A Stochastic Model of Crystallization in an Emulsion - Laboratoire de ...

5.3.3 Conclusion
So, according to (16), lim
T T
2 2at e E J u dx dt + 2 a
2at
2
e E|| u|| dt+
0
T T
lim
2at N 2
e E|| u u || dt +
2at
e E
2
( B b( u, ϕ)) dx dt.
0
T
T
2at 2
2at
N 2
+ e E ( B b( u, ϕ)) dx dt + lim e E|| u u || dt.
0
In order to conclude, using (13) and (17 ), we get : L
⎡
T T
⎤
(1 2 a + 2 c( J, , b)) ⎣ 2at N 2
e E|| u u || dt +
2at
N 2
e E|| ϕ ϕ || dt ⎦ .
Then, for a 1 + c( J, , b ) , (14) and (18) lead to :
N N
T
2at
2
0 + + 2 e E ( B b( u, ϕ)) dx dt+
T T
lim
2at N 2
e E|| ϕ ϕ || dt + lim
2at
N 2
e E|| u u || dt.
0
So ( u ) and ( ϕ ) converge to u and ϕ in H and B = b( u, ϕ)
.
0
As , and J are Lipschitz functions, it is possible to identify ( u ) , (1 ϕ) and J( u)
as the
N N limits of the sequences ( ( u )) N , ( (1 ϕ )) N
and H respectively such that :
N and ( J( u )) N . Then, there exists u and ϕ in V
∀v ∈ V, ∀t ∈ [0 , T ];
t t
( u( t) u ) v dx + ∇( u) ∇v dx ds +
∞
( u u ) v d ds =
0
N
In order to prove that ( u, ϕ) is a solution, we have to show that 0 ϕ 1 i.e. 0 ϕ( t)
1.
2
So, considering a C ( R, R)
function , the Itô formula gives :
Remark 2 Thanks to lemma (6), if u ∈ V then G( u) ∈ L ( , H ( Q))
.
0
0
0 0
t t
ϕ( t) ϕ = (1 ϕ) J( u) ds + b( u, ϕ) dw( s ) .
0
t t
′
2
[ ( ( ))] [ ( 0)]
= (1 ) ( ) ( ) + 1
E ϕ t E ϕ E ϕ J u ϕ ds E b ( u, ϕ) ”( ϕ) ds.
2
0
0
0
0 0
0 0
0
0
N N
( ϕ( t) ϕ ) v dx
Thus, if we suppose that ( x) = 0 for any x in [0, 1] , E[ ( ϕ( t))] = 0 for any t.
Then, for any t in [0 , T ] , 0 ϕ( t ) 1 , and ( u, ϕ)
is a solution.
20
+
0
2 1
(18)