Modélisation, analyse mathématique et simulations numériques de ...

tel-00656013, version 1 - 3 Jan 2012
2.3 Well-posedness of the multilayer model 63
where C5 = C ′ 2 +C4. Finally, we add (2.3.10) and (2.3.16), and control the exponentials
to obtain (2.3.14).
This gives the uniqueness for the solution. Let us now prove the existence of solution.
On the one hand, the existence ofUfollows from Proposition 3.4, and we have the regularity
which gives
U ∈ C(0,T;H 2 (R))∩C 1 (0,T;L 2 (R))∩L 2 (0,T;H 3 (R)),
f ∈ C(0,T;H 1 (R))∩C 1 (0,T;L 2 (R))∩L 2 (0,T;H 2 (R)).
So we can apply Proposition 3.5 for k = 1 to obtain the existence of h in C(0,T;H 1 (R)).
To get more regularity in space, we differentiate the problem with respect to x:
⎧
∂t(∂xU)−µ∂xx(∂xU) = ∂x
⎪⎨
˜ S,
∂t(∂xh)+ũN ∂x(∂xh) = ∂xf −∂xũN ∂xh,
Noticing that
⎪⎩
∂xU 0 , ∂xh 0 ∈ H 1 (R).
∂xf 1 CE∂xU1,
we can solve this problem by the same iteration process as in the lattest part of Proposition
3.5, this concludes the proof.
In order to obtain the solution to the nonlinear initial value problem (2.3.2), we will build
a convergent sequence, this is the last part of the proof of Theorem 2.3.
2.3.3 Iterative scheme
We construct a recursive sequence (U (j) ,h (j) ) = (u (j)
1 ,...,u(j)
N ,h(j) )j∈N as follows.
∀(t,x) ∈ [0,T]×R, (U (0) ,h (0) )(t,x) = (U 0 ,h 0 )(x) ∈ H 2 (R),
and for all j ∈ N, (U (j+1) ,h (j+1) ) solves the initial value problem:
⎧
⎪⎨
⎪⎩
(∂t −µ∂xx)
L u (j)
N
U (j+1)
= S (j) ,
h (j+1)
= F (j,j+1) ,
(U (j+1) ,h (j+1) )(t = 0) = (U 0 ,h 0 ),
where the sequence of source terms is given by, for any j ∈ N:
⎧
S
⎪⎨
⎪⎩
(j) = S U (j) ,h (j) ,∂xU (j) ,∂xh (j) ,
F (j,j+1) N−1
= − ∂x hiu (j+1)
i −∂xu (j+1)
N h (j) .
i=1
(Pj+1)