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

and,
e E ϕ ( T ) E ϕ 2 a e E ϕ dt+
2 2
0
2
T
aT
||
N
|| || ||
0
2at ||
N 2
||
N
1
k=0
2akt 2
a( k+1)t T T
2at N N N
2at
N N 2
2 e E J( u ) (1 ϕ ) ϕ dx dt + e E b( u , ϕ ) dx dt.
0
0
Step 3 : Lower semi-continuity
T T
Since applications u ↦→ u ds and u ↦→ u dw( s)
are linear continuous functions from H to
H,
0 0
T T T T
N N
J( u ) (1 ϕ ) ds +
N N
b( u , ϕ ) dw( s) ⇀ J ds + B dw( s)
H
0 0 0 0
and thanks to (10), ϕ ( T ) ⇀ ϕ( T ) in H.
Let us set
Remark that 0 and that (11) implies,
2aT N 2 2aT
2
T
T
+ E|| ϕ || 2 a
2as 2
e E|| ϕ|| ds +
2as
e E 2
2
J ϕ dx + || B|| ds =
0 2
0
2aT 2 2aT
N 2
+ e E|| ϕ( T ) || = e lim E|| ϕ ( T ) || L
where, we note L = E|| ϕ ||
+ lim { 2 a e E|| ϕ || dt+
0 2
N∈
2 N 2
T
0
2at
N 2
T T
2at N N N
2at
N N 2
2 e E J( u ) (1 ϕ ) ϕ dx dt + e E b( u , ϕ ) dx dt } .
Since the sequences
0
0
T
T
2at
N 2
2at
N N N
( e E|| ϕ || dt ) , ( e E J( u ) (1 ϕ ) ϕ dx dt ) ,
0
0
T
T
2at
N N 2
2at
N 2
( e E b( u , ϕ ) dx dt) and e E|| ϕ ϕ || dt
0
N
2C t ( e e ) + C t+
Thus,
N∈
are bounded in R,
a subsequence can be found, still indexed by N,
such that all of the above
sequences converge.
consequently,
T
2at
N 2
L = E|| ϕ || 2 a lim e E|| ϕ || dt+
0 2
T
2 lim
2t
e E
N N N
J( u ) (1 ϕ ) ϕ dx dt+
0
T
lim
2at
e E
N N 2
b( u , ϕ ) dx dt.
0
E|| ϕ( T ) || lim E|| ϕ ( T ) || .
= e lim E|| ϕ ( T ) || e E|| ϕ( T ) || .
12
0
0
0
N∈
(12)