Antoine Benoit
email: antoine.benoit@univ-littoral.fr

WKB expansions for hyperbolic boundary value problems in a strip: selfinteraction meets strong well posedness

Keywords:

In this article we are interested in the rigorous construction of WKB expansions for hyperbolic boundary value problems in the strip R d-1 × [0, 1]. In this geometry, a new inversibility condition has to be imposed to construct the WKB expansion. This new condition is due to selfinteraction phenomenon which naturally appear when several boundary conditions are imposed. More precisely, by selfinteraction we mean that some rays can regenerated themselves after some rebounds against the side of the strip. This phenomenon is not new and has already been studied in [Benoit, 2016]-[Sarason and Smoller, 7475] in the corner geometry. In this framework the existence of such selfinteracting rays is linked to the geometry of the characteristic variety. However for the strip geometry such rays become generic.

The new inversibility condition, used to construct the WKB expansion, is a microlocalized version of the one characterizing the uniform in time strong well-posedness [Benoit, ].

Introduction

This article deals with the geometric optics expansion for the following highly oscillating hyperbolic problem in the strip

R d-1 × [0, 1]            L(∂)u ε := ∂ t u ε + d j=1 A j ∂ j u ε = 0 for (t, x , x d ) ∈ R × R d-1 × ]0, 1[ , B 0 u ε |x d =0 = g ε for (t, x ) ∈ R × R d-1 , B 1 u ε |x d =1 = 0 for (t, x ) ∈ R × R d-1 , u ε |t≤0 = 0 for (x , x d ) ∈ R d-1 × [0, 1] , (1)
where the coefficients in the interior, namely the A j s, are in M N ×N (R), the ones on the boundaries, namely B 0 and B 1 are respectively in M p×N (R) and M N -p×N (R) (the value of p will be made precise in Assumption 2.2). Consequently the solution u ε of (1) lies in R N . In (1) the only non zero source term 1 is on the boundary R d-1 × {0} and is highly oscillating with respect to the parameter 0 < ε 1 (we refer to Section 4 for more details about the precise expression of g ε ).

The aim of geometric optics expansions is to construct an approximate solution of (1) in the high frequency asymptotic. Then we expect that some qualitative phenomena can be easily observed on this approximate solution whereas they are not easily readable on the solution of (1).

Before to give some more comments about the strong well-posedness of (1) let us recall some elements about the analogous (well-known) situation in the half space. We consider the following boundary value problem in the half space geometry:

     L(∂)u ε = f ε for (t, x , x d ) ∈ R × R d-1 × R * + , Bu ε |x d =0 = g ε for (t, x ) ∈ R × R d-1 , u ε |t≤0 = 0 for (x , x d ) ∈ R d-1 × R + .
(2)

From the seminal work of [Kreiss, 1970] it is known that the strong well-posedness (here by strong wellposedness we mean existence, uniqueness and an energy estimate in some weighted (in time) L 2 -norm) of ( 2) is equivalent to the fulfillment of the so-called uniform Kreiss-Lopatinskii condition. Roughly speaking this condition ensures that in the normal mode analysis no stable mode is solution of the homogeneous boundary condition on R d-1 × {0}. With more details, the uniform Kreiss-Lopatinskii condition states that for all (time-space) frequency parameter ζ in the normal mode analysis we have the decomposition

E s (ζ) ⊕ ker B = C N , (3)
or equivalently that the restriction of B to the stable subspace E s (ζ) is an isomorphism.

1 We could also consider problems (1) with a non zero source term in the interior (and by linearity also on the boundary R d-1 × {1}). However, we are here mainly interested in the influence of the boundaries on the behaviour of the solution of (1).

That is why we decided, in order to simplify the computations, to set homogeneous source terms on the boundary R d-1 × {1} or/and in the interior.

Then when one wants to construct the geometric optics expansion for (2) (see for example ()) then he has to impose a microlocalized version of the uniform Kreiss-Lopatinskii condition. To explain this, in a formal setting, let us consider a situation in which a compact supported interior source term f ε induces some waves travelling (with fixed frequency ζ) to the boundary R d-1 × {0}. Then by finite speed of propagation arguments these waves will hit the boundary (this kind of travelling waves will be refered as outgoing waves) after a finite travel time and will be reflected back. To determine the reflections that go from the boundary to the interior (they will be refered as incoming waves) one needs to express the new incoming waves in terms of the outgoing ones (at frequency ζ) and it is exactly a microlocalized version of the uniform Kreiss-Lopatinskii condition at the frequency ζ. Indeed in such a situation incoming waves are elements of E s (ζ) so that (3) microlocalized in ζ = ζ permits to invert B in the boundary condition.

The aim of this article is roughly speaking to determine if such a situation also occurs in the strip geometry. That is does some condition (or a microlocalized version of a condition), coming from the characterization of the strong well-posedness of (1) can be observed in the construction of the WKB expansion of (1). This question has already been adressed for hyperbolic boundary value problems in a corner (see [Sarason and Smoller,7475]- [Benoit, 2016]) About the strong well-posedness of (1) we first observe that from the result of [Kreiss, 1970], localization and stability by zero order terms arguments it is easy to show that if the strip problem (1) satisfies the uniform Kreiss-Lopatinskii condition on each side then there exists a unique solution u of (1) with bounded exponential growth in time (we refer to Definition 7.1 or to [Benoit, ] for more details).

For uniform in time results we refer to [Benoit, ] in which the author gives a new characterization of uniformly strongly well-posed hyperbolic boundary value problems in a strip in terms of the inversibility of some trace operators, reading under the form (I -T (ζ)), that in input takes the value of the trace of the solution on R d-1 × {0} (resp. R d-1 × {1}) and gives in output the value of the trace of the solution on R d-1 × {1} (resp. R d-1 × {0}).

Let us explain why such a condition is so natural. Consider two decoupled transport equations one traveling to the "right" and the second to the "right". Choose boundary conditions in (1) coupling these two transport phenomenon together. Inductively the non trivial source term g ε induces a wave traveling to the side R d-1 × {1}. This wave will be reflected against this side and travel back to R d-1 × {0} and after some time more travel time the same process is repeated periodically in time (this kind of phases will be refered as selfinteracting phases). If we denote by R the coefficient of amplification during the two reflections needed to regenerate back the first considered transport phenomenon, then intuitively the growth of the source term g with respect to time should behave like R t and we expect to have exponential growth in time when |R| > 1. The conditions in [Benoit, ], even if they seem to be somewhat technical, are linked to this simple energy observation (we refer to [Benoit, ] Paragraph 3.3.3 (first part by "hand") for more details).

In this article for the construction of the geometric optics expansion for (1) we ewhibit the fact that a microlocalized version of one of the condition of [Benoit, ] is necessary to initialize the resolution of the cascade of equations. With more details we ask the inversibility of an operator reading (I -T (ζ)) (where ζ is a (micro)-localization of the frequency) on some spaces H ∞ ,γ (where γ stands for the maximal exponential growth in time of the solution. In particular the geometric optic expansions if lower exponentally growthing in time if and only if γ = 0 and in this framework we can explicit some results of [Benoit, ].

An other point of interest is while one of the inversibility condition in [Benoit, ] is asked to hold on the full subspace E s (ζ), the one in this article only has to hold on the hyperoblic component of E s (ζ)). This observation will be explained through this article and is linked to the fact that non-hyperbolic modes are linked to boundary layers so that they do not propagate information from one side to the other. We postnone to Section 9 for more details.

The article is organized as follows, Section 2 contains some notations and the main assumptions. In Section 3 we give a formal analysis of the phase generation process and in particular we explain in a formal setting why selfinteraction becomes generic in the strip geometry.

The construction of the geometric optics expansion is performed in Section 4-6 and justified in Section 7. As already noticed this construction is made under a new inversibility condition which is studied in more details in Section 8.

Finally Section 9 contains some examples of application and gives some comments about the described results.

Notations and definitions

For simplicity we introduce the following notations for the strip and the time/space strip:

Γ := R d-1 × [0, 1] , ∂Γ 0 := Γ ∩ {x d = 0} , ∂Γ 1 := Γ ∩ {x d = 1} Ω := R t × Γ, ∂Ω 0 := R t × ∂Γ 0 and ∂Ω 1 := R t × ∂Γ 1 .
The frequency space and its boundary are defined by:

Ξ := ζ := (σ = γ + iτ, η) ∈ C × R d-1 \ γ ≥ 0 and Ξ 0 := Ξ ∩ {γ = 0} .
In order to state the energy estimates used in this article we define the following weighted Sobolev spaces.

Let s ∈ N, X ⊂ R t × R d
x and χ > 0, the H s -weighted (in time) Sobolev spaces is defined by:

H s χ (X) := u ∈ D (X) \ ue -χt ∈ H s (X) , equipped with the norm • H s χ (X) := • e -χt H s (X)
. We also denote H ∞ χ (X) := ∩ s∈N H s χ (X) and finally for s ∈ N ∪ {∞} we define H s ,χ (X) as the set of functions of H s χ (X) that vanish for negative times.

In all this article we make the following assumptions on the strip problem (1). The first assumption ensures that the operator L(∂) is hyperbolic in the following sense: Assumption 2.1 (Constantly hyperbolic operator) The system (1) is constantly hyperbolic that is there exists q ≥ 1, real valued analytic functions λ 1 , ..., λ q on R d \ {0} and positive integers µ 1 , ..., µ q such that:

∀ξ ∈ S d-1 , det   τ + d j=1 ξ j A j   = q j=1 (τ + λ j (ξ)) µj , with λ 1 (ξ) < • • • < λ q (ξ
) and the eigenvalues λ j (ξ) of d j=1 ξ j A j are semi-simple. The second one imposes that the boundaries are not characteristics for L(∂) and that the number of boundary conditions imposed on each side of the boundary gives rise to a well-determined problem Assumption 2.2 (Non characteristic boundary conditions) The matrix A d is invertible. Let p be the number of positive eigenvalues (counted with multicplicity) of

A d then B 0 ∈ M p×N (R) and B 1 ∈ M N -p×N (R).
With Assumption 2.2 in hand we can perform a Laplace transform in time (t σ) and a Fourier transform in the tangential space variable (x η) so that (1) reads in the resolvent form:

     d dx d u(ζ, x d ) = A (ζ) u(ζ, x d ), for x d ∈ ]0, 1[ , B 0 u(ζ, 0) = g(ζ), B 1 u(ζ, 1) = 0, (4)
in which ζ ∈ Ξ acts like a parameter and where the so-called resolvent matrix A (ζ) is defined by:

A (ζ) = A -1 d   σI + i d-1 j=1 η j A j   . (5)
The following classical result due to Hersh [Hersh, 1963] ensures that as soon as the Laplace parameter σ has non vanishing real part then the elements in the spectrum of A (σ, η) are well-separated.

