Modélisation, analyse mathématique et simulations numériques de ...

tel-00656013, version 1 - 3 Jan 2012
2.1 Introduction and Main Result 49
The formal derivation of this set of equations in 2D will be obtained in Section 2.2. Moreover,
we will study in this paper the local in time existence of strong solution for the 1D
version of the system, that is:
⎧
N
⎪⎨
∂tH +∂x
i=1
∂t(hN uN)+∂x
hiui
hnu 2 N +g h2
N
2
= 0,
= µ
∂x(hN ∂xuN)+DU z N+1/2 −DUz
N−1/2
−ghN ∂xzb +w N−1/2u N−1/2 −w N+1/2u N+1/2,
∂t(hiui)+∂x hiu 2
i +ghi∂xhN = µ hi∂xxui +DU z i+1/2 −DUz
i−1/2 −ghi∂xzb
⎪⎩
+wi−1/2ui−1/2 −wi+1/2ui+1/2, 1 i N −1.
(2.1.10)
where we drop the Coriolis terms which have no meaning in 1D. In order to state the result,
we introduce the following notations.
For any function f, we note f ( resp. fk ) the L2-norm (resp. Hk-norm ) of f. If
f = (f1,...,fn) is multidimensional, we define its Hk-norm by
fk :=
n
fik.
i=1
Let B be a Banach space, k a non-negative integer and T some positive constant. We
denote by Lk ∞ (0,T;B) the Banach space of functions f on [0,T] which have their values
in B and are k times differentiable with respect to t and all the derivatives are bounded
in B. We can now state our main result.
Theorem 1.1. Consider the system (2.1.10) where w i+1/2, u i+1/2 and DU z i+1/2
by the 1D versions of (2.1.7), (2.1.8) and (2.1.9), with initial data
are defined
(U,hN)(0,x) = (U 0 (x),h 0 N(x)) ∈ H 2 (R), (2.1.11)
where U = (u1 ... uN) T is the vector of velocities. Suppose
inf
x∈R h0 (x) η0 > 0,
for some constant η0, and note E = 2(U 0 ,h 0 N)2. Assume the topography has the regularity
zb ∈ C 2 (R). Then, there exists a positive constant T such that the Cauchy problem
(2.1.10)-(2.1.11) has a unique strong solution (U,hN) satisfying:
U ∈ C(0,T;H 2 (R))∩C 1 (0,T;L 2 (R))∩L 2 0,T;H 3 (R) ,
hN ∈ C(0,T;H 2 (R))∩C 1 (0,T;H 1 (R)).