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

T T
2at N N N
2at
N N 2
2 e E J( u ) (1 ϕ ) u dx dt + e E b( u , ϕ ) dx dt } .
Since the sequences
0
0
T
T
2at
N 2
2at
N N
( e E|| u || dt ) , ( e E ∇( u ) ∇u
dx dt ) ,
0
N∈
T T
( E
2at
e
N 2
( u ) d dt ) , (
2at
e E
N N N
J( u ) (1 ϕ ) u dx dt)
0
N∈
T
2at
N 2
and ( e E|| u u || dt)
are bounded in R,
a subsequence can be found, still indexed by N,
such that the above sequences
converge and consequently,
Let us set
N∈
T
2at ∞
= E|| u || + 2 e E u u d dt
N∈
T
T
2at N 2
2at
N N
2a lim e E|| u || dt 2 lim e E ∇( u ) ∇u dx dt
0
T T
2 limE
2at e
N 2
( u ) ddt + 2 lim
2at
e E
N N N
J( u ) (1 ϕ ) u dxdt
0
T
2at
N N 2
+ lim e E b( u , ϕ ) dx dt.
T T
= 2a 2at N 2
e E|| u u || dt 2
2at
N N
e < Au Au , u u > dt+
N
0
T
2
2at
e E
N N N
[ J( u ) (1 ϕ ) J( u) (1 ϕ)]( u u) dx dt+
0
T
T
2at N N
2
2at
N 2
e E ( b( u , ϕ ) b( u, ϕ)) dx dt + e E|| u u || dt.
0
T
Remark that c( J, , b) 2at
N 2
e E|| ϕ ϕ || dt+
Moreover, since
N
0
0
0 2
0
0
0
0
0
T
2at
N 2
(1 2 a + c( J, , b)) e E|| u u || dt.
0
T T
= 2a 2at N 2
e E|| u || dt 2
2at
N N
e E < Au , u > dt+
N
0
18
0
0
0
0
N∈
(17)