Lemma 2.1 [Hersh, 1963]Under Assumptions 2.1 and 2.2, for all frequency parameter ζ ∈ Ξ \ Ξ 0 , the resolvent matrix A (ζ) only admits eigenvalues with non-zero real part, and thus does not have purely imaginary eigenvalues. We denote by E s (ζ) (resp. E u (ζ)), the stable (resp. unstable) space of A (ζ) that is the eigenspace associated with the negative (resp. positive) real part eigenvalues. Then independently of ζ ∈ Ξ \ Ξ 0 , dim E s (ζ) = p and dim E u (ζ) = N -p and we have the following decomposition:

C N = E s (ζ) ⊕ E u (ζ). ( 6
)
However for ζ ∈ Ξ 0 then generically Lemma 2.1 is not satisfied anymore because of the possible degeneracy of some real parts of the eigenvalues. In this setting the result allowing to describe the situation is the so-called block structure Theorem firstly shown by [Kreiss, 1970] for strictly hyperbolic systems and then extended by [Métivier, 2000] for constantly hyperbolic systems (see also [START_REF] Métivier | Hyperbolic boundary value problems for symmetric systems with variable multiplicities[END_REF] for systems with non constant multiplicities) Theorem 2.1 (Block structure) Under Assumptions 2.1 and 2.2, for all ζ ∈ Ξ, there exists a neighborhood

V of ζ in Ξ, an integer L ≥ 1, a partition N = µ 1 + • • • + µ L ,
with µ 1 , ..., µ L ≥ 1 and an invertible matrix T , regular on V such that:

∀ζ ∈ V , T -1 (ζ)A (ζ)T (ζ) = diag (A 1 (ζ), • • • , A L (ζ))
where the blocks A j (ζ) ∈ M µj ×µj (C) satisfy one of the following alternatives:

i) all the elements in the spectrum of A j (ζ) have positive real part.

ii) All the elements in the spectrum of

A j (ζ) have negative real part. iii) µ j = 1, A j (ζ) ∈ iR, ∂ γ A j (ζ) ∈ R \ {0} and A j (ζ) ∈ iR for all ζ ∈ Ξ 0 ∩ V . iv) µ j > 1 and there exists k j ∈ iR such that A j (ζ) =    k j i 0 . . . i 0 k j    , the coefficient in the lower left corner of ∂ γ A j (ζ) ∈ R \ {0} and for all ζ ∈ Ξ 0 ∩ V , A j (ζ) ∈ iM µj ×µj (R).
Consequently Theorem 2.1 permits to give the following decomposition of the boundary of the frequency space.

Definition 2.1 For ζ ∈ Ξ 0 we define:

• E the elliptic area which is the set of ζ such that Theorem 2.1 is satisfied with blocks of type i) and ii) only.

• EH the mixed area which is the set of ζ such that Theorem 2.1 is satisfied with blocks of type i), ii) and at least one block of type iii).

• H the hyperbolic area which is the set of ζ such that Theorem 2.1 is satisfied with blocks of type iii) only.

• G the glancing area which is the set of ζ such that Theorem 2.1 is satisfied with at least one block of type iv).

We thus have the following decomposition of Ξ 0 :

Ξ 0 = E ∪ EH ∪ H ∪ G. (7)
Moreover for all ζ ∈ Ξ 0 \ G the decomposition (6) still holds and we write:

C N = E s (ζ) ⊕ E u (ζ), (8)
where

E s (ζ) (resp. E u (ζ)) is the extension by continuity of E s (ζ) (resp. E u (ζ)) up to the boundary Ξ 0 .
These spaces admit the following decompositions:

E s (ζ) = E s e (ζ) ⊕ E s h (ζ) and E u (ζ) = E u e (ζ) ⊕ E u h (ζ), (9)
where E s e (ζ) (resp. E u e (ζ)) is the generalized eigenspace associated to eigenvalues of A (ζ) with negative (resp. positive) real part and where the E s h (ζ), E u h (ζ) are sums of eigenspaces associated to some purely imaginary eigenvalues of A (ζ).

However for ζ ∈ G the decomposition (8) does not hold anymore because at glancing frequencies we have

E s (ζ) ∩ E u (ζ) = {0}.
In this setting we define the following decompositions of the stable and unstable spaces E s (ζ) and E u (ζ).

E s (ζ) = E s e (ζ) ⊕ E s h (ζ) ⊕ E s g (ζ) and E u (ζ) = E u e (ζ) ⊕ E u h (ζ) ⊕ E u g (ζ), ( 10
)
where E s g (ζ) and E u g (ζ) are sum of eigenspaces associated to the Jordan's block of type iv) of A (ζ) in Theorem 2.1 and consequently satisfying

E s g (ζ) ∩ E u g (ζ) = {0}.
Geometric optics expansions involving glancing frequencies (that is to say frequencies such that

E s g (ζ) ∩ E u g (ζ) = {0}
) have been studied in the half space geometry by [Williams, 1996]- [Williams, 2000]. In these papers, in order to define a bounded projector on E s g (ζ) associated to the decomposition (10) (which is needed in order to define the boundary layer induced by glancing modes), the author assumes that

E s g (ζ) = M j=1 G s j (ζ)
, where for all j = 1, [Kreiss, 1970]- [Métivier, 2000], this assumption is equivalent to the fact that Theorem 2.1 is satisfied with block of type iv) of size at most three. Indeed in this case the contribution in

• • • , M , dim G s j (ζ) = 1. Following
E s g (ζ) (resp. E u g (ζ)
) of one block of type iv) is one dimensional (resp. one dimensional if the associated block is of size two, two dimensional if the associated block is of size three). Consequently the projector upon E s g (ζ) remains bounded.

In the following we shall define the projectors on both E s g (ζ) and E u g (ζ) so that we make the following assumption:

Assumption 2.3 Let ζ ∈ G then Theorem 2.1 is satisfied with blocks of type iv) of size two only. In this setting we have that there exists M ∈ N, M ≤ N 2 and (e j ) j=1,••• ,M ∈ C N such that:

E s g (ζ) = E u g (ζ) = M j=1 G j (ζ) where G j (ζ) := vect {e j } .
We can give some precisions about the spaces

E s h (ζ), E u h (ζ), E s g (ζ) and E u g (ζ)
. Let iξ m ∈ iR be a purely imaginary eigenvalue of A (ζ) (possibly with multiplicity more than two except for glancing modes for Assumption ?? then det

  τ I + d-1 j=1 η j A j + ξ m A d   = 0.
From Assumption 2.1 there exists an index k m such that τ + λ km (η, ξ m ) = 0, where λ km is smooth in both variables. This motivates the following definition: Definition 2.2 The set of incoming (resp. outgoing) phases for the side ∂Γ 0 denoted by I 0 (resp. O 0 ) is the set of indices m such that the group velocity

v m := ∇λ km (η, ξ m ) satisfies ∂ ξ λ km (η, ξ m ) > 0 (resp. ∂ ξ λ km (η, ξ m ) < 0).
The set of incoming (resp. outgoing) phases for the side ∂Γ 1 denoted by I 1 (resp. O 1 ) is the set of indices m such that the group velocity

v m := ∇λ km (η, ξ m ) satisfies ∂ ξ λ km (η, ξ m ) < 0 (resp. ∂ ξ λ km (η, ξ m ) > 0).
The set of glancing phases for the side ∂Γ 0 (or equivalently for the side ∂Γ 1 ) denoted by G is the set of indices m such that the group velocity v m := ∇λ km (η, ξ m ) satisfies ∂ ξ λ km (η, ξ m ) = 0.

Clearly we have I 0 = O 1 and O 0 = I 1 . So that in the following we will use the convention that an incoming (resp. outgoing) phase is incoming if it is incoming (resp. outgoing) for the side ∂Γ 0 . Thus, with this convention in mind we set:

I := I 0 = O 1 and O := O 0 = I 1 ,
and for simplicity we also define H := I ∪ O, the set of indeces associated to hyperbolic modes.

With this definition in hand we can give the following description of the spaces

E s h (ζ), E u h (ζ), E s g (ζ) and E u g (ζ).
Lemma 2.2 For all ζ ∈ Ξ 0 we have:

E s h (ζ) = k∈I ker L (τ , η, ξ k ), E u h (ζ) = k∈O ker L (τ , η, ξ k ) and E s g (ζ) = E u g (ζ) = k∈G ker L (τ , η, ξ k ), ( 11
)
where L stands for the symbol of L(∂) defined for all ω = (ω 0 ,

• • • , ω d ) ∈ R d+1 by L (ω) := ω 0 I + d j=1 ω j A j . Consequently for ζ ∈ G, (10) reads: E s (ζ) = k∈I ker L (τ , η, ξ k ) k∈G ker L (τ , η, ξ k ) ⊕ E s e (ζ), (12)
E u (ζ) = k∈O ker L (τ , η, ξ k ) k∈G ker L (τ , η, ξ k ) ⊕ E u e (ζ). ( 13
)
We now turn to the definition of the uniform Kreiss-Lopatinskii condition which is the condition ensuring the strong well-posedness of the boundary value problem in the half space. It is not difficult to show (and to be convincing) that the strong well-posedness of (1) requires that each boundary condition on ∂Γ 0 and ∂Γ 1 satisfies the uniform Kreiss-Lopatinskii condition. So that in the WKB expansion construction we should assume that these conditions hold. More precisely we assume the following Assumption 2.4 (Uniform Kreiss-Lopatinskii condition) Under Assumptions 2.1 and 2.2 let ζ ∈ Ξ and as previously we still denote by E s (ζ) (resp. E u (ζ)) the extension by continuity of E s (ζ) up to ξ ∈ Ξ 0 of the well-defined stable (resp. unstable) subspace of A (ζ). Then each of the boundary ∂Γ 0 and ∂Γ 1 satifies the uniform Kreiss-Lopatinskii condition that is to say that:

∀ζ ∈ Ξ, ker B 0 ∩ E s (ζ) = ker B 1 ∩ E u (ζ) = {0} .
In other words, the restriction of B 0 (resp. B 1 ) to E s (ζ) (resp. E u (ζ)) is invertible and we denote its inverse by φ 0 (ζ) := B -1 ζ) ). We conclude this Section by defining some projectors that will be useful in the construction the WKB expansion.

0 | E s (ζ) (resp. φ 1 (ζ) := B -1 1 | E u (
Definition 2.3 (Interior projectors) Under Assumptions 2.1 and 2.2 for ζ = iτ + η ∈ Ξ 0 we define:

• Π s e := Π s e (ζ) (resp. Π u e := Π u e (ζ)) the spectral projector on E s e (ζ) (resp. E u e (ζ)). • For k ∈ I ∪ O ∪ G , Π k := Π k (ζ) the orthogonal projector on ker L (τ , η, ξ k ). • For k ∈ I ∪ O ∪ G , we define Υ k := Υ k (ζ) the partial inverse of L (τ , η, ξ k ) characterized by the relations: ΥL (τ , η, ξ k ) = I -Π k , Υ k Π k = Π k Υ k = 0. ( 14
)
Definition 2.4 (Boundary projectors) Under Assumptions 2.1, 2.2 and 2.3 for ζ = iτ + η ∈ Ξ 0 we define:

• P s e := P s e (ζ) (resp. P u e := P u e (ζ)) the projector on E s e (ζ) with respect to (12) (resp. (13)).

• For k ∈ I (resp. k ∈ O), P k h := P k h (ζ) the projector on ker L (τ , η, ξ k ) with respect with the sums (12) (resp.( 13)).

• For k ∈ G we define P k g,s := P k g,s (ζ) (resp. P k g,u := P k g,u (ζ)) the projector on ker L (τ , η, ξ k ) with respect with the sum (12) (resp. (13))

Formal analysis

In this paragraph, we give a formal analysis describing the phases appearing in the WKB expansion as well as the selfinteraction phenomenon between the oscillating ones.

As the reader will notice, in comparison with the expansions for the quarter space geometry (see for example [Benoit, 2016]), on the one hand the phase generation process in the strip geometry will not be richer than the one in the half space geometry. Indeed the number of generated phases will be the same as the one for the problem in the half space. This was not the case for the corner problem for which the number of considered phases was generically greater than the one in the half space (this number can even be infinite).

However on the other hand, the selfinteraction phenomenon (meaning that a phase can regenerate itself after a suitable number of rebounds against the sides of the domain of resolution), which can be seen as a somewhat anecdotal behaviour in the quarter space geometry (because it requires strong constraints on the geometry of the characteristic variety) becomes generic in the strip geometry. Indeed an incoming phase coming from the side ∂Γ 0 will always be reflected back against the side ∂Γ 1 and will always regenerate itself after two reflections.

