Modélisation, analyse mathématique et simulations numériques de ...

tel-00656013, version 1 - 3 Jan 2012
58 A dynamic multilayer shallow water model
Lemma 3.3. We assume uN ∈ H 2 (R) with
uN2 E
for some positive constant E, and define the differential operator
LuN := ∂t +uN ∂x.
Then, for any h ∈ L 0 ∞ (0,T;H2 (R)), we have:
∂xx h
∂xxh CEh2,
Proof. We only compute:
h
∂xx
LuN
LuN
Hence, using Lemma 3.1 yields:
∂xx h
LuN
− LuN
− LuN
− LuN
∂xxh = 2∂xuN ∂xxh+∂xxuN ∂xh.
∂xxh 2E∂xxh+E∂xh1.
In the next subsection, we obtain energy estimates and study a linearized version of
the multilayer system.
2.3.2 Study of the linearized problem
Let us introduce a linearized version of system (2.3.2):
⎧
⎨
⎩
∂tU−µ∂xxU = S( Ũ,˜ h,∂x Ũ,∂x ˜ h) := ˜ S,
LũN (h) = F −˜ h∂xuN := f.
(2.3.8)
In order to study the well-posedness of this coupled linear parabolic-hyperbolic problem,
we first solve the parabolic system, and next the transport equation on h by considering
the right hand side
f = −˜ N−1
h∂xuN − ∂x(hiui)
as a known function. Thus, we will first study separately the following Cauchy problems,
one parabolic system ⎧
⎨ (∂t −µ∂xx)
⎩
U = ˜ S,
(A)
and one hyperbolic scalar equation:
⎧
⎨
⎩
i=1
U(0,x) = U 0 ∈ H 2 (R),
LũN
h = f ,
h(0,x) = h 0 ∈ H 2 (R).
(B)