvalue of A), then Uo is an isolated solution of the equa› tion Su = Suo’ and one can define the index i(S, u 0 ). One has i(S, uo)
= it] -
A, 0)
= i(1 -
A).
(iv) If A is a linear compact operator in Vand if 1 - A is one to one, the index of 1 - A is – 1.(1) Similarly the index of 1 - M is defined on any interval A’ < A < A" containing no eigenvalue of A; the index is constant on such intervals and is equal to –1. In particular i(l- M) = 1 on the interval (0, AI)’ where Al is the smallest positive eigenvalue of A. When A crosses a spectral value Ai of A, the index i(1 - M) is multi› plied by (_I)m where m is the degree of Ai’ i.e. the dimension of Ker(1 - AiA)k which is independent of k, when k is sufficiently large. (1) One can define the index of I - A if and only if I - A is one to one; when it is defined, the index is the same at every point u u ’
244
Steady-state Navter-Stokes equations
Ch. II, §4
4.4 The non-uniqueness theorem.
Our purpose is to prove the following result. Theorem 4.1. For A sufficiently large, and for suitable values of L, the problem (4.2), (4.3), (4.6) possesses z-periodic solutions of period L which are different from the trivial solution (4.7). Proof. We will prove that the equation (4.20) has a non-trivial solu› tion in V, when A is sufficiently large. According to Lemma 4.12, we can choose L so that Al is a simple eigenvalue of B in V: these are the values of L mentioned in Theorem 4.1. It is known from Proposition 4.2 that (4.20) possesses only the trivial solution for A sufficiently small (A Al . Lemma 4.14. Let w be some open ball of V centered at O. There exists some 6 > 0 such that ¢ :II: A T ¢ has no solution on the boundary aw of t», for each A in the interval [AI’ Al + 6]. Proof. We argue by contradiction. If this statement is false, there exists a sequence of An decreasing to AI, and a sequence of Uk belonging to aw. such that
Since the sequence un is bounded, the sequence T un is relatively compact (by Lemma 4.4), and there exists a subsequence T uni con› verging to some limit v in V. Then u ni = Ani T u ni converges to Al v. Since T is continuous, we must have
Al v = Al T(AI v). Thus Al V is a solution of ¢ = Al T¢, and because of (4.44), Al v = 0, v = 0. This contradicts the fact that I!A I v II v is equal to the radius of the ball w (llu n II = radius of w, V n).
Ch. II, §4
Bifurcation theory and non-uniqueness results
245
Lemma 4.15. Under the assumption (4.44), if wand 8 are as in Lemma 4.14, the equation cP = A T cP has no solution on aw, for any A E [0, Al + 8). Obvious Corollary of Lemma 4.14. This lemma allows us to define the degree d(l - AT, w, 0) for A E [0, x, + 8). Lemma 4.16. With 8 and w as before, d(I - AT, w, 0)
= 1,
for A E [0, Al + 8).
Proof. It follows from the property (iii) of the degree that d(l - AT, w, 0) = di], w, 0) = i(l) and this index is equal to one (the index of the identity). Lemma 4.17. Under the assumption (4.44), there exists for any A E (AI’ Al + 8) at least one non-trivial solution of cP = A T cp. Proof. According to Lemma 4.13, and the properties (iv) of the index, i(I - AB) is equal to 1 on [0, AI) and is equal to -Ion (A. 1 , Al + 8). According to the property (iii) of the index, i(l - AT, 0) is + 1 for A E (0, AI) and -1 for A E (AI’ Al + 8).’ If A E (AI’ Al + 8) and if zero is the only solution of cp = A. T cp in w, we should have d(I - AT, w, 0)
= i(l -
AT, 0)
according to the property (i) of the index. But we proved that d(l - AT, w, 0)
= +1,
i(l - AT, 0)
= -1,
A E (AI’ Al + 8).