Source term induced phases

The first point of our discussion is to determine the source term induced phases. Note that the system (1) is hyperbolic, so that it satisfies the finite speed of propagation property, and that the only non trivial information in (1) lies on the side ∂Γ 0 . Consequently, this information can not hit the side ∂Γ 1 immediately and we can (in a formal setting and at least during a short time) neglect the boundary condition on the side ∂Γ 1 . By doing this we shall consider the following system of equations:

     L(∂)u ε = 0 for (t, x , x d ) ∈ ]-∞, T ] × R d-1 × R * + , B 0 u ε |x d =0 = g ε for (t, x ) ∈ ]-∞, T ] × R d-1 , u ε |t≤0 = 0 for (x , x d ) ∈ R d-1 × R + , (15)
for T > 0 (possibly small) and it is thus natural to choose for ansatz the one for the problem in the half space (15). More precisely, g ε reads

g ε (t, x ) := e i ε (t,x )•(τ ,η) g(t, x ),
where the amplitude g ∈ H ∞ (∂Ω 0 ) is given and where the frequency parameters τ ∈ R, η ∈ R d-1 are fixed.

The ansatz reads

u ε app ≈ K k=1 e i ε ((t,x )•(τ ,η)+x d ξ k ) u ε k (t, x), (16)
where u ε k = n≥0 ε n u n,k , the ξ k are roots in the ξ variable of the so-called dispersion relation det L (τ , η, ξ) = 0 where we recall that L stands for the symbol of L(∂).

The behaviour of the amplitude u ε k in ( 16) is thus given by the kind of phases that we are considering which depends on ξ k and we have to discuss several cases:

ξ k ∈ C, Im ξ k = 0.
In this case the factor e i ε ((t,x )•(τ ,η)+x d ξ k ) has a (real) exponential behaviour with respect to the sign of Im ξ k . More precisely:

• Im ξ k > 0 (evanescent for the side ∂Γ 0 ). In this subcase the factor e i ε ((t,x )•(τ ,η)+x d ξ k ) induces an exponential decrease with respect to the normal variable x d . The associated amplitude has exponential decrease so that when it hits the side ∂Γ 1 its contribution is O(ε ∞ ) with respect to ε and it will not contribute to the boundary condition on ∂Γ 1 . Consequently it will not be reflected back.

• Im ξ k < 0 (explosive for the side ∂Γ 0 ). In this subcase the factor e i ε ((t,x )•(τ ,η)+x d ξ k ) induces an exponential growth with respect to the normal variable x d . As in the half space geometry we decide, to simplify the discussion, to initially neglect these amplitudes in the ansatz ( 16) (recall that we are interested in L 2 γ (Ω) for some γ > γ 0 ≥ 0).

v k,d > 0 v k,d = 0 {x d = 0} {x d = 1}
evanescent ξ k ∈ R. In this case the factor e i ε ((t,x )•(τ ,η)+x d ξ k ) induces an oscillating behaviour. Moreover Lax's Lemma [Lax, 1957] should apply and the leading order term in the amplitude u ε k , namely the u 0,k are expected to solve the transport equations:

∂ t u 0,k + v k • ∇ x u 0,k = 0, (17)
where the velocity v k is the so-called group velocity for ξ k (we refer to Paragraph 2 Definition 2.2 for a precise definition). Depending on the sign of v k,d the transport equation ( 17) has to be completed by some boundary conditions, which lead us to the following study of subcases:

• v k,d < 0 (outgoing for the side ∂Γ 0 ). In this subcase the transport in the equation ( 17) is made from the "right to the left". So that the transported informations can be the ones in the interior or the ones on the side ∂Γ 1 . But in (1) these source terms are chosen to be zero. Consequently, u 0,k is zero and this amplitude is initially neglected in (16).

• v k,d = 0 (glancing for the side ∂Γ 0 ). In this subcase the transport equation ( 17) reads

∂ t u 0,k + v k • ∇ x u 0,k = 0, (18)
equation which does not require any boundary condition on ∂Γ 0 or on ∂Γ 1 . The only transportable information is the one in the interior, it is zero, so that the associated amplitude u 0,k shall be zero and shall be neglected in ( 16). However, to solve the boundary conditions for the WKB expansion of (1), with an accurate enough error, it will be necessary to consider a boundary layer (around ∂Γ 0 ) for u 0,k . We refer to [Williams, 1996] and [Williams, 2000] for more details. Consequently, the u 0,k are not neglected in ( 16). However, due to the special form of the transport equation ( 18), this boundary layer can not be propagated to the side ∂Γ 1 , it will not be reflected against this side and will not contribute to the boundary condition on ∂Γ 1 .

• v k,d > 0 (incoming for the side ∂Γ 0 ). Finally in this subcase the transport is made from the "left to the right". Consequently, the non zero information on the side ∂Γ 0 , is transported. The associated amplitude u 0,k is not zero, it is not neglected in the ansatz (16). Moreover, this non trivial information will hit the side ∂Γ 1 after some travel time. It will be reflected and we have to determine its reflections. It is the aim of the following paragraph.

In conclusion, the source term induced phases are the glancing ones, the incoming ones and the evanescent ones. Only the incoming ones spread some non trivial information from the side ∂Γ 0 to the side ∂Γ 1 and only their reflections have to be considered. The situation is summarized in Figure 1 3

.2 The first reflection

We assume that there exists at least one incoming phase2 , that is that det L (τ , η, ξ) = 0 admits at least one root ξ k such that the associated group velocity v k satisfies v k,d > 0.

We have justified in the previous paragraph that the amplitude u 0,k , after some travel time, induces a non trivial information on the side ∂Γ 1 . Once again by finite speed of propagation arguments, this information can not go back to the side ∂Γ 0 immediately, so that, in a formal setting and at least during a short time, we can consider the problem L(∂)u ε = 0 defined on the half space {x d ≤ 1} with a boundary condition on ∂Γ 1 involving the amplitude u 0,k |x d =1 and with homogeneous initial condition. We shall describe the amplitudes induced by the source term on ∂Γ 1 .

Note that because we are still working in a half space indexed by x d , the possible induced amplitudes satisfy the same dispersion relation as the one for the source term induced phases3 . That is the ξ k are roots in the ξ variable of the dispersion relation det L (τ , η, ξ) = 0. So that the discussion of the previous paragraph can also be performed to determine the reflections during the first rebound.

However, due to the change of orientation in the x d variable, the sign in the discussion have to be reverse. More precisely, let ξ k be such that det L (τ , η, ξ k ) = 0 we distinguish:

ξ k ∈ C, Im ξ k = 0.
Then the amplitude u 0,k is associated to a non trivial real exponentional factor. And depending on the sign of Im ξ k we have:

• Im ξ k < 0 (evanescent for the side ∂Γ 1 ). These amplitudes have been initially neglected in the amplitudes induced by the source term. But at this step of the analysis they are evanescent for the side ∂Γ 1 (or equivalently explosive for the side ∂Γ 0 ) so that we reintroduce these amplitudes in the ansatz ( 16). They are associated to boundary layer around the side ∂Γ 1 which propagate to ∂Γ 0 and hit this side as O(ε ∞ ) so that they are not reflected against ∂Γ 0 and do not contribute in the boundary condition on ∂Γ 0 .

• Im ξ k > 0 (explosive for the side ∂Γ 1 ). These amplitudes are evanescent for the side ∂Γ 0 . So that they are still present in the ansatz ( 16) and there is no need to add them.

ξ k ∈ R Then the associated amplitude is oscillating, Lax's Lemma [Lax, 1957] applies and we expect to solve the transport equation ( 17) and we have to reiterate the discussion of the previous paragraph depending on the sign of the d th component of the group velocity v k :

• v k,d > 0 (outgoing for the side ∂Γ 1 ). These amplitudes are already present in (16) because they are incoming for the side ∂Γ 0 .

• v k,d = 0 (glancing for the side ∂Γ 1 (or equivalently for ∂Γ 0 )). In this case, once again the transport equation ( 17) degenerates in (18) and we already justified that even if this equation is homogeneous we chose to keep u 0,k as a boundary layer in the neighborhood of ∂Γ 0 (in order to solve the boundary conditions up to an acceptable error term). In order to solve the boundary condition on ∂Γ 1 (which at this step of the analysis is not homogeneous anymore because it depends on u 0,k |x d =1 ) we will introduce in u 0,k a boundary layer in the neighborhood of ∂Γ 1 . However this new layer can not be propagate to ∂Γ 0 (because of the degeneracy of the transport equation) so that it will not contribute to the boundary condition on ∂Γ 0 and will not be reflected against this side.

• v k,d < 0 (incoming for the side ∂Γ 1 ). We recall that these amplitudes have initially been neglected in ( 16) and that they are associated to transport equation for the "right to the left". But at this step of the discussion, the information lying on the side ∂Γ 1 is not trivial anymore) so these amplitudes propagate this information from ∂Γ 1 to ∂Γ 0 and are not zero anymore. Consequently they have to be considered in ( 16). These phases will hit the side ∂Γ 0 after some positive travel time and we have to determine their rebounds. This is done in the next paragraph.

To sum up, the first rebound makes us consider the explosive and outgoing (for the side ∂Γ 0 ) phases which has been initially discarded. So that all the possible phases are now taken into account in ( 16). Moreover, we also add a boundary layer in the neighborhood of ∂Γ 1 to deal with glancing modes. However the only phases carrying some non trivial information from the side ∂Γ 1 to the side ∂Γ 0 are the outgoing (for the side ∂Γ 0 ) ones. The generated phases during the first reflection are described on Figure 2

v k,d > 0 v k,d < 0 v k,d = 0 {x d = 0} {x d = 1}
explosive Figure 2: The first rebound.

Selfinteraction phenomenon

Once again we assume that there exists an outgoing (for the side ∂Γ 0 ) phase associated to some ξ satisfying det L (τ , η, ξ ) = 0 and v ,d < 0. Then the information carried by the amplitude u 0, hits the side ∂Γ 0 after some travel time and we have to determine its reflections against this side. However reiterating exactly the same arguments as in Paragraph 3.1 (that is finite speed of propagation property to restrict the problem to the study of the problem ( 15)), we obtain that the reflections are associated to the ξ k satisfying det L (τ , η, ξ k ) = 0 and one of the following alternatives:

i) Im ξ k > 0, ii) ξ k ∈ R, v k,d = 0, iii) ξ k ∈ R, v k,d > 0.
Recall that all of these phases are already considered in the ansatz ( 16). Consequently we do not have to add any phase in (16). However, let us remark that the amplitude u 0,k considered at the beginning of Paragraph 3.2 satisfies iii) so that this phase has regenerated itself after two rebounds. It is what we mean by selfinteraction. This phenomenon will be crucial in the construction of the geometric optics expansion for (1) and will lead to an inversibility condition in order to initialize the resolution of the cascade of equations.

Let us make some other remarks. In this discussion we followed the path of phases k → but if one changes the choice of and considers a path of phases k → then during the second rebound the phase k is still generated. So that each path of the form k → , where is associated to an outgoing phase (for the side ∂Γ 0 ) gives a contribution to the regeneration of the phase associated to k.

Moreover a path of phases of the form k → where k = k is associated to an incoming phase (for the side ∂Γ 0 ) and to an outgoing phase (for the side ∂Γ 0 ) will also generate the phase associated to k.

Consequently, compared to the corner geometry see [Benoit, 2016], the selfinteraction phenomenon is here a bit more complicated because there is a priori more than one path of phases that regenerate a fixed phase. Moreover, once again compared to the corner geometry, here the selfinteraction phenomenon is generic because to hold it only requires the existence of an incoming phase and an outgoing phase. Whereas in the corner geometry, some really restrictive assumptions has to be made on the geometry of the characteristic variety (we refer to [Benoit, 2016] for more details).

