153
By (l.6.3b), we have G" h" = G" fl" onto 2:-. As Gnfl" -" Gil into £f (2:-), by the
uniqueness of the limit, we obtain:
Gh=Gp
onto 2:1.6.2 Momentum equation
We're going to give the lemma that allows us to obtain the limit in the Momentum
equation. We can look for the proof of this lemma in Orenga (1995). We give the result
with w = ° to simplify.
Lemma 6 .-. Let {'PI,'" ''Pn''''} be a basis of the space V, 'Pn belongs to H3(OY and
let Vn be the set of linear combinations of the n first elements of the basis. Let Vn and hn
the two sequences verifYing V<.p E Vn :
(Vn,r,'P) +Aa(vn,y) -- ~(v;.,diV<.p) + (curlvna(vn),'P)
(I.6.4a)
+ D(vn Iv"I,'P) = (hn,div'PJ+ (f,y)
(I.6.4b)
vn(t = 0) = va,n E Vn, va./l ---> Va into V
(I.6.4c)
(I.6.4d)
hn .."' .,. h into LY)(O, T;L2(rl))
f E L2(0, T;H-J(fl))
and the following estimate :
(I.6.4e)
Then we can extract a sequence from v'" always noted v,,, such as :
(I.6.4f)
(I.6.4g)
(1.6.4h)
(l.6.4i)
(1.6.4))
Vn weakly - * converge to v into L2( 0, T; V) n L 00 (0, T; L2( IW)
curl V/l 0'( vn) weakly converge to curl v a( v) into d (0, T; L} (fl)2)
\7v~ weakly converge to \7v 2 into LJ (0, T; L} (fl n
Vn,t is bounded into LJ (0, T; H- 3 (fl)2) and v(t = 0) = Va
v verifies (6.4a), 'v'P E V
We can find the proof of this lemma into Orenga (1995).