Thus the equation cp = A T cp has a non-trivial solution for any A E (A.1’ Al + 8). The proof of Theorem 4.1 is complete. Remark 4.3. The condition "A sufficiently large", amounts to saying that the angular velocity a is large or that the viscosity v is small (for fixed r1’ ’2)’ Remark 4.4. Under condition (4.44), there exists for each A E (AI’ Al + 6) a non-trivial solution Cp", of (4.20). One can prove that CPA -+ 0 in V, as A decreases to AI’ This is the bifurcation
246
Steady-state Navier-Stokes equations
Ch.II, §4
In case of the Benard problem the situation is very similar, but it can be proved that there only exists the trivial solution for A E [0, AI]’ Thus the assumption (4.44) is unnecessary, and one does prove the occurrence of a bifurcation (see V. I. Iudovich [2] , Rabinowitz [1], Velte [1] ). A study of the Taylor problem by analytical methods is developed in Rabinowitz [5]. 0 Acknowledgment. The author gratefully acknowledges useful remarks of P. H. Rabinowitz on Section 4.
CHAPTER III THE EVOLUTION NAVIER-STOKES EQUATION Introduction This final chapter deals with the full Navier-Stokes Equations; i.e., the evolution nonlinear case. First we describe a few basic results con› cerning the existence and uniqueness of solutions, and then we study the approximation of these equations by several methods. In Section I we briefly examine the linear evolution equations (evolution Stokes equations). This section contains some technical lemmas appropriate for the study of evolution equations. Section 2 gives compactness theorems which will enable us to obtain strong con› vergence results in the evolution case, and to pass to the limit in the non› linear terms. Section 3 contains the variational formulation of the problem (weak or turbulent solutions, according to J. Leray [1], [2], [3]; E. Hopf [2]) and the main results of existence and uniqueness of solution (the dimension of the space is n = 2 or 3); the existence is based on the construction of an approximate solution by the Galerkin method. In Section 4 further existence and uniqueness results are pre› sented; here existence is obtained by semi-discretization in time, and is valid for any dimension of the space. In the final section we study the approximation of the evolution Navier-Stokes equations, in the two- and three-dimensional cases. Several schemes are considered corresponding to a classical discretiza› tion in the time variable (implicit, Crank-Nicholson, explicit) asso› ciated with any of the discretizations in the space variables introduced in Chapter I (finite differences, finite elements). We conclude with a study of the nonlinear stability of these schemes, establishing sufficient conditions for stability and proving the convergence of all these schemes when they are stable.
§ 1. The Linear Case In this section we develop some results of existence, uniqueness, and regularity of the solutions of the linearized Navier-Stokes equations. After introducing some notation useful in the linear as well as in the 247
248
The evolution Navier-Stokes equations
Cli. III, § 1
nonlinear case (Section 1.1), we give the classical and variational formu› lations of the problem and the statement of the main existence and uniqueness result (Section 1.2); the proofs of the existence and of the uniqueness are then given in Sections 1.3 and 1.4.
1.1. Notations Let 0 be an open Lipschitz set in eRn ; for simplicity we suppose 0 bounded, and we refer to the remarks in Section 1.5 for the unbounded case. We recall the definition of the spaces "Y, V; H, used in the pre› vious chapters and which will be the basic spaces in this chapter too: "1/= {u E ~(O),
div u = O}
V = the closure of fin Hfi (0),
H
= the closure
of fin L 2 (0).
(1.1) (1.2) (1.3)
The space H is equipped with the scalar product (. , .) induced by L 2(0); the space V is a Hilbert space with the scalar product n
«u, v)
= 2: i= 1
(DiU, Div),
(1.4)
since 0 is bounded. The space V. is contained in H, is dense in H, and the injection is con› tinuous. Let H’ and V’ denote the dual spaces of H and V, and let i denote the injection mapping from V into H. The adjoint operator t is linear continuous from H’ into V’, and is one to one since i( V) = V is dense in Hand i’(H’) is dense in V’ since i is one to one; therefore H’ can be identified with a dense subspace of V’. Moreover, by the Riesz representation theorem, we can identify H and H', and we arrive at the inclusions VCH=H'CV',
(1.5)
where each space is dense in the following one and the injections are continuous. As a consequence of the previous identifications, the scalar product in H of f E Hand u E V is the same as the scalar product of f and u in the duality between V’ and V: (f, u) = if, u), V fEH, VuE V.