We conclude this section by Figure 3 illustrating the several amplitudes in the WKB expansion and the selfinteraction phenomenon.

v k,d > 0 v ,d < 0 v k,d > 0 v k,d = 0 {x d = 0} {x d = 1} ξ k ∈ C

The cascades of equations

We consider the following system of equations

           L(∂)u ε = ∂ t u ε + d j=1 A j ∂ j u ε = 0 in Ω, B 0 u ε |x d =0 = g ε on ∂Ω 0 , B 1 u ε |x d =1 = 0 on ∂Ω 1 , u ε |t≤0 = 0 on Γ, (19)
Let ζ := (iτ , η) ∈ Ξ 0 be a fixed frequency parameter. We define the phases functions

ψ(t, x ) := τ t + η • x and for k ∈ H ∪ G , ϕ k (t, x) := ψ(t, x ) + ξ k x d , (20)
where the ξ k stands for the real roots of det L (τ , η, ξ) in the ξ variable. In ( 19) the source term on the boundary ∂Ω 0 reads:

g ε := g ε (t, x ) := e i ε ψ(t,x ) g(t, x ), (21)
where the amplitude g ∈ H ∞ (∂Ω 0 ).

We define the ansatz4

u ε (t, x) ∼ k∈H e i ε ϕ k (t,x) n≥0 ε n u ε h,n,k (t, x) + k∈G e i ε ϕ k (t,x) 1 n=0 ε n u ε g,n,k (t, x) (22)
+ n≥0 e i ε ψ(t,x ) ε n U ε ev,n t, x, x d ε + n≥0 e i ε ψ(t,x ) ε n U ε ex,n t, x, x d -1 ε ,
where for all 0 < ε 1, for all n ∈ N and for all k ∈ H (resp. k ∈ G ) the profiles u h,n,k (resp. u g,n,k ) are in H ∞ γ, (Ω) for all γ > γ 0 for some γ 0 ≥ 0 to be determined and where the evanescent (resp. explosive) profiles U ε ev,n (resp. U ε ex,n ) are looking in the following profile spaces:

Definition 4.1 The space P ev (resp. P ex ) of evanescent (resp. explosive) profiles is the set of functions

U (t, x, X d ) ∈ H ∞ (Ω × R + ) (resp. H ∞ (Ω × R -)) moreover satisfying that there exists δ > 0 such that e δX d U (t, x, X d ) ∈ H ∞ (Ω × R + ) (resp. H ∞ (Ω × R -)).
In the ansatz ( 22) let us stress that depending on the kind of the frequency ζ some (but not all) sums can be zero. Indeed for example if ζ ∈ E then the sums on H and on G are zeros, and so one depending on the kind of ζ considered. We also insist on the fact that the sum on H can always be zero when ζ / ∈ H.

Plugging the ansatz ( 22) in the evolution equation of ( 19) leads, by identification on the ε n , to the following cascade of equation

                             L ( dϕ k )u ε h,0,k = 0 ∀k ∈ H , iL ( dϕ k )u ε h,n+1,k + L(∂)u ε h,n,k = 0 ∀k ∈ H , ∀n ∈ N, L ( dϕ k )u ε g,0,k = 0 ∀k ∈ G , iL ( dϕ k )u ε g,1,k + L(∂)u ε g,0,k = 0 ∀k ∈ G , L(∂)u ε g,1,k = 0 ∀k ∈ G L(∂ X d )U ε ev,0 = L(∂ X d )U ε ex,0 = 0, L(∂ X d )U ε ev,n+1 + L(∂)U ε ev,n (t, x, X d ) = 0 ∀n ∈ N, X d > 0, L(∂ X d )U ε ex,n+1 + L(∂)U ε ex,n t, x, X d = 0 ∀n ∈ N, X d < 0, ( 23
)
where the operator of differentiation with respect to the fast variable is defined by

L(∂ X d ) = A d ∂ X d -A (ζ) .
Then plugging the ansatz ( 22) in the boundary conditions of ( 19) gives

B 0 k∈H u ε h,n,k (t, x , 0) + k∈G u ε g,n,k (t, x , 0) + U ε ev,n (t, x , 0, 0) + U ε ex,n t, x , 0, - 1 ε = δ n,0 g, (24)
and

B 1 k∈H e i ε ξ k u ε h,n,k (t, x , 1) + k∈G e i ε ξ k u ε g,n,k (t, x , 1) + U ε ev,n t, x , 1, 1 ε + U ε ex,n (t, x , 1, 0) = 0, ( 25
)
where δ n,p stands for the Kronecker's symbol. However by definition of the evanescent and explosive set of profiles, the terms U ε ex,n t, x , 0, -1 ε and U ε ev,n t, x , 1, 1 ε appearing in ( 24) and ( 25) are respectively O(ε ∞ ) so that the boundary conditions ( 24) and ( 25) can be simplified in

B 0 k∈H u ε h,n,k (t, x , 0) + k∈G u ε g,n,k (t, x , 0) + U ε ev,n (t, x , 0, 0) = δ n,0 g, (26)
and

B 1 k∈H e i ε ξ k u ε h,n,k (t, x , 1) + k∈G e i ε ξ k u ε g,n,k (t, x , 1) + U ε ex,n (t, x , 1, 0) = 0. ( 27
)
Finally plugging the ansatz (22) in the initial condition of (19) leads to

           u ε h,n,k |t≤0 = 0 ∀k ∈ H , ∀n ∈ N, u ε g,n,k |t≤0 = 0 ∀k ∈ G , ∀n ∈ N, U ε ev,n |t≤0 = 0 ∀n ∈ N, U ε ex,n |t≤0 = 0 ∀n ∈ N. (28)
So that to construct an approximate solution of ( 19) one shall solve the cascade of equations ( 23)-( 26)-( 27) and ( 28) up to some order. The construction of the leading order term, that is the one associated to ε 0 , in the expansion is performed in the following section.

Construction of the leading order term

To initialize the construction of the leading order term of the geometric optics expansion we study the behaviour of the oscillating and glancing amplitudes in (22), that is the u h,0,k and the u g,0,k . The first (resp. the third) equation of ( 23) implies that for all k ∈ H (resp. k ∈ G ), u h,0,k ∈ ker L ( dϕ k ) (resp. u g,0,k ∈ ker L ( dϕ k )). Consequently we have the well-known polarization condition

∀k ∈ H ∪ G , Π k u h,0,k = u h,0,k (29)
where we recall that the projectors Π k are introduced in Definition 2.3. Using the polarization condition (29) and composing the second (resp. the fourth) equation of ( 23) by

Π k gives Π k L(∂)Π k u h,0,k = 0 resp. Π k L(∂)Π k u g,0,k = 0 ,
so that we are in a position to apply Lax's lemma [Lax, 1957]:

Lemma 5.1 ( [Lax, 1957]) Under Assumption 2.1 we have the equalities

∀k ∈ H ∪ G , Π k L(∂)Π k = (∂ t + v k • ∇ x )Π k , ( 30
)
where we recall that v k is the group velocity associated to k introduced in Definition 2.2.

As a consequence the leading order oscillating and glancing amplitudes are expected to satisfy transport equations and we have to consider several cases depending on k:

•k ∈ I . By definition of I the group velocity v k satisfies v k,d > 0 so that the transport equation ( 30) only requires a boundary condition on the side ∂Γ 0 and an initial condition to be solved (the value of the trace on ∂Γ 1 is deduced by integration along the characteristics).

•k ∈ O. By definition of O the group velocity v k satisfies v k,d < 0 so that the transport equation ( 30) only requires a boundary condition on the side ∂Γ 1 and an initial condition to be solved (the value of the trace on ∂Γ 0 is deduced by integration along the characteristics).

•k ∈ G . By definition of G the group velocity v k satisfies v k,d = 0 so that the transport operator in (30) reads ∂ t + v k • ∇ x and no boundary conditions on ∂Γ 0 or on ∂Γ 1 are required and we just need the initial condition.

These remarks lead us to study the boundary conditions ( 26) and ( 27) written for n = 0, they read:

B 0 k∈I u ε h,0,k |x d =0 + k∈G u ε g,0,k |x d =0 + U ε ev,0 |x d =X d =0 = g -B 0 ∈O u ε h,0, |x d =0 , (31)
and

B 1 ∈O e i ε ξ u ε h,0, |x d =1 + ∈G e i ε ξ u ε g,0, |x d =1 + U ε ex,0 |x d =X d =1 = -B 1 k∈I e i ε ξ k u ε h,0,k |x d =1 . (32)
We remark, from (12) (resp. ( 13)), that the term in the left hand side of (31) (resp. ( 32)) lies in

B 0 E s (ζ) (resp. B 1 E u (ζ))
, so that, by the uniform Kreiss-Lopatinskii condition (see Assumption 2.4), we can multiply (31) (resp. ( 32)) by φ 0 (ζ) (resp. φ 1 (ζ)) and then by P k h , P k g,s , P ev respectively (resp. P k h , P k g,u , P ex respectively) (recall that these projectors are defined in Definition 2.4) to obtain

         u ε h,0,k |x d =0 = P k h φ 0 (ζ) g -B 0 ∈O u ε h,0, |x d =0 ∀k ∈ I , u ε g,0,k |x d =0 = P k g,s φ 0 (ζ) g -B 0 ∈O u ε h,0, |x d =0 ∀k ∈ G , U ε ev,0 |x d =X d =0 = P ev φ 0 (ζ) g -B 0 ∈O u ε h,0, |x d =0 , (33)
and

       u ε h,0, |x d =1 = -e -i ε ξ P h φ 1 (ζ)B 1 k∈I e i ε ξ k u ε h,0,k |x d =1 ∀ ∈ O, u ε g,0, |x d =1 = -e -i ε ξ P g,u φ 1 (ζ)B 1 k∈I e i ε ξ k u ε h,0,k |x d =1 ∀ ∈ G , U ε ex,0 |x d =X d =1 = -P ev φ 1 (ζ)B 1 k∈I e i ε ξ k u ε h,0,k |x d =1 , (34)
where we used the fact that

P k h Π k = Π k , P k g,s Π k = P k g,u Π k = Π k and the polarization condition (29).
The main observation is that in ( 33) and ( 34), to determine the traces of the glancing amplitudes, the evanescent amplitude or the explosive amplitude it is sufficient to first determine the traces of the amplitudes associated to indeces in H . Consequently we shall determine the amplitudes associated to indeces in H before the other ones to initialize the resolution of the cascade of equations.

However in (33) to determine the traces associated to the indeces in I we have to determine the traces of the amplitudes for the indeces in O, which depend on the traces of the amplitudes for the indeces in I by (34). So that ( 33) and (34) show that the traces of the amplitudes for the indeces in I (or O) depend on themselves which agree with the selfinteraction phenomenon described formally in Section 3. The rigorous determination of these amplitudes is made in the next paragraph.

