Modélisation, analyse mathématique et simulations numériques de ...

tel-00656013, version 1 - 3 Jan 2012
2.3 Well-posedness of the multilayer model 55
while Sl = S1 l ,...,SN T l and Snl = S1 nl ,...,SN T nl are respectively the linear and the
non linear sources, that is:
⎧
S i l = −g∂xh, 1 i N
⎪⎨
⎪⎩
S 1 nl
S i nl
= 2µ
= 2µ
u2 −u1 κ
− u1
h1(h1 +h2) h1
2 1
−∂x u1 −
h1
ui+1 −ui
hi (hi +hi+1) −2µ
2 1
−∂x ui −
hi
u3/2w3/2 −u1w 1/2 ,
ui −ui−1
hi (hi +hi−1)
S N nl = −2µ uN −uN−1 ∂xh∂xuN
+µ
h (h+hN−1) h
− 1
2 ∂x
2 1
uN +
h
ui+1/2wi+1/2 −ui−1/2w i−1/2 , 2 i N −1,
uN−1/2 −uN wN−1/2.
Hence, we can sum up by considering that the source term Snl is roughly composed of
three kinds of non linearities, that is
u/h, u∂xu/h, ∂xh∂xu/h.
Consequently, in order to simplify the next calculations, we will only consider the simpler
hyperbolic-parabolic problem
⎧
⎨ ∂tU−µ∂xxU = Sb +Sl +Snl,
(2.3.2)
⎩
∂th+∂x(huN) = F ,
where F, Sb, Sl are not changed, while the nonlinear source is simplified as
where
Snl =
3
Sk,
k=1
S1 = U
h , S2 = uN
h ∂xU, S3 = 1
h ∂xh∂xU.
The proof of Theorem 2.3 is divided into three parts. In the first subsection we perform
some estimates on the source terms for the simpler problem (2.3.2). Next we solve a linearized
version of system (2.3.2) and derive energy estimates. Finally, we build a recursive
sequence of solutions of linear systems, and show a convergence to the strong solution we
are looking for.