For each u in V, the form
(1.6)
Thelil1ear case
Ch. III, § 1
249
v E V-+ ((U, V)) E lR
(1.7)
is linear and continuous on V; therefore, there exists an element of V’ which we denote by Au such that
= «U, v)), V v E V.
(1.8)
It is easy to see that the mapping u -+ Au is linear and continuous, and, by Theorem 1.2.2 and Remark 1.2.2, is an isomorphism from V onto V’.
If n is unbounded, the space V is equipped with the scalar product [u, v]
= «u,
v)) + (u, v);
(1.9)
the inclusions (1.5) hold. The operator A is linear continuous from V into V’ but it not in general an isomorphism; for every e > 0, A + el is an isomorphism from Vonto V’. Let a, b be two extended real numbers, -00 ~ a < b ~ 00, and let X be a Banach space. For given ex, I ~ ex < +00, LI:I.(a, b; X) denotes the space of LI:I.-integrable functions from [a, b] into X, which is a Banach space with the norm
L I b
11!tt) 11’jC cit
11/1:1.
(1.10)
The space L (a, b; X) is the space of essentially bounded functions from [a, b] into X, and is equipped with the Banach norm GO
Ess Sup "f(t) [a, bl
"x'
(1.11)
The space ~([a, b] ; X) is the space of continuous functions from [a, b] into X and if -00 < a < b < 00 is equipped with the Banach norm Sup
tE
[a. bl
"f(t)
"x'
(1.12)
°
Most often the interval [a. b] will be the interval [0, T], T> fixed; when no confusion can arise, we will use the following more con› densed notations, LI:I.(X) = LI:I.(O, T; X), I ~(X)
=0
be some function
J: ~o(t)dt
Any function
rJ>
in~«O,
T»,
such that
= I.
in.@«O,
T»
can be written as
J T
~ =1-4>0 + «, A = 0 ~(t)dt,
r
'iJ E ~«O,
T));
(1.21)
indeed since
(if>(t) -
o
A~o(t) dt
= 0,
the primitive function of rJ> - ’ArJ>0 vanishing at 0, belongs to ’@«O, T», and I/J is precisely this primitive function. According to (1.20) and (1.21),
J
T 0 (> is arbitrarily small, w is zero on the whole interval [0, T]. 0
°
1.2. The Existence and Uniqueness Theorem
°
Let n be a lipschitz open bounded set in &in and let T> be fixed. We denote by Q the cylinder n X (0, n. The linearized Navier-Stokes equations are the evolution equations corresponding to the Stokes problem: To find a vector function
u : n X [0,
T] -+
lR n
and a scalar function p :
n X [0, T]
-+
lR,
respectively equal to the velocity of the fluid and to its pressure, such that
Ch.IIl, §1
Thelinear case
auat - VlYl + grad p =I in Q = n X (0, T), div u
= a in Q,
253
(1.23) (1.24)
= a on an X [0, T] , u(x, 0) = uo(x), in n, u
(1.25) (1.26)
where the vector functions I and Uo are given, I defmed on n X [0, T] , Uo defined on n; the equations (1.25) and (1.26) give respectively the boundary and initial conditions. Let us suppose that u and p are classical solutions of (1.23)-( 1.26), say u EfG2(Q), P E ~1 (Q). If v denotes any element of ~ it is easily seen that
(
~"
) + I’«u
))
=if, v),
(1.27)
By continuity, the equality (1.27) holds also for each v E V; we observe also that
) at ’ v ( au
d
= dt
(u. v).