Construction of the leading order selfinteracting amplitudes

In this paragraph we show that the determination of the amplitudes associated to the indeces in I necessitates a new inversibility condition. We consider ∈ O so that the group velocity v is outgoing and the resolution of the transport equation ( 30) only requires a boundary condition on ∂Γ 1 . More precisely from (34), the equation to solve is:

     (∂ t + v • ∇ x )u ε h,0, = 0, u ε h,0, |x d =1 = -e -i ε ξ P h φ 1 (ζ)B 1 k∈I e i ε ξ k u ε h,0,k |x d =1 , u ε h,0, |t≤0 = 0. ( 35
)
Let us assume that in (35) the right hand side of the boundary condition, namely k∈I e

i ε ξ k u ε h,0,k |x d =1
, is a known function. Then is it easy to solve (35) by integration along the characteristics to determine u ε h,0, . More precisely, we have:

u ε h,0, (t, x) = -e -i ε ξ P h φ 1 (ζ)B 1 k∈I e i ε ξ k u ε h,0,k |x d =1 t + 1 v ,d (1 -x d ), x + v v ,d (1 -x d ) , (36)
where we used the notation

v = (v , v ,d ) ∈ R d-1 × R -. We easily determine the value of u ε h,0, |x d =0 for ∈ O u ε h,0, |x d =0 (t, x ) = -e -i ε ξ P h φ 1 (ζ)B 1 k∈I e i ε ξ k u ε h,0,k |x d =1 t + 1 v ,d , x + v v ,d . (37)
Using (37) we can compute the right hand side of the first equation of (33). For k ∈ I we have:

u ε h,0,k |x d =0 (t, x ) =P k h φ 0 (ζ)g(t, x ) (38)
+ P k h φ 0 (ζ)B 0 ∈O e -i ε ξ P h φ 1 (ζ)B 1 k ∈I e i ε ξ k u ε h,0,k |x d =1 t + 1 v ,d , x + v v ,d ,
and from the fact that for k ∈ I , u ε h,0,k solves the incoming transport equation ( 30), we deduce, by integration along the characteristics, that:

u ε h,0,k (t, x) =P k h φ 0 (ζ)g t - 1 v k,d x d , x - v k v k,d x d (39) +P k h φ 0 (ζ)B 0 ∈O e -i ε ξ P h φ 1 (ζ)B 1 k ∈I e i ε ξ k u ε h,0,k |x d =1 t + 1 v ,d - 1 v k,d x d , x + v v ,d - v k v k,d x d ,
from which we immediately deduce the value of the trace of u ε h,0,k on ∂Γ 1 :

u ε h,0,k |x d =1 (t, x ) =P k h φ 0 (ζ)g t - 1 v k,d , x - v k v k,d (40)
+P k h φ 0 (ζ)B 0 ∈O e -i ε ξ P h φ 1 (ζ)B 1 k ∈I e i ε ξ k u ε h,0,k |x d =1 t + 1 v ,d - 1 v k,d , x + v v ,d - v k v k,d .
Equation ( 40) holds for all k ∈ I so that we can multiply by e i ε ξ k and sum over k ∈ I to derive the following condition on

U ε I := k∈I e i ε ξ k u ε h,0,k |x d =1 : (I -T ε (ζ))U ε I = G ε (ζ)g, (41)
where we set for f a function defined on

R t × R d-1 x (T ε (ζ)f )(t, x ) := k∈I e i ε ξ k P k h φ 0 (ζ)B 0 ∈O e -i ε ξ P h φ 1 (ζ)B 1 f t + 1 v ,d - 1 v k,d , x + v v ,d - v k v k,d , (42)
and

(G ε (ζ)f )(t, x ) := k∈I e i ε ξ k P k h φ 0 (ζ)f t - 1 v k,d , x - v k v k,d . (43)
Note that in the definitions of T ε (ζ) and G ε (ζ) the evaluations in the time variable are of the form t-α k, where α k, > 0 because by definition for k ∈ I , v k,d > 0 and for ∈ O, v ,d < 0. Consequently, the form of the operator T ε (ζ) agrees with the intuition given in Section 3 that the selfinteraction phenomenon needs some time (more precisely at least the minimum of the times needed to make two reflections) to appear. We will give more comments about the operator T ε (ζ) in Section 8.

Moreover from the definitions of

T ε (ζ) and G ε (ζ) it is clear that they left invariant the sets H ∞ ,γ (R t × R d-1 x ).
Equation ( 41) combined with the fact that

U ε I ∈ E s h (ζ) and G ε (ζ)g ∈ E s h (ζ)
lead us to the following assumption:

Assumption 5.1 There exists γ 0 ≥ 0 such that for all 0 < ε 1 the operator

(I -T ε (ζ)), where T ε (ζ) is defined in (42) is invertible from H ∞ ,γ (∂Ω 1 , E s h (ζ)) to H ∞ ,γ (∂Ω 1 , E s h (ζ)
) for all γ > γ 0 . We refer to Section 8 for a study of Assumption 5.1. More precisely by giving a particular condition involving the coefficients of reflection which is sufficient for Assumption 5.1 to hold. But we will also show in Section 9 to show that Assumption 5.1 is satisfied on explicit examples.

With Assumption 5.1 in hand it is now easy to determine the amplitudes associated to indeces in H . From Assumption 5.1 we obtain:

U ε I = (I -T ε (ζ)) -1 |E s h (ζ) G ε (ζ)
g, and we can use this expression in ( 36) and ( 39) to obtain that for all ∈ O:

u ε h,0, (t, x) = -e -i ε ξ P h φ 1 (ζ)B 1 (I -T ε (ζ)) -1 |E s h (ζ) G ε (ζ)g t + 1 v ,d (1 -x d ), x + v v ,d (1 -x d ) , (44)
and for all k ∈ I :

u ε h,0,k =P k h φ 0 (ζ)g t - 1 v k,d x d , x - v k v k,d x d (45) +P k h φ 0 (ζ)B 0 ∈O e -i ε ξ P h φ 1 (ζ)B 1 (I -T ε (ζ)) -1 |E s h (ζ) G ε (ζ)g t + 1 v ,d - 1 v k,d x d , x + v v ,d - v k v k,d x d ,
equations which uniquely determine u ε h,0,k for k ∈ H in terms of the known source term g. Also note that due to the fact that g ≡ 0 for negative times, the initial condition (28) written for n = 0 is satisfied for k ∈ H . This concludes the construction of the leading order amplitudes for selfinteracting phases. It remains to consider the other kinds of phases. The construction is made in the following paragraphs.

To sum up we give the following proposition:

Proposition 5.1 Under Assumptions 2.1, 2.2, 2.4 and 5.1. For all k ∈ H and for all 0 < ε 1 there exists u ε h,0,k ∈ H ∞ ,γ (Ω) for all γ > γ 0 (the one given by Assumption 5.1) satisfying the cascade of equations ( 23),( 26),( 27),(28) written for n = 0.

Construction of the leading order glancing amplitudes

To simplify the following we introduce the notations:

U ε n,I := k∈I e i ε ξ k Π k u ε h,n,k |x d =1 and U ε n,O := ∈O Π u ε h,n, |x d =0 (46)
Let k ∈ G then the associated amplitude u g,0,k shall solve the transport equation (see ( 30),( 33),( 34) and ( 28))

           (∂ t + v k • ∇ x )u ε g,0,k = 0 for (t, x) ∈ Ω, u ε g,0,k |x d =0 = P k g,s φ 0 (ζ) g -B 0 U ε 0,O on ∂Ω 0 , u ε g,0,k |x d =1 = -e -i ε ξ k P k g,u φ 1 (ζ)B 1 U ε 0,I on ∂Ω 1 , u ε g,0,k |t≤0 = 0 on Γ, (47)
where from Paragraph 5.1 the right hand side in the boundary conditions of ( 47) are known functions in

H ∞ ,γ (R t × R d-1
x ), for all γ > γ 0 depending on g (their precise expression in terms of g can be made precise from ( 44) and ( 45) but is of little interest in the following).

As noticed in [Williams, 1996]- [Williams, 2000], the main issue in ( 47) is that due to the fact that the group velocity v k is tangent to the boundary the couple of equations:

(∂ t + v k • ∇ x )u ε g,0,k = 0, u ε g,0,k |t≤0 = 0, (48)
already determines the solution u ε g,0,k (and witth homogeneous initial condition and interior forcing term it shall be zero). Consequently with the boundary conditions the system (47) is overdetermined (and the boundary conditions can not be satisfied because u ε g,0,k ≡ 0).

However we stress that we need to solve theses boundary conditions to obtain a suitable error on the boundary in the energy estimate.

To overcome this difficulty induced by glancing modes, we follow the method of [Williams, 1996] that is we decompose u ε g,0,k = u ε, g,0,k + u ε, g,0,k where u ε, g,0,k solves the transport equation ( 48) (in our study we can choose u ε, g,0,k ≡ 0) and where u ε, g,0,k is a boundary layer satisfying the boundary conditions of (47). Indeed, note that if u ε, g,0,k does not satisfy the boundary conditions ( 33) and (34) then because boundary conditions ( 26) and ( 27) are decoupled compared to n the error on the boundaries for glancing modes will be O(1) with respect to ε which is not a suitable error rate for the justification of the WKB expansion (see Section 7).

Following [Williams, 1996] let χ ∈ C ∞ (]0, 1[) satisfying χ(x) = 1 for x ≤ 1 4 and χ(x) = 0 for x ≥ 3 4 , we define5 :

u ε g,0,k (t, x) = u ε, g,0,k (t, x) :=χ x d √ ε P k g,s φ 0 (ζ) g -B 0 U ε 0,O (t, x ) (49) -1 -χ x d √ ε e -i ε ξ k P k g,u φ 1 (ζ)B 1 U ε 0,I (t, x ).
It is clear that such a u ε g,0,k satisfies the boundary conditions ( 33) and (34). Moreover, by construction u ε g,0,k ∈ ker L (dϕ k ), so that the third equation of ( 23) is satisfied and by definition of g, the initial condition (28) written for n = 0 is satisfied for k ∈ G .

The construction of a corrector tem is postnone to Paragraph 6.3. In the last Paragaph of this Section we conclude the construction by the one of evanescent/explosive amplitudes.

Construction of evanescent and explosive leading order amplitudes

The only remaining leading order amplitudes to be constructed are the evanescent and the explosive ones. We recall that the evanescent amplitude of leading order satisfies the equations (see ( 23),( 33) and ( 28))

       L(∂ X d )U ε ev,0 (X d ) = 0 for X d ≥ 0, U ε ev,0 |x d =X d =0 = P ev φ 0 (ζ) g -B 0 U ε 0,O , U ε ev,0 |t≤0 = 0, (50)
and that the explosive amplitude of leading order satisfies the equations (see ( 23),( 34) and ( 28))

     L(∂ X d )U ε ex,0 ( X d ) = 0 for X d ≤ 0, U ε ex,0 | x d = X d =0 = -P ex φ 1 (ζ)B 1 U ε 0,I , U ε ex,0 |t≤0 = 0, (51)
where we set

X d = X d -1, x d = x d -1
, and that from Paragrah 5.1 (see Proposition 5.1) for all 0 < ε the right hand side of the boundary condition in (50) (resp. ( 51)) is a known function in H ∞ ,γ (∂Ω 0 ) (resp. H ∞ ,γ (∂Ω 1 ) for all γ > γ 0 depending only on g.

To solve these systems of equation we follow the method introduced by [Lescarret, 2007] that is we firstly determine the value of the double traces x d = X d = 0 and x d = X d = 0 and then we extend these traces for x d = 0 and x d = 0 as boundary layers in the normal variable.

The following Lemma is a trivial generalization of the one dealing only with evanescent modes in [Lescarret, 2007] to explosives modes: Lemma 5.2 ( [Lescarret, 2007]) We define for X d ≥ 0

P ev U (X d ) :=e X d A (ζ) Π s e U (0), (52)
Q ev F (X d ) := X d 0 e (X d -s)A (ζ) Π s e A -1 d F (s)ds - ∞ X d e (X d -s)A (ζ) Π u e A -1 d F (s)ds, ( 53
)
and for X d ≤ 0

P ex U ( X d ) :=e X d A (ζ) Π u e U (0), (54)
Q ex F ( X d ) := X d -∞ e ( X d -s)A (ζ) Π s e A -1 d F (s)ds - 0 X d e ( X d -s)A (ζ) Π u e A -1 d F (s)ds, (55)
then for all F ∈ P ev (resp. F ∈ P ex ) the equation

L(∂ X d )U = F for X d ≥ 0, resp. L(∂ X d )U = F for X d ≤ 0 , admits a solution reading U = P ev U + Q ev F (resp. U = P ex U + Q ex F ).
Lemma 5.2 combined with equations ( 50) and ( 51) implies that we have the conditions U ε ev,0 = P ev U ε ev,0 and U ε ex,0 = P ex U ε ex,0 which are comparable to the polarization condition (29) for oscillating amplitudes. We describe in the following the way to construct the evanescent amplitude U ε ev,0 the arguments are exactly the same for the explosive amplitude U ε ex,0 . From the definition of P ev and the "polarization" condition U ε ev,0 = P ev U ε ex,0 to determine U ε ev,0 it is sufficient to determine its trace on {X d = 0}. However by (50) we do not know this trace but only the double one on {x d = X d = 0}, so that we follow the method of [Lescarret, 2007] consisting in extending this double trace for x d > 0 as a boundary layer. Consequently

U ε ev,0 (t, x, X d ) = χ(x d )e X d A (ζ) P ev U ε ev,0 (t, x , 0, 0) = χ(x d )e X d A (ζ) P ev φ 0 (ζ) g -B 0 U ε 0,O ,
where χ ∈ C ∞ (]0, 1[) satisfies6 χ(x) = 1 for x ≤ 1 4 and χ(x) = 0 for x ≥ 3 4 is a solution of (50). The same kind of formula also holds for explosive amplitudes. Moreover clearly by definitions of P ev (resp. P ex ) is it clear U ε ev,0 (t, x, X d ) ∈ P ev (resp. U ε ex,0 (t, x, X d ) ∈ P ex ).

