1. INTRODUCTION 7

to the limit hyperbolic problem, and extend continuously to 7 = 0. The uniform

stability condition implies that for all (6, u) G C and (r, 7,77) 7^ 0 with 7 0:

(1.20) E_ (6, u, C) p | ({0} x C") = {0} ,

and for all (f, 7, fj) ^ 0 with 7 0:

(1.21) E°„(6,

U

,f,r?,7)n({0}xC J V ) = {0}.

This will be shown in Appendix A.

d) The stability condition also involves a uniform behavior as (r, 77,7) tends

to infinity. Indeed, one can show that the spaces E_(y, 6, u, r, 77,7) have limits

as E?°(y, 6, w,f,7,77) when (r, 7,77) = (A2f, A27, A77) and A tends to infinity, with

(f,7,77) 7^ 0, 7 0. The uniform stability condition (1.18) for large values of £ is

equivalent to the transversality condition E^° n kerT = {0}. It turns out that this

condition is equivalent to the well-posedness of the parabolic Dirichlet boundary

value problem, as can be seen by a standard rescaling/asymptotic ODE argument

(see [Z], Lemma 4.28 and also the proof of Lemma 2.14 below). In particular, it

is satisfied when the parabolic operator is symmetric, i.e. when there is a smooth

definite positive S(b,u) such that Re (^Z^j^kSBj^) is positive definite for £ ^ 0

(see Remark 2.15 below).

The following useful relation was established by Rousset [Rou] via Evans func-

tion calculations. A proof of the second part of the assertion (i.e., satisfaction of

the uniform Kreiss-Lopatinski condition) is given in Appendix A; see also Remark

c) above.

PROPOSITION

1.6. [Rou] Under Assumptions 1.1, 1.27 the uniform stability

condition Assumption 1.4 implies both that the limiting boundary condition (1.8) is

transversal and that the resulting limiting hyperbolic boundary value problem (1.1)

(1.8) satisfies the uniform Kreiss-Lopatinski condition. (Indeed, these two state-

ments are together equivalent to Assumption lA(ii).)

Therefore, under Assumptions 1.1, 1.2 and 1.4, one can solve the mixed prob-

lem (1.1) (1-8) for initial conditions which satisfy sufficiently many compatibility

conditions (see [Maj], [Ra-Ma], [Mok], [Me2]).

REMARK

1.7. For outer initial data sufficiently close to some particular value

uo for which there exists a boundary layer satisfying the uniform stability condition

1.4 (or, more generally, a compact set Uo of such values), Remark 1.5 (b) above,

together with the above proposition, implies that Assumption 1.4 is automatically

satisfied for such time as the outer solution remains near uo (resp. Uo). This gives

a simple situation in which these assumptions are verifiable a priori for solutions

with large boundary layer. Note that this does not preclude the possibility of

multiple (but locally unique), stable boundary layers, with associated distinct valid

boundary layer expansions. For analogs in the shock layer setting, see [AMPZ].

Consider a solution uo in Hs°([—To, To] x Q) of the hyperbolic boundary value

problem (1.1) (1.8), with b a given smooth enough function in Hs°([—T0, To] x Q).

The index so is large enough, and how large will be made precise later. In any case,

so 1 + ^2^

s o t n a t

functions in

Hs°

are Lipschitz continuous. By definition of

the boundary condition, there is a profile

( L22 ) ™o(*, V, Z) = wb(t,y),u0(t,y) (z) •