This leads to the following weak formulation of the problem (1.23)› (1.26): For I and Uo given,
IE £2(0, T; V’)
(1.28)
uOEH,
(1.29)
to find u, satisfying
u E £2(0, T; V)
(1.30)
and d dt (u. v) + v«u. v» u(O)
= uo'
= (f, v>,
Vv E V.
(1.31) (1.32)
If u belongs to £2(0, T; V) the condition (1.32) does not make sense in general; its meaning will be explained after the following two remarks: (i) The spaces in (1.28), (1.29), and (1.30) are the spaces for which existence and uniqueness will be proved; it is clear at least that a smooth
Theevolution Navier-Stokes equations
254
Ch. III, § 1
solution u of (1.23)-(1.26) satisfies (1.30). (ii) We cannot check now that a solution of (1.30)-(1.32) is a solution, in some weak sense, of (1.23)-(1.26); hence we postpone the investigation of this point until Section 1.5. By (1.6) and (1.8), we can write (1.30) as d dt tu, v) = (f - vAu, v), Vv E V.
(1.33)
Since A is linear and continuous from V into V’ and u E L 2 (V), the function Au belongs to L 2(V '); hencef - vAu E L2(V') and (1.33) and Lemma 1.1 show that u' E L 2(0, T; V’)
(1.34)
and that u as a.e. equal to an absolutely continuous function from [0, T] into V'. Any function satisfying (1.30) and (1.31) is, after modi› fication on a set of measure zero, a continuous function from [0, T] into V’, and therefore the condition (1.32) makes sense. Let us suppose again that j' is given in L2(V ') as in (1.28). Ifu satisfies (1.30) and (1.31) then, as observed before, u satisfies (1.34) and (1.33). According to Lemma 1.1 the equality (1.33) is itself equivalent to u ’ + vAu
=f.
(1.35)
Conversely if u satisfies (1.30), (1.34), and (1.35), then u clearly satisfies (1.31), V v E V. An alternative formulation of the weak problem is the following: Given f and Uo satisfying (1.28)-(1.29), to find u satisfying u E L 2(0, T; V), u’ E L2(0, T; V'), (1.36) u’ + vAu = f, on (0, T),
(1.37)
u(O) = uo'
(1.38)
Any solution of (1.36)-(1.38) is a solution of (1.30)-(1.32) and conversely. Concerning the existence and uniqueness of solution of these problems, we will prove the following result.
Theorem 1.1. For given f and Uo which satisfy (1.28) and (1.29), there exists a unique function u which satisfies (1.36)-(1.38). Moreover
255
The linear case
Ch. III, § 1
(1.39)
T] ; H).
U E~([O,
The proof of the existence is given in Section 1.3, that of the unique› ness and of (1.39) are in Section 1.4.
1.3. Proof of the Existence in Theorem 1.1 We use the Faedo-Galerkin method. Since V is separable there exists a sequence of linearly independent elements, WI, ... , W m , ... , which is total in V. For each m we define an approximate solution um of 0.37) or (1. 31) as follows: m
um
=
2:
(1.40)
Kim (t)Wi’
i= 1
and (U:n, Wj) + U m (0)
v«u m Wj»
= (f,
Wj), j
= 1, ... , m,
= UOm ’
(1.41) (1.42)
where "Om is, for example, the orthogonal projection in H of Uo on the space spa nne db y wI’ ... , wm • (l) The functions Kim’ 1 ~ i ~ m, are scalar functions defined on [0, T], and 0.41) is a linear differential system for these functions; indeed we have m
m
2:
2:
i= 1
i= 1
= 1/I(I)dI,
(1.64)
for each v which is a finite linear combination of the w;’s. Since each term of (1.64) depends linearly and continuously on v, for the norm of V, the equality (1.64) is still valid, by continuity, for each v in V. Now, writing in particular (1.64) with VJ = ~ E9}«O, T», we find the following equality which is valid in the distribution sense on (0, T): d dt (u, v) + lI«U, v»
= (f,
v), V v E V;
(1.65)
The evolutionNavier-Stokel equations
260
Ch. III, § 1
which is exactly (1.31). As proved before the statement of Theorem 1.1, this equality and (1.59) imply that u' belongs to L 2(0, T; V’) and
u' + vAu
=f.