By sum up the results of this Section in the following proposition:

Proposition 5.2 Under Assumptions 2.1, 2.2, 2.4, assume that ζ / ∈ G and that Assumption 5.1. For all k ∈ H and for all 0 < ε 1 there exists u ε h,0,k ∈ H ∞ ,γ (Ω) for all γ > γ 0 (the one given by Assumption 5.1) satisfying the cascade of equations ( 23),( 26),( 27),( 28) written for n = 0. There exist U ε ev,0 ∈ P ev and U ε ex,0 ∈ P ex satisfying the cascade of equations ( 23),( 26),( 27),( 28) written for n = 0. So at this step we have determined all the amplitudes of the leading order in the ansatz ( 22). The following section aims to show that this construction can be repeated to higher order in order to obtain an approximate solution of ( 19) (we postnone the justification to Section 7).

Construction of higher order terms

In this paragraph we first sketch the construction of the amplitudes of order one in the WKB expansion. As the reader will notice, the construction for selfinteracting amplitudes is rather classical, that is we firstly determine the unpolarized part of the amplitudes (which only depend on the leading order amplitude) and then reiterate the construction described in Paragraph 5.1 to determine the polarized part. The determination of the evanescent or explosive amplitudes follows more or less the same ideas. More precisely we decompose the evanescent/explosive amplitude in some "unpolarized part" depending only on the leading order evanescent/explosive amplitude and some "polarized" part which is determined as described in Paragraph 5.3.

Finally we show in Paragraph 6.4 that these constructions can be performed at any order for selfinteracting and evanescent/explosive amplitudes if ζ / ∈ G.

However, the situation is not so ideal when glancing modes exist. Indeed as mentioned in Paragraph 5.2, the glancing amplitudes can not solve simultaneously the boundary conditions and the equation in the interior. This fact implies that we are able to define a corrector ensuring a suitable rate of convergence but that we will not be able to define these correctors at any order. We refer to Paragraph 6.3 for more details.

Selfinteracting amplitudes of order one

Firstly, in a classical setting see for example [Rauch, 2012], we determine the unpolarized part of the hyperbolic amplitudes of order one, namely the u ε h,1,k for k ∈ H . In order to do so, we apply the pseudo-inverse Υ k (see Definition 2.3) to the second equation of ( 23) written for n = 0. By definition of Υ k we obtain that

∀k ∈ H , (I -Π k )u ε h,1,k = iΥ k L(∂)u ε h,0,k . ( 56
)
The right hand side of ( 56) has been determined in Paragraph 5.1, so that (56) uniquely determines the unpolarized part of the selfinteracting amplitudes (moreover they are in H ∞ ,γ (Ω) for all γ > γ 0 ). So to conclude the construction it only remains to determine the polarized parts, namely the Π k u ε h,1,k for k ∈ H . Consider the second equation of ( 23) written for n = 1, composing by Π k and using the trivial decomposition

I = I -Π k + Π k leads to: Π k L(∂)Π k u ε h,1,k = -Π k L(∂)(I -Π k )u ε h,1,k ⇐⇒ Π k L(∂)Π k u ε h,1,k = -iΠ k L(∂)Υ k h L(∂)u ε h,0,k .
We can apply Lax's Lemma [Lax, 1957] to rewrite this equation as:

(∂ t + v k • ∇ x )Π k u ε h,1,k = -iΠ k L(∂)Υ k h L(∂)u ε h,0,k . ( 57
)
We again have to solve a transport equation so we reiterate the discussion depending on the type of the amplitude. k ∈ I . In that case the transport phenomenon is incoming so that to be solved (57) only requires a boundary condition on ∂Γ 0 . To determine this boundary condition we consider (26) written for n = 1 that we write under the form:

B 0 k∈I Π k u ε h,1,k + k∈G Π k u ε h,1,k + U ε ev,1 |X d =0 |x d =0 = -B 0 k∈I (I -Π k )u ε h,1,k + ∈O (I -Π )u ε h,1, (58)
+ k∈G (I -Π k )u ε g,1,k + ∈O Π u ε h,1, |x d =0
, and we remark that all the terms in the right hand side of ( 58), except the last one, are known functions in H ∞ ,γ (∂Ω 0 ) for all γ > γ 0 . Consequently applying the uniform Kreiss-Lopatinskii condition (see Assumption 2.4) and the projector P k h (see Definition 2.4) to ( 58) shows that the polarized part Π k u ε h,1,k for k ∈ I satisfies the transport equation (note that

P k h Π k = Π k ):        (∂ t + v k • ∇ x )Π k u ε h,1,k = F ε 1,k,I , Π k u ε h,1,k |x d =0 = -P k h φ 0 (ζ)B 0 U ε 1,O + G ε 1,I , Π k u ε h,1,k |t≤0 = -(I -Π k )u ε h,1,k |t≤0 = 0, ( 59
)
where we recall that U ε 1,O is defined in ( 46) and where the source terms are given by:

F ε 1,k,I := -iΠ k L(∂)Υ k L(∂)u ε h,0,k , G ε 1,I := k∈I (I -Π k )u ε h,1,k + ∈O (I -Π )u ε h,1, + k∈G (I -Π k )u ε g,1,k |x d =0
.

∈ O. In that case the transport phenomenon is outgoing so that to be solved (57) only requires a boundary condition on ∂Γ 1 . Reiterating essentially the same computations as the ones for the case k ∈ I we easily obtain that the polarized part of the outgoing amplitudes satisfy the transport equation:

       (∂ t + v • ∇ x )Π u ε h,1, = F ε 1, ,O , Π u ε h,1,k |x d =1 = -e -i ε ξ P h φ 1 (ζ)B 1 U ε 1,I + G ε 1,O , Π u ε h,1, |t≤0 = 0, ( 60
)
where the source terms are given by:

F ε 1, ,O := -iΠ L(∂)Υ L(∂)u ε h,0, , G ε 1,O := k∈I e i ε ξ k (I -Π k )u ε h,1,k + ∈O e i ε ξ (I -Π )u ε h,1, + k∈G e i ε ξ k (I -Π k )u ε g,1,k |x d =1
.

We can repeate the same arguments as the ones described in Paragraph 5.1 to obtain a compatibility condition on

U ε 1,I = k∈I e i ε ξ k Π k u ε h,1, |x d =1 .
Integrating (60) along the characteristics gives:

Π u ε h,1, (t, x) = -e -i ε ξ P h φ 1 (ζ)B 1 U ε 1,I t + 1 v ,d (1 -x d ), x + v v ,d (1 -x d ) (61) -e -i ε ξ P h φ 1 (ζ)B 1 G ε 1,O t + 1 v ,d (1 -x d ), x + v v ,d (1 -x d ) - 1-x d 0 F ε 1, ,O t + 1 v ,d (1 -x d -s), x + v v ,d (1 -x d -s), 1 -s ds,
and consequently the right hand side of the boundary condition of ( 59) depends on U ε 1,I , G ε 1,O and F ε 1, ,O . Integrating again along the characteristics the transport equation gives (by linearity):

Π k u ε h,1,k (t, x) =P k h φ 0 (ζ)B 0 ∈O e -i ε ξ P h φ 1 (ζ)B 1 U ε 1,I t + 1 v ,d - 1 v k,d x d , x + v v ,d - v k v k,d x d (62) + P k h φ 0 (ζ)B 0 ∈O e -i ε ξ P h φ 1 (ζ)B 1 G ε 1,O t + 1 v ,d - 1 v k,d x d , x + v v ,d - v k v k,d x d -P k h φ 0 (ζ)B 0 G ε 1,I t - 1 v k,d x d , x - v k v k,d x d + P k h φ 0 (ζ)B 0 ∈O 1 0 F ε 1, ,O t k, (s, x d ), x k, (s, x d ), 1 -s ds + x d 0 F ε 1,k,I t - 1 v k,d (x d -s), x + v k v k,d (x d -s), s ds,
where we defined:

t k, (s, x d ) := t + 1 v ,d (1 -s) + 1 v k,d
x d and x k, (s,

x d ) := x + v v ,d (1 -s) + v k v k,d x d .
Multiplying ( 62) by e i ε ξ k and suming for k ∈ I we obtain the compatibility condition:

(I -T ε (ζ))U ε I ,1 =T ε (ζ)G ε 1,O - k∈I e i ε ξ k P k h φ 0 (ζ)B 0 G ε 1,I t - 1 v k,d , x - v k v k,d (63)
+

k∈I e i ε ξ k P k h φ 0 (ζ)B 0 ∈O 1 0 F ε 1, ,O t k, (s, x d ), x k, (s, x d ), 1 -s ds + k∈I e i ε ξ k 1 0 F ε 1,k,I t - 1 v k,d (1 -s), x + v k v k,d (1 -s), s ds,
where T ε (ζ) is defined in (42). Remark that all the terms in the right hand side of ( 63) are in E s h (ζ) (recall that by definition F ε 1,k,I reads F ε 1,k,I = Π k F ε 1,k,I for k ∈ I ) so we can use Assumption 5.1 in (63) to determine the value of U ε I ,1 (in terms of some known functions in

H ∞ ,γ (∂Ω 1 , E s h (ζ)), namely G ε 1,O , G ε 1,I , F ε 1,k,I and F ε 1, ,O
). Plugging this value in ( 61) and ( 62) gives the value of the polarized part of the amplitude for selfinteracting phases namely the Π k u n,1,k for k ∈ H . This concludes the construction of the selfinteracting amplitudes of order one.

Evanescent and explosive amplitudes of order one

We now turn to the determination of evanescent and explosive amplitudes of order one. Considering (23) written for n = 0 we obtain that

L(∂ X d )U ε ev,1 = -L(∂)U ε ev,0 for X d ≥ 0 and L(∂ X d )U ε ex,1 = -L(∂)U ε ex,0 for X d ≤ 0. (64)
From Proposition 5.2 the known function U ε ev,0 ∈ P ev (resp. U ε ex,0 ∈ P ex ) and it is clear that P ev (resp. P ex ) is stable by L(∂). Consequently the right hand sides in (65) are in P ev and P ex respectively, and we can apply Lemma 5.2 to obtain the decompositions:

U ε ev,1 = P ev U ε ev,1 -Q ev L(∂)U ε ev,0 and U ε ex,1 = P ex U ε ex,1 -Q ex L(∂)U ε ex,0 . (65)
To determine U ε ev,1 (resp. U ε ex,1 ) it is sufficient to determine P ev U ε ev,1 (resp. P ex U ε ex,1 ). This is done mainly in the same way that in Paragraph 5.3. We briefly sketch the construction of P ev U ε ev,1 for completedness. Recall that by definition of P ev (see ( 52)) P ev U ε ev,1 is known if and only if we know the value of its trace on {X d = 0} to determine this trace we consider the cascade of equation ( 26) written for n = 1 and then we extend the double trace on {X d = x d = 0} as a trace on {X d = 0} only. The second equation of ( 26) written for n = 1 reads (after decomposition on the stable/unstable subspaces, by the uniform Kreiss-Lopatinskii condition on ∂Γ 0 and the composition by P s e ):

U ε ev,1 |x d =X d =0 = -P s e φ 0 (ζ)B 0 k∈O u ε h,1,k |x d =0 ,
where from Paragraph 6.1 the right hand side is a known function in H ∞ ,γ (Ω) for all γ > γ 0 . Consequently by definition of P ev we obtain (by using a function χ as before) that:

P ev U ε ev,1 (t, x, X d ) = -χ(x d )e X d A (ζ) P s e φ 0 (ζ)B 0 k∈O u ε h,1,k |x d =0 ,
which concludes the construction of evanescent amplitude of order one and once again in which of the definitions of P ev and Q ev it is clear that U ε ev,1 ∈ P ev . The same permits to show that U ε ex,1 ∈ P ex .

