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

Remark 1 For technical reasons (lost of the maximum principle for ( ϕn)
), we need to use
J( un) (1 ϕn) instead of J( un) (1 ϕ n)
. We then have to prove that, at the limit when N goes
to innity, the solution ϕ obtained satises 0 ϕ 1.
Proposition 1 There exists a sequence ( u n, ϕn) n∈ , ( u n, ϕn)
Ϝnt
-measurable, with values in
V H,
satisfying (6) and (7).
Proof. The result leads from a classical xed point argument (see Grecksch et al [8] for example
in the context of stochastic calculus).
5.2 A priori estimates
Lemma 2
N
2
2
∀n ∈ { 0, 1, .., N } , E|| ϕ || M so t
E|| ϕ || M,
N
1
Proof. Since [( ϕ ) ( ϕ ) + ( ϕ ϕ ) ] = ( ϕ ϕ ) ϕ ,
tJ( un) (1 ϕn) ϕn+1 + ϕn+1 b( u n, ϕn)( wn+1 w n)
.
As, ϕ b( u , ϕ ) is Ϝ -measurable, E ϕ b( u , ϕ )( w w ) =
E ( ϕ ϕ ) b( u , ϕ )( w w ) + E { ϕ b( u , ϕ )( w w ) } =
E ( ϕ ϕ ) b( u , ϕ )( w w )
1
2
E( ϕn+1 ϕn) + Ct. 4
Since, E tJ( u ) (1 ϕ ) ϕ C t E( ϕ ) + 1 ,
2 2
2
E( ϕ ) E( ϕ ) + E( ϕ ϕ ) C t E( ϕ ) + 1 .
n n
2
E( ϕ ) + E( ϕ ϕ ) C t E( ϕ ) + C
n
2
C t E( ϕ ) + C + C tE( ϕ ) .
n
2
Whereas t 0,
∃M > 0, ∀n ∈ { 0, 1, .., N 1}
,
n
and E|| ϕ ϕ || M.
n=0
1 2 2
2
[( ϕn+1) ( ϕn) + ( ϕn+1 ϕn)
] =
2
∀ n, E|| ϕ || Ce Ce
E|| ϕ ϕ || M t.
n=0
1 2 2
2
2 n+1 n n+1 n n+1 n n+1
n n n n t n+1 n n n+1 n
Whereafter,
n+1 n n n n+1 n n n n n+1 n
n+1 n n n n+1 n
n n n+1 n+1
2
n+1 n n+1 n n+1
2
n+1
2
n+1
2
k=0
k=0
Gronwall lemma (cf. Bainov [1]) since :
k+1 k
k=0
2
n+1 n
k=0
k n+1
2
k
2 Cnt CT
n+1
2
n+1 n
6
n
k+1
2