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

we get,
T T
2at 2
2at
2
e E ( B b( u, ϕ)) dx dt e E B dx dt,
T T
2at 2
2at
2
lim L = L E|| ϕ || + 2a
e E|| ϕ|| dt e E B dx dt
N
0
T T
2
2at e E J ϕ dx dt + lim
2at
N 2
e E|| ϕ ϕ || dt+
5.3.2 Study of the parabolic equation :
According to the lemma 7, it is possible to extract a subsequence of ( u ) ∈ , still noted ( u ) ∈ ,
N N ∞
such that u (resp. ( u ) N∈
) converges weakly towards u (resp. ) in V as well as in L ( V)
weak- .
1
N N
Let us consider v ∈ H ( Q ) , and set v the step function and v
the piecewise affine continuous
function from [0 , T ] to V, built from the sequence v = v( n t ) as introduce in (8) .
Step 1 : energy equality
According to (6), for any n, one has :
0
T
2at
2
e E ( B b( u, ϕ)) dx dt.
N
T
T
2at 2
2at
N 2
+ e E ( B b( u, ϕ)) dx dt + lim e E|| ϕ ϕ || dt.
0
n 1
v +1 v
unvn u0v0 dx t
uk t
∇ ∇
n 1
n 1
t ( u ) v dx + t ( u u ) v d =
k=0
0 2
Following (12), one nds that lim L
Thus,
0
k+1 k+1
k=0
n
1
∞
k+1 k+1 k+1
v +1 v
ϕnvn ϕ0v0 dx t
ϕk t
N N
dx+
∇ ∇
nt N
N N N
N
u ( n t) v ( n t) u0v (0) dx u
0
+
nt n 1
N N
( ) +
N
( k+1 k)
+
∂v
∂t
dx ds
u v dx ds t u u ∂v
∂t dx
0
nt
N ∞,N
N
( u u ) v d ds =
N N
ϕ ( n t) v
( n t) dx
nt N
n 1
N
N
N
ϕ v (0) dx ϕ + ( k+1 k)
∂v
∂v
dx ds t ϕ ϕ
∂t ∂t dx.
0
t [0 , T ] n ( n 1)t < t nt,
t
N N N
u ( t) v ( t) u0v (0) dx
N
N
u +
∂v
Therefore, for any in and for such that we get,
∂t
dx ds
0
n
0
k=0
k=0
0
k=0
0
0
14
0
0
0
k k
k k
k=0
dx.
N N
(14)