A corrector for glancing amplitudes

In this paragraph we follow the method of [Williams, 1996] to construct a corrector for glancing modes such that the geometric optics expansion is a good approximation of the exact solution up to an admissible rate of convergence (that is O(ε 1/4 )).

To this aim we recall the equations governing glancing amplitudes of order one in (23) (namely the fourth and the fifth equations of ( 23) in which we reintroduced the power of ε for convinience) that is:

ε 0 iL ( dϕ k )u ε g,1,k + L(∂)u ε g,0,k = 0 ∀k ∈ G , εL(∂)u ε g,1,k = 0 ∀k ∈ G . (66)
We decompose the first equation of (66) as:

ε 0 iL ( dϕ k )u g,1,k + L(∂)Π k u ε g,0,k = ε 0 iL ( dϕ k )u ε g,1,k + Π k L(∂)Π k u ε g,0,k + (I -Π k )L(∂)u ε g,0,k = 0
. This equation has exactly the same form as the one for hyperbolic amplitudes except that we chose the leading order glancing mode in such a way that it satisfies the boundary condition (to ensure an error at least of size O(ε) on the boundary) but not the interior equation so that compared to hyperbolic modes Π k L(∂)Π k u g,0,k is not zero and gives rise to an extra error in the interior.

However as for oscillating modes we compose the first equation of (66) by Υ k the partial inverse of L ( dϕ k ) and we define:

u ε g,1,k = (I -Π k )u ε g,1,k := iΥ k L(∂)u ε g,0,k . (67)
By doing this we obtain that for all k ∈ G

L(∂)(e i ϕ k ε (u ε g,0,k + εu ε g,1,k )) = Π k L(∂)Π k u ε g,0,k + εL(∂)u ε g,1,k . (68)
The term of order ε 0 in the right hand side of (68) may seem to be alarming to obtain a good error estimate for glancing modes but thanks to the choice of the boundary layer in (49) it is not. Indeed, from (49) and using the fact that for all

k ∈ G , Π k L(∂)Π k = ∂ t + v k • ∇ x (so that Π k L(∂)Π k does not act on the x d variable) we have that: Π k L(∂)Π k u ε g,0,k = χ ε (x d )B 0 (t, x ) + (1 -χ ε (x d ))B 1 (t, x ),
where χ ε (x d ) := χ(ε -1/2 x d ) and where from Proposition 5.1

B 0 (t, x ), B 1 (t, x ) are in H ∞ ,γ (R t × R d-1 x ) for all γ > γ 0 . So a simple change of variables shows that Π k L(∂)Π k u ε g,0,k is O(ε 1/4 ) in L 2 γ (Ω) for all γ > γ 0 .
We now turn to the term εL(∂)u ε g,1,k in the right hand side of (68). From ( 49) and ( 67), u ε g,1,k reads under the form:

u ε g,1,k = ε -1/2 χ ε (x d ) B 0 (t, x ) + (1 -χ ε (x d )) B 1 (t, x ) + h.o.t, where B 0 , B 1 ∈ H ∞ ,γ (R t × R d-1
x ) for all γ > γ 0 . From which we immediately deduce that εL(∂)

u ε g,1,k is O(ε 1/4 ) in L 2 γ (Ω) for all γ ≥ γ 0 . Using the fact that by construction B 0 u ε g,0,k | x d =0 = g ε and B 1 u ε g,0,k | x d =1 = 0 it follows:
Proposition 6.1 Assume that the hyperbolic strip problem (19)7 satisfies Assumptions 2.1, 2.2, 2.4, 2.3 and 5.1 for some γ 0 ≥ 0. Then with u ε g,0,k defined in (49) and u g,1,k defined in (67) we have

                 L(∂) k∈G e i ϕ k ε (u ε g,0,k + εu ε g,1,k ) = O Ω (ε 1/4 ) in Ω, B 0 k∈G e i ϕ k ε (u ε g,0,k + εu ε g,1,k ) |x d =0 = O ∂Ω0 (ε) on ∂Ω 0 , B 1 k∈G e i ϕ k ε (u ε g,0,k + εu ε g,1,k ) |x d =1 = O ∂Ω1 (ε) on ∂Ω 1 , k∈G e i ϕ k ε (u ε g,0,k + εu ε g,1,k ) |t≤0 = 0 on Γ, (69)
where O X (•) are in L 2 γ (X) for all γ > γ 0 .

Higher order non glancing amplitudes

As mentioned in Paragraph 6.3 when the frequency ζ admits glancing modes then we can construct a first order corrector such that the error (in the interior) is O(ε 1/4 ) in L 2 γ (Ω). However it seems difficult to reiterate this method to construct a second order corrector giving rise to an admissible error (the reason remains that glancing modes can not solve the interior and the boundary equations simultaneously).

However when G = ∅ we can repeat the construction made in Paragraphs 6.1 and 6.2 to define an arbitrary number of correctors. In this paragraph we briefly describe the way to proceed.

Assume that all the terms u ε h,n,k , k ∈ H and U ε ev,n , U ε ex,n appearing in ( 22) have been constructed up to some order n 0 ≥ 1. We sketch the construction of the u ε h,n0+1,k , k ∈ H , U ε ev,n0+1 and U ε ex,n0+1 .

• Firstly the second equation of ( 23) written for n = n 0 gives the unpolarized part of the hyperbolic amplitude u ε h,n0+1,k (so that it is sufficient to determine the polarized part) and the 7 th (resp. 8 th ) equation of ( 23) combined with Lemma 5.2 implies that to determine U ε ev,n0+1 (resp. U ε ex,n0+1 ) it is sufficient to determine P ev U ε ev,n0+1 (resp. P ex U ε ex,n0+1 ) (see ( 52) and ( 54)).

• From Lax's Lemma [Lax, 1957] and Lemma 5.2 each of the terms mentioned above require only a boundary condition (on ∂Γ 0 for the u ε h,n0+1,k , k ∈ I and U ε ev,n0+1 and on ∂Γ 1 for the u ε h,n0+1,k , k ∈ O and U ε ex,n0+1 ). Identify in ( 26) and ( 27) (written for n = n 0 ) the stable and the unstable parts of the traces shows that the "double trace" of evanescent and explosive amplitudes only depends on the trace of the hyperbolic amplitudes. Consequently we shall determine the traces of the hyperbolic amplitudes first.

• To determine the trace of the oscillating amplitudes we remark that by the uniform Kreiss-Lopatinskii condition on each side the boundary conditions ( 26) and ( 27) (written for n = n 0 ) can be written under the form:

Π k u ε h,n0+1,k | x d =0 = -P k n φ 0 (ζ)B 0 U ε n0+1,O + F 0 k ∈ I Π u ε h,n0+1, | x d =1 = -P n φ 1 (ζ)B 1 U ε n0+1,I + F 1 ∈ O,
where F 0 and (resp. F 1 ) is a given source terms that depends on the

u ε h,n,k for k ∈ H and n ≤ n 0 , on (I-Π k )u ε h,n0+1,k | x d =0 (resp. (I-Π )u ε h,n0+1, | x d =1 ) but not on Π k u ε h,n0+1,k | x d =0 (resp. Π u ε h,n0+1, | x d =1 ).
Reiterate exactly the same kind of computations as the ones described in Paragraphs 5.1 and 6.1 lead to the compatibility condition:

(I -T (ζ))U ε I ,n0+1 = F n0+1 , where F n0+1 is a known function in H ∞ ,γ (∂Ω 1 , E s h (ζ)
) for all γ > γ 0 . From Assumption 5.1 we determine U ε I ,n0+1 and then each oscillating amplitude u ε h,n0+1,k for k ∈ H by resolution of transport equations.

• The final step is to construct the "polarized" parts of the evanescent/explosive amplitude (that is P ev U ε ev,n0+1 and P ex U ε ex,n0+1 ). Is it done exactly as it as been done in Paragraphs 5.3 and 6.2. More precisely the knowledge of the traces of the oscillating amplitudes gives the knowledge of the "double" traces of U ε ev,n0+1 , U ε ex,n0+1 and then we are free to extend these double traces in simple ones thanks to the cutt-off function χ.

This concludes the construction of the amplitudes at any order in the particular framework where G = ∅ to sum up we give the following proposition: Proposition 6.2 Under Assumptions 2.1,2.2 and 2.4 also assume that G = ∅ and that Assumption 5.1 holds for some γ 0 ≥ 0. Then for all n ∈ N, for all k ∈ H there exist u ε h,n,k ∈ H ∞ ,γ (Ω) for all γ > γ 0 and U ε ev,n ∈ P ev , U ε ex,n ∈ P ex satisfying the cascades of equations ( 23), ( 26), ( 27) and (28).

Justification of the WKB expansion

In this section we give two justifications of the WKB expansion depending on the kind of the frequency ζ.

Firstly let us introduce what we mean by a strongly well-posed hyperbolic boundary value problem in the strip Ω.

Definition 7.1 Let f ∈ L 2 (Ω), g 0 ∈ L 2 (∂Ω 0 ) and g 1 ∈ L 2 (∂Ω 1 ) be given source terms. The hyperbolic boundary value problem in the strip Γ

         L(∂)u = f in Ω, B 0 u |x d =0 = g 0 on ∂Ω 0 , B 1 u |x d =1 = g 1 on ∂Ω 1 , u |t≤0 = 0 on Γ,
is said to be strongly well-posed if its admits a unique solution u ∈ L 2 (Ω) with traces u |x d =0 ∈ L 2 (∂Ω 0 ) and u |x d =1 ∈ L 2 (∂Ω 1 ) satisfying the energy estimate that there exist C > 0 and γ 0 ≥ 0 such that for all γ > γ 0 :

γ u 2 L 2 γ (Ω) + u |x d =0 2 L 2 γ (∂Ω0) + u |x d =1 2 L 2 γ (∂Ω1) ≤ C 1 γ f 2 L 2 γ (Ω) + g 0 2 L 2 γ (∂Ω0) + g 1 2 L 2 γ (∂Ω1) . (70)
In the particular setting where γ 0 = 0 the strip problem is said to be lower exponentially strongly well posed.

As mentioned in the introduction show that the there exists γ 0 > 0 such that the strip problem is strongly well posed in the sense of Definition (7.1) only requires the uniform Kreiss-Lopatinskii condition on each side of the boundary so that under Assumptions2.1, 2.2 and 2.4 the strip problem ( 19) is automatically strongly well-posed in the sense of Definition 7.1.

The question of the lower exponential strong well posedness of ( 19) is studied in [Benoit, ]. In this article the author describes a particular framework, namely the one of strictly dissipative boundary conditions in which the lower exponential strong well posedness of (19) as well as a full characterization of lower exponentially strongly well posed problems. More precisely this characterization asks the inversibility of some trace operators reading under the form

where we stress that C and C do not depend on ζ. The precise expressions of T (ζ) and T (ζ) can be find in [Benoit, ] and we will also give it in Paragraph 8 in order to compare T (ζ) with T ε (ζ). However these expressions are of little interest for the justification of the WKB expansion which only requires the (lower exponential) strong well-posedness of ( 19). We sum up the known results about the strong well-posedness of (19) in the following Theorem Theorem 7.1 (Strong well-posedness of (19))

• Under Assumptions 2.1, 2.2 and 2.4 there exists γ 0 ≥ 0 such that the strip problem (19) is strongly well-posed in the sense of Definition 7.1.