(1.66)
Finally, it remains to check that u(O) = Uo (the continuity of u is proved in Section 1.4). For this, we multiply (1.65) by 1/1(1), (the same 1/1 as before), integrate with respect to t, and integrate by parts:
~ We get
T
d
dt (u(t), v)1/I(t)dt
o·
=- JT (u(t), v)1/I'(t)dt 0
+ (u(O),
-J
T
v)1/I(0).
J T
0 (u(I), _l\!/(I)dl
+ v 0 «u(I), v))>/I(ljdl
J T
= (u(O), _)>/1(0) + 0 (/(1), v)>/I(t)dl.
(1.67)
By comparison with (1.64), we see that (uo - u(O), v)1/I(0)
= 0,
for each v E V, and for each function 1/1 of the type considered. We can choose 1/1 such that 1/1(0) =1= 0, and therefore (u(O)-uo, v)=O, VvE V.
This equality implies that u(O)
= Uo
and achieves the proof of the existence.
0
1.4. Proof of the Continuity and Uniqueness This proof is based on the following lemma which is a particular case of a general theorem of interpolation of Lions-Magenes [I] :
Lemma 1.2. Let V; H, V’ be three Hilbert spaces, each space included
Thelinear case
Ch. Ill, § 1
261
in the following one as in (l.5), V’ being the dual of V If a function u belongs to L 2(0, T; V) and its derivative u’ belongs to L2(0, T; V’), then u is almost everywhere equal to a function continuous from [0, T] into H and we have the following equality, which holds in the scalar distribution sense on (0, D: d
dt lul 2 = 2(u’, u).
(1.68)
The equality (1.68) is meaningful since the functions t -+ lu(t)1 2 , t -+ (u’(t), u(t»
are both integrable on [0, T]. An alternate elementary proof of the lemma is given below. If we assume this lemma, (1.39) becomes obvious and it only remains to check the uniqueness. Let us assume that u and v are two solutions of (1.36)-(1.38) and let w = u - v. Then w belongs to the same spaces as u and v, and w’ + vAw = 0, w(O) = 0.
(1.69)
Taking the scalar product of the first equality (1.69) with w(t), we find
=
(w’ (t), w(t» + v II w(t) 11 2
°
a.e.
Using then (1.68) with u replaced by w, we obtain d -lw(t)1 2 + 2vllw(t) 11 2 dt
=
°
Iw(t)1 2 '" Iw(0)1 2 = 0, t E [0, T] . and hence .,(t) = v(t) for each t,
0
Proof of Lemma 1.2. The elementary proof of Lemma 1.2 which was announced before, is now given in the two following lemmas. Lemma 1.3. Under the assumptions of Lemma 1.2, the equality (1.68) is satisfied. Proof. By regularizing the function U, from cR into V, which is equal to u on [0, T] and to 0 outside this interval, we easily obtain a sequence of functions urn such that V m, urn is infmitely differentiable from [0, T] onto V, (1.70)
The evolutionNavier-Stokes equations
262
as m ~OO,
Ch. III, § 1
Llo c ( ] 0, T[; V),
Um
~U in
u~
~u’inLfoc(]O,T[;
V').
(1.71)
Because of (1.6) and (1.70), the equality (1.68) for um is obvious: -
d
dt As m ~
IU m (r)]
2 -_ 2(u’m (t), U m (t)) _ - 2(u’m (t), u m (t) ) ,V m.
