Modélisation, analyse mathématique et simulations numériques de ...

tel-00656013, version 1 - 3 Jan 2012
64 A dynamic multilayer shallow water model
We define the constants ⎧ ⎪⎨
⎪⎩
E = 2(U 0 ,h 0 )2,
η0 = 1
2 inf
x∈R h0 (x).
The following lemma gives the existence of the whole sequence.
Lemma 3.7. For suitably small T > 0, the sequence (U (j) ,h (j) )j∈N is well defined and
satisfies, for any t ∈ [0,T] and any j ∈ N:
U (j) ∈ C(0,T;H 2 (R))∩C 1 (0,T;L 2 (R))∩L 2 (0,T;H 3 (R)),
h (j) ∈ C(0,T;H 2 (R))∩C 1 (0,T;H 1 (R)).
Moreover, for all (t,x) in [0,T]×R and all j ∈ N, we have:
(U (j) ,h (j) )(t)2,
t
0
U (j) (τ) 2 1/2 3 dτ
(2.3.18)
E, (2.3.19)
h (j) (t,x) ≥ η0 > 0. (2.3.20)
Proof. First we initialize the recursion. (U (0) ,h (0) ) verifies the good conditions by definition.
Applying Proposition 3.6, we obtain existence of (U (1) ,h (1) ) in C(0,t;H 2 (R)) for any
t > 0. Moreover, applying the characteristic formula to h (1) , we get, for any t > 0:
h (1) (t,y) = h 0 t
(X(0,t,y)) +
0
≥ 2η0 +C(E)t
≥ η0 if t T1 small enough,
F (0,1) (s,X(s,t,y))ds
where T1 = T1(η0,E), which yields (2.3.20) for j = 1. It remains to prove (2.3.19). To
do so, we write the inequality (2.3.14) given by Proposition 3.6 for (U (1) ,h (1) ), that is, for
any t T1:
(U (1) ,h (1) )(t)2,
t
U
0
(j) (τ) 2 1/2
3 dτ
Ke C (1+E)2 t
Hence, applying Lemma 3.2 (2.3.3) to S (0) , we obtain
(U (1) ,h (1) )(t)2,
t
U
0
(j) (τ) 2 3 dτ
1/2
E/2+
t
S
0
(0) (τ) 2 1/2
1 dτ
Ke C (1+E)2 t
E/2+C(η0)E(1+E) √
t .
Therefore, we can find 0 < T2 = T2(η0,E) T1 such that (2.3.19) is satisfied for any
t T2. We choose T := T2.
Next we pass from j to j +1. If for any j in N, (U (j) ,h (j) ) satisfies (2.3.18), (2.3.19)
and (2.3.20) for any t T2, the existence of (U (j+1) ,h (j+1) ) follows again from Proposition
.