v(0) = 0 has no nontrivial bounded solutions, equivalently that E_ is transverse
to kerT := {0} x C ^ , where T is the mapping (u,v) ^ u from C ^ x C ^ to C ^ .
This is clearly necessary for linear stability, its violation implying the existence of
local time-exponentially growing modes. The strong or uniform stability condition
requires in addition some uniform behavior as £ tends to zero and also as £ tends to
infinity. In particular, the uniform behavior near the origin is needed to recover the
uniform stability of the hyperbolic boundary value problem. The uniform behav-
ior at infinity is equivalent to the well-posedness of the Dirichlet boundary value
problem for the parabolic part of the equation.
Because E_ and kerT both have dimension TV in a space of dimension 27V,
there is a determinant
(1.15) D(b, u, C) = det (E_(6, u, C), ker r )
obtained by taking orthonormal bases in each space, and the result is independent
of the choice of the bases. This is the Evans' function (see [Z], [S]). D vanishes if
and only if E_ D kerT is not reduced to {0}.
To deal properly with the high frequencies, some appropriate scaling is required.
A (
C ) = ( l + r 2 +
2 + |7
| 4 ) i
introduce the space E_(6, u, Q J A E _ ( 6 , U,Q where J\ is the mapping (u,v) »—»
(it, A'1!;) in C ^ x CN and the "scaled" Evans' function
(1.17) 5(6 , u, C) = det (E_(6, u, (), ker T) .
Note that kerT is invariant by J A SO that D vanishes if and only if D vanishes.
Moreover, for bounded values of £, there is C such that ^\D\ \D\ C\D\, since,
in the computation of the Evans's functions, the introduction of J A only amounts
to a change of scalar product in C2N.
The weak stability condition requires that D ^ 0 for (6, u) G C and ( ^ 0 with
7 0. The strong or uniform reads
A S S U M P T I ON 1.4 (Uniform stability condition). There is a constant c 0 such
that for all for all (6, u) G C and C = (T, 7,77) ^ 0 with 7 0
(1.18) | 5 ( M , c ) | c
R E M A R K S 1.5. a) The weak and uniform stability conditions are conditions
on the "frozen coefficient" planar boundary value problems associated with the
inner layer solution. They are natural analogs of those defined in [Z] for the planar
shock case. In the one-dimensional boundary layer case, Assumption 1.4 reduces
to the condition imposed by Grenier and Rousset [Gr-Ro].
b) The uniform stability condition is equivalent to saying that
(1.19) \v\ CA\u\ for U = t(u,v)eE-(b,u,T,rj,'y)
uniformly with respect to (6, u) and (r, 77,7) bounded, with (r, 77,7) 7^ 0 and 7 0.
c) Under Assumptions 1.1 and (1.2) the spaces E_(6, u, r, 7,77) have limits
E°_(6, u, f,77,7) when (r, 7,77) = p(f,7),7) and p tends to zero, with (f, 77,7) ^ 0,
7 0. In addition, the spaces~E°_are closely related to the similar spaces associated
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