(1.72)
it follows from (1.71) that lu ml 2 ~lul2 in Lloc(] 0, T[) (u~, u m ) ~ (u', u) in Lloc(] 0, T[) 00,
These convergences also hold in the distribution sense; therefore we are allowed to pass to the limit in (1.72) in the distribution sense; in the limit we find precisely (1.68). Since the function t ~ (u'(t), u(t)
is integrable on [0, T] , the equality (1.68) shows us that the function u of Lemma 1.3 satisfies (1. 73) u E L" (0, T; H). In the particular case of the function u satisfying (1.36)-(1.38), this was proved directly in Section 1.3. According to Lemma 1.1., u is continuous from [0, T] into V'. Therefore, with this and (1.73), the following Lemma 1.4 shows us that u is weakly continuous from [0, T] into H, i.e., V v EH, the function t .... (u(t), v) is continuous.
(1.74)
Admitting temporarily this point we can achieve the proof of Lemma 1.2. We must prove that for each to E [0, T], lu(t)-u(to)1 2 ~O,
ast~to’
(1.75)
Expanding this term, we find lu(t)1 2
+ lu(to)1 2
- 2(u(t), u(to))'
When t ~ to, lu(t)1 2 ~ lu(to)1 2 since by (1.68), lu(t)1 2
= lu(to)1
2
+
2J
t
(u'(s), u(s)>ds; to
The linear case
Ch. III, § 1
263
and because of (1.74) (u(t), u(to)) -+ lu(to)/2,
so (1.75) is proved. The proof of Lemma 1.2 is achieved as soon as we prove the next lemma. This lemma is stated in a slightly more general form.
Lemma 1.4. Let X and Y be two Banach spaces, such that
XeY
(1.76)
with a continuous injection. If a function tP belongs to L"" (0, T; X) and is weakly continuous with values in Y, then tP is weakly continuous with values in X. Proof. If we replace Y by the closure of X in Y, we may suppose that X is dense in Y. Hence the dense continuous imbedding of X into Y gives by duality a dense continuous imbedding of Y’ (dual of Y), into X’ (dual of X):
y'ex'.
(1.77)
By assumption, for each 11 E y’, (tP(t), 11)-+ (tP(to), Tl), ast-+to,Vto,
0.78)
and we must prove that (1.78) is also true for each 11 EX’. We first prove that tP(t) E X for each t and that IItP(t)lI x ~ IItPIIL""co,T;X)' VtE[O,T].
(1.79)
°
Indeed, by regularizing the function 1> equal to tP on [0, T] and to outside this interval, we find a sequence of smooth functions tP m from [0, T] into X such that IItPm (t)lI x ~ IItPll
ee
L
(X)
,V m, V t E [0, T]
and (tP m (t), 11) -+ (tP(t), 11), m -+ 00, V 11 E y’.
Since l(tP m (t), 11)1 ~ IltP II
L
co
(X)
1I11l1x', V m, V t,
we obtain in the limit l(tP(t), 11)1 ~ IItP ll
L
ee
(X)
1I11 l1x" V t E [0, T], V11 E y’.
This inequality shows that tP(t) EX and that (1.79) holds.
264
Theevolution Navier-Stoke« equations
Ch. III, § 1
Finally let us prove (1.78) for 'TI in X’. Since y’is dense in X’, there exists, for each > 0, some 'TIe E y’ such that
1I'T1 - 'TIe"x' We then write
l' to, since 'TIe E y’, the continuity assumption implies that
I ~ 2e II cP IlL "(X)·
°
Since e > is arbitrarily small, the preceding upper limit is zero, and (1.78) is proved. 0 1.5. Miscellaneous Remarks We give in this section some remarks and complements to Theorem 1.1. An Extension 01 Theorem 1.1. Theorem 1.1 is a particular case of an abstract theorem, involving abstract spaces V and H, and an abstract operator A; see Lions› Magenes [I] . If instead of (1.28) we assume that (1.80) 1=/1 +/2,11 EL 2(0,T;V’), h E L 1(0, T; H), then all the conclusions of Theorem 1.1 are true with only one modifi› cation: U’ E L2(0, T; V’) + L 1(0, T; H). (1.81) In the proof of the existence, we write after (1.47): d dt IU m (t)1 2 + 2vllu m (t)1I 2 ~ 211t1 (t)lI v' lIum (t)1I
+ 2Ih(t)llum (t)1 ~ vllu m (t)1I2 I +-
V
IIfl (t) II}. + Ih(t)1
{l + Iu m (t)12 }.
