1. INTRODUCTION 5
Consider a point (b,u) G C, b (£, x, b0). To derive the Kreiss-Lopatinski
condition for hyperbolic problems, the idea is to approximate Q near x_ by the half
space {v(x)-x 0}, and to linearize the equation (1.1) around the constant solution
u. This leads to a constant coefficient problem which is analyzed using a tangential
Fourier-Laplace transform. We proceed in a similar way for (1.4). To (6, u) G C
we associate the profile w(z) Wb^u(z). The substitute for the constant state u is
the "planar" boundary layer w(x) = w{y x/e) that interpolates between 0 on the
boundary v x = 0 and the inner state u . Next, we linearize the equation (1.4)
around w to get the linear operator
d d ..
(1.10) dtv + ] T A)d3v -sYl Bj,kdlkv + -Eh
where
d d
Aj-t; = AjV - ^T vk(dzw VuBk,3)v -^vk(v - VuBjik)dzw
k=l k=l
(1.11) Eh = ^ i/fc(v VttAfc)dzW - ^ v3vk[y
VuB3,k)d2z*W
d
~ ] C VjVk(VlB3,k(v,dzw))dzw
where / stands for the function / evaluated at (b,w). In (1.10), the coefficients are
functions of v x only, and thus we can again perform a Fourier-Laplace transform in
(£, y) where y are the variables in the tangent space T^l. This leads us to introduce
the following symbols, where 7 7 G T*dtt is a Fourier tangential frequency and r ij
a Fourier-Laplace time frequency:
(1.12) M = (iT +
7
)Id + X ^ 4 + J2 WkBjtk + E*
j = l lj,fcd
(1.13) A =
J2^Al-
E ivMBjj + Bkj)
j = l li,fcd
The symbols M and A are functions of z G [0, oof which depend on the parameters
(b,u) G C and (r, 77,7) G R x T^9Qx]0, oo[. Introducing the fast variable z and
scaling the frequency variables properly (see section 2 below), the Fourier-Laplace
transform of (1.10) reads
(1.14) Cv :=-Bnp^ + A^ + Mv
dzz dz
This is an ordinary differential system in 2, depending on the parameters (6, w, £)
withC := (r,??,7).
Let E _ ( M , 0 denote the set of initial data (u(0),g(0)) G C N x CN such
that the corresponding solution of Cv = 0 is bounded as z tends to infinity. Un-
der Assumptions 1.1 and 1.2, for all (6, u) G C and £ = (r, 77,7) ^ 0 with 7 0,
E_(6, w, C) has dimension iV and depends smoothly on the parameters (6, u, Q (see
Corollary 2.7 below). The wea/c stability condition states that the problem Cv 0,
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