Modélisation, analyse mathématique et simulations numériques de ...

tel-00656013, version 1 - 3 Jan 2012
66 A dynamic multilayer shallow water model
Therefore, the solution to system (Dj) satisfies the following energy estimate, for any t T
(where T is given by Proposition 3.7):
U (j+1) −U (j) ,h (j+1) −h (j)
(t)
C(E)e Cb (1+E) 2 t
1
t
0
U (j) −U (j−1) ,h (j) −h (j−1)
Hence there exists a subsequence, still labelled U (j) ,h (j)
U (j) ,h (j)
j
such as
−→
j→∞ (U,h) strongly in C(0,T;H1 (R)).
Moreover, Lemma 3.7 gives, up to a subsequence, the convergence:
U (j) ⇀ U weakly in L
j→∞ 2
0,T;H 3
(R) ,
while, for every fixed t T:
U (j) ,h (j)
(t) ⇀
j→∞ (U,h)(t) weakly in H 2 (R).
(τ) 2
1 dτ
Thus we have a solution(U,h) to system (2.3.2), lying inC(0,T;H 1 (R))∩L ∞
0,T;H 2 (R)
satisfying for any t,x:
h(t,x) ≥ η0 > 0,
t
(U,h)(t)2 , U(τ)
0
2 1/2
3 dτ E.
Finally, we show that (U,h) ∈ C(0,T;H 2 (R)) by regularizing: we consider (U ε ,h ε ) =
(ρε ∗U,ρε ∗h), where ρε∗ is the Friedrichs’mollifier with respect to x. Thus, applying ρε∗
to system (2.3.2) we obtain
⎧
⎪⎨
⎪⎩
∂tU ε −µ∂xxU ε = S ε +C ε 0 ,
∂th ε +u ε N ∂xh ε = F ε +C ε 1 ,
(U ε ,h ε )(t = 0) = (ρε ∗U 0 ,ρε ∗h 0 ) ∈ C ∞ ,
where Sε = ρε ∗S, Fε = ρε ∗F and
⎧
⎨ C ε 0 = (∂t −µ∂xx) (U ε )−ρε ∗(∂t −µ∂xx) (U),
⎩
C ε 1 = {∂th ε −∂x(h ε u ε N)}−ρε ∗{∂th−∂x(huN)} .
By classical arguments on mollifiers [146, 174], we have, as ε goes to zero:
⎧
⎨ → 0,
⎩
C ε 0 , Cε 1
(U ε , h ε ) → (U,h).
1/2
.
,
(2.3.21)
Therefore, at the uniform limit we have (U,h) ∈ C(0,T;H 2 (R)). Uniqueness follows from
the energy estimate and this concludes the proof of Theorem 2.3.