(1.82)
Thelinear case
Ch. III, § 1
Hence, in particular d I - {I + Iurn (t)1 2 } :E;;; - IItl (t)II~, dt v
265
+ Ih(t)/ {I + Iurn (t)1 2 }.
(1.83)
Multiplying this by
we obtain
d~
I
exp ( - (
.;;
~
If,(a)lda)- (I + IU m (tll')
nfl (t)II}. exp ( - (
I
If,(a)Ida) .
°
Integrating this inequality from to s, s > 0, we obtain a majoration similar to (1.50) which implies (1.51). Then integrating (1.82) from to Twe obtain (1.53). The proof of the existence is then conducted exactly as in Section 1.3. Concerning the derivative u’, we have u’
= -vAu + t 1 + h
E L 2(0, T; V’)
°
+ L 1 (0, T; H).
(1.84)
It is easy to see that Lemma 1.2 is also valid if
u E L 2(0, T; V) n ir (0, T; H), u’ E L2(0, T; V’) + Ll (0, T; H).
(1.85)
Noting this, we can prove the uniqueness and the continuity of u, u E 0, there exists some constant c Tl depending on 11 (and on the spaces X o, X, Xl) such that:
IIv IIx
:eo;; 11llv IIxo’ + c Tl IIv II x 1,
Vv E X o.
(2.3)
Proof. The proof is by contradiction. Saying that (2.3) is not true amounts to saying that there exists some 11 > 0 such that for each c in lR,
IIv IIx
~
11 llv II xo + cIIv IIx 1,
for at least one v. Taking c = m, we obtain a sequence of elements V m satisfying
IIvm II x
~
11 IIv m II x 0 + mIIv m II x l , V m.
We consider then the normalized sequence Wm
= II Vm IIXo
'
which satisfies
IIwm IIx
~
11 + m Ilwm IIx 1, V m.
Since IIw m II xo = I, the sequence that
IIw m IIx 1 -+ 0, as m -+ 00.
Wm
(2.4)
is bounded in X and (2.4) shows (2.5)
Compactness theorems
Ch. III, § 2
271
In addition, by (2.2), the sequence wm is relatively compact in X; hence we can extract from W m a subsequence w~ strongly convergent in X. From (2.5) the limit of w~ must be 0, but this contradicts (2.4) as:
IIx ~
IIw~
f/
> 0,V IJ..
2.2. A Compactness Theorem in Banach Spaces
Let Xo, X, Xl, be three Banach spaces such that (2.6)
Xo CXCXI,
where the injections are continuous and: Xi
is reflexive,
i
the injection Xo
= 0, -+
I,
(2.7)
X is compact.
(2.8)
Let T> 0 be a fixed finite number, and let ao, al , be two finite numbers such that ai > I, i = 0, I. We consider the space :v=,~(0,
j’ =
1.
T; ao, al; X o, Xl) E L" (0, T; X 0), .’ = :; E La, (0, T,' Xl)
!
(2.9) (2.10)
The space Y is provided with the norm
IIvlLw = IIvllL ~
(O,T;Xo)
+ IIv'lIL
Ci
1(O,T;X 1)
'
(2.11)
which makes it a Banach space. It is evident that :VC LO 0 is arbitrarily small in Lemma 2.1, this upper limit is 0 and thus (2.14) is proved. (iii) To prove (2.15) we observe that (2.18)
T] ;Xf},
~C