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

A : V → V
and A ∈ V :
∞
∞
T t
N
E u ∂v
∂t
′
′
0 0
∞
′
V ′ ,V
V ′ ,V
∞
0 0
∞ ′ ∞ ∞
∈
=
T t
N N
r( t) b dx ds dt + E ∇( u ) ∇v
r( t) b dx ds dt+
T t T
E
N ∞,N
N
( u u ) v r( t) b d ds dt = E
N N
ϕ ( t) v ( t) r( t) b dx dt
0 0
T
T t N
N
N
E ϕ v (0) r( t) b dx dt E ϕ ( )
∂v
r t b dx ds dt
∂t
0
T
+ E ε( t) r( t) b dt.
T T
E u( t) v( t) r( t) b dx dt E u v(0) r( t) b dx dt
T t T t
E u ( ) + ∇ ∇ ( ) +
0 0 0 0
T t T
∞
( ) ( ) = ( ) ( ) ( )
∂v
r t b dx ds dt E v r t b dx ds dt
∂t
E u u v r t b d ds dt E ϕ t v t r t b dx dt
T T t
E ϕ v(0) r( t) b dx dt E ϕ ( )
∂v
r t b dx ds dt.
∂t
the monotone operator :
< Au, v > = ∇( u) ∇v
dx + uv d,
< A , v > = ∇ ∇v
dx + uv d.
t t
∞
u( t) u + A ds = F ds + ϕ( t) ϕ ,
where F V is dened by < F , v > u v d.
N
N
0
Thus, it follows that u ( T ) converges weakly in H towards some U and
T T
∞
U = u A ds + F ds + ϕ( T ) ϕ .
∞
0 0
But, since one can choose u continuous, it ensures that u ( T ) ⇀ u( T ) in H.
2at
2
Then, by applying the Itô formula to (15) with f( t, x) = e || x||
and since
N
0
0 0
0 0
Then, passing to the limit to innity with N,
one obtains :
Let us note :
0 0
0 0 0
0 0
0
0
t t
ϕ( t) ϕ(0) = J ds + B dw( s ) ,
N
0
0 0
0
Then A is the weak limit in V of ( Au ) and one has,
0 0
16
0
0
(15)