• [Benoit, ] Under Assumptions 2.1, 2.2 also assume that the matrices A j are symmetric for all j = 1, ..., d and that the boundary conditions on ∂Γ 0 and ∂Γ 1 are strictly dissipative that is there exist

C 0 , C 1 , ε 0 , ε 1 > 0 such that ∀u ∈ R N ε 0 |u| 2 + A d u, u -C 0 |B 0 u| 2 < 0 and ε 1 |u| 2 + A d u, u -C 1 |B 1 u| 2 > 0, ( 73
)
then the strip problem (19) is lower exponentially strongly well-posed in the sense of Definition 7.1.

• [Benoit, ] Under Assumptions 2.1, 2.2, 2.4 also assume that the matrices A j are symmetric for all j = 1, ..., d and that ker B 0 ∩ ker B 1 = {0}. Then the strip problem (19) is lower exponentially strongly well-posed in the sense of Definition 7.1 if and only if the inversibility conditions (71) and (72) hold.

As explained in Paragraph 6.3, when the frequency ζ involves glancing modes (that is to say G = ∅) then the error between the approximate solution given by the WKB expansion and the exact solution of ( 19) is O(ε 1/4 ) because the glancing amplitudes of order one, namely the u g,1,k can not solve simultaneously the boundary conditions ( 27) (written for n = 1) and the equation in the interior (that is the fourth equation of ( 23) written for n = 0).

Whereas when the frequency ζ does not involve glancing modes, the arguments described in Paragraphs 6.1, 6.2 and 6.4 show that we can construct the geometric optics expansion at any order so that the error between this expansion and the exact solution of ( 19) is of order O(ε N0+1 ), where N 0 stands for the number of terms in the geometric optics expansion.

We first consider the case where G = ∅. In this framework we define an approximate solution of u ε by: for

N 0 ∈ N u ε app,N0 := N0 n=0 k∈H e i ε ϕ k (t,x) ε n u ε h,n,k (t, x) + N0 n=0 e i ε ψ(t,x ) ε n U ε ev,n t, x, x d ε + U ε ex,n t, x, x d -1 ε , (74)
where the terms appearing in the right hand side of (74) are defined in Proposition 6.2. The result is the following.

Theorem 7.2 Under Assumptions 2.1-2.2 and 2.4 also assume that (19) is strongly well-posed in the sense of Definition 7.1 for some γ 0 ≥ 0 and that Assumption 5.1 holds for some treshold 0 ≤ γ 0 ≤ γ 0 . Then u ε app,N0 defined in (74) is an approximate solution of u ε the solution of (19) in the sense that: there exists C > 0 such that for all γ > γ 0

u ε -u ε app,N0 L 2 γ (Ω) ≤ Cε N0+1 . (75)
In particular if (19) is lower exponentially strongly well-posed in the sense of Definition 7.1 and if Assumption 5.1 holds for γ 0 = 0, then estimate (75) is uniform in time.

Proof : By construction of u ε app,N0 , u ε app,N0+1 -u ε satisfies the hyperbolic boundary value problem

         L(∂)(u ε app,N0+1 -u ε ) = ε N0+1 f ε N0+1 in Ω, B 0 (u ε app,N0+1 -u ε ) |x d =0 = 0 on ∂Ω 0 , B 1 (u ε app,N0+1 -u ε ) |x d =1 = 0 on ∂Ω 1 , (u ε app,N0+1 -u ε ) |t≤0 = 0 on Γ,
where we defined

f ε N0+1 := k∈H e i ε ϕ k L(∂)u ε h,N0+1,k + e i ε ψ L(∂)U ε ev,N0+1 t, x, x d ε + L(∂)U ε ex,N0+1 t, x, x d -1 ε .
By construction and from Assumption 5.1 the terms composing f ε N0+1 are H ∞ ,γ (Ω) for all γ > γ 0 (because the u ε h,N0+1,k ∈ H ∞ ,γ (Ω) for all γ > γ 0 independently on ε). Consequently f ε N0+1 is in H ∞ ,γ (Ω) for all γ > γ 0 ( and consequently for all γ > γ 0 ) so that from the energy estimate (70) we obtain:

u ε -u ε app,N0+1 2 L 2 γ (Ω) ≤ Cε N0+1
, for all γ > γ 0 . We then conclude to (75) by the triangle inequality.

We now turn to the case where G = ∅. We include in (74) the contribution of glancing modes and restrict the expansion to the order one to define:

u ε app := 1 n=0 k∈H e i ε ϕ k (t,x) ε n u ε h,n,k (t, x) + 1 n=0 k∈G e i ε ϕ k (t,x) ε n u ε g,n,k (t, x) (76)
+

1 n=0 e i ε ψ(t,x ) ε n U ε ev,n t, x, x d ε + U ε ex,n t, x, x d -1 ε ,
The result giving the justification of the WKB expansion with glancing modes is the following Theorem 7.3 Under Assumptions 2.1, 2.2, 2.3 and 2.4 also assume that ( 19) is strongly well-posed in the sense of Definition 7.1 for some γ 0 ≥ 0 and that Assumption 5.1 holds for some treshold 0 ≤ γ 0 ≤ γ 0 . Then u ε app defined in (76) is an approximate solution of u ε the solution of (19) in the sense that: there exists C > 0 such that for all γ > γ 0 u ε -u ε app L 2 γ (Ω) ≤ Cε 1/4 . Proof : This theorem is an immediate corollary of Theorem 7.2 and Proposition 6.1.

Study of Assumption 5.1

In this paragraph we study Assumption 5.1, that is to say that there exists some γ 0 ≥ 0 such that the operator

(I -T (ζ)) is invertible on H ∞ ,γ (R t × R d-1 x , E s h (ζ)) with values in H ∞ ,γ (R t × R d-1 x , E s h (ζ)) for all γ > γ 0 . For convenience we recall that T ε (ζ) is defined by: (T ε (ζ)f )(t, x ) := k∈I , ∈O e i ε (ξ k -ξ ) R k, (ζ)f (t -α k, , x + β k, ) (77)
where we set

R k, (ζ) := P k h φ 0 (ζ)B 0 P h φ 1 (ζ)B 1 , α k, := 1 v ,d - 1 v k,d and β k, := v v ,d - v k v k,d . Clearly when f ∈ H ∞ ,γ (R t × R d-1 x , E s h (ζ)
) then so do (I -T (ζ))f (because the derivatives only apply on f and because by definition v ,d < 0 for ∈ O and v k,d > 0 for k ∈ I so that t -α k, < t).

The simplest way to show that (

I -T (ζ)) is invertible over H ∞ ,γ (R t × R d-1 x , E s h (ζ)) with values in H ∞ ,γ (R t × R d-1 x , E s h (ζ)) is of course to show that T (ζ) is a contraction on H ∞ ,γ (R t × R d-1 x , E s h (ζ))
. In Paragraphs 9.1.1 and 9.1.2 we give some examples of such a situation.

From the particular expression of

T ε (ζ) it is sufficient to consider the L 2 γ (R t × R d-1 x )-norm. Proposition 8.1 Let γ 0 be such that 8 k∈I , ∈O R k, (ζ) 2 < e γ 0 min k∈I , ∈O α k,l , ( 78
) then T ε (ζ) is a contraction on H ∞ ,γ (R t × R d-1 x , E s h (ζ)
) for all γ > γ 0 and consequently Assumption 2.1 is satisfied.

Proof : As already mentioned it is sufficient to consider the L

2 γ (R t × R d-1 x )-norm of T ε (ζ). We have for f ∈ L 2 γ (R t × R d-1 x ) T ε (ζ)f 2 L 2 γ (Rt×R d-1 x ) ≤ k∈I , ∈O R+×R d-1 e -2γt |R k, (ζ)f (t -α k, , x -β k, )| 2 dt dx , ≤ k∈I , ∈O e -2α k, γ R+×R d-1 e -2γt |R k, (ζ)f (t, x )| 2 dt dx , ≤ f 2 L 2 γ (Rt×R d-1 x ) k∈I , ∈O e -2α k, γ R k, (ζ) 2 . So that if we choose γ 0 ≥ 0 large enough such that k∈I , ∈O R k, (ζ) 2 < e γ 0 min k∈I , ∈O α k,l then T ε (ζ) is a contraction on H ∞ ,γ (R t × R d-1 x , E s h (ζ)
) for all γ > γ 0 .

Remark

• We note that if γ 0 = 0 in (78) then Assumption 5.1 holds with γ 0 = 0 and consequently the approximate solution given by the geometric optics expansion (76) or (74) admits a lower exponential growth in time (so that it can be a good approximation of a solution which is lower exponentially strongly well posed).

• In (78) the term min k∈I , ∈O α k, is the minimal time to perform a full regenerating reflection.

• Moreover in the particular setting where #I = #O = 1 (meaning that there is only one selfinteraction path of phases) then (78) becomes R(ζ) < e αγ 0 , where α is the time needed to perform a full regenerating reflection. In particular when γ 0 = 0 this condition is nothing but asking that the coefficient of reflection for a complete circuit is less than one so that the energy decreases after a complete circuit.

This condition agrees with the intuition that if the energy increases after one complete circuit then the associated solution should have an exponential growth in time depending on the time needed to perform a complete circuit.

• In [Benoit, ] one of the condition characterizing the lower exponentially strongly well-posed problems, namely the uniform inversibility of (I -T (ζ)) on E s (ζ) , can be explicit as: ζ) .

T (ζ) = φ 0 (ζ)B 0 e -A (ζ) φ 1 (ζ)B 1 e A (
So that from this expression it immediately follows that the condition used to construct the WKB expansion (that is Assumption 5.1) is a microlocalized versoin of the condition (71) on hyperbolic modes (and only on hyperbolic modes).

9 Examples and comments 9.1 Examples

The wave equation in two dimensions

In this first example we consider the wave equation in two dimensions

             ∂ t u ε + A 1 ∂ 1 u ε + A 2 ∂ 2 u ε = 0 for (t, x) ∈ Ω, B 0 u ε |x2=0 := 1 -α 0 u ε |x2=0 = g ε on (t, x 1 ) ∈ ∂Ω 0 , B 1 u ε |x2=1 := -α 1 1 u ε |x2=1 = 0 on (t, x 1 ) ∈ ∂Ω 1 , u ε |t≤0 = 0 for x ∈ Γ, (79)
where α 0 , α 1 ∈ R and where the coefficients A 1 , A 2 are given by:

A 1 := 1 0 0 -1 and A 2 := 0 1 1 0 .

In (79) the source term g ε reads

g ε (t, x 1 ) := e i ε (τ t+ηx1) g(t, x 1 ), (80)
where g ∈ H ∞ (∂Ω 0 ) and where τ , η ∈ R are fixed frequency parameters.

We can easily check that the boundary conditions in (79) are strictly dissipative (see ( 73)) if and only if the parameters α 0 , α 1 satisfy α 0 < 0, α 1 > 0. So for such parameters Theorem 7.1 applies and ( 79) is exponentially strongly well-posed. We also recall that from [Hersh, 1963], as in the case N = 2 the uniform Kreiss-Lopatinskii condition is equivalent to the strict dissipativity of the boundary condition. Consequently the restrictions α 0 < 0, α 1 > 0 are the only ones leading to an exponentially strongly well-posed problem.

We are now interested in the fullfilment of Assumption 5.1 in order to construct a geometric optics expansion by Theorem 7.2 or Theorem 7.3 (depending on the frequency (τ , η)).

The resolvent matrix associated to (79

) for ζ = (σ, η) is A (ζ) = 0 -σ + iη -(σ + iη) 0 .
So that for ζ = (iτ , η) we deduce that if X is an eigenvalue of A (ζ) then it satisfies the dispersion relation

X 2 = η 2 -τ 2 .
Consequently the partition of the boundary of the frequency space Ξ 0 in (7) reads:

E = (τ, η) ∈ R 2 \ |η| > |τ | , H = (τ, η) ∈ R 2 \ |τ | > |η| , G = (τ, η) ∈ R 2 \ |η| = |τ | and EH = ∅.
Without loss of generality let us assume that τ > 0 and in order to study Assumption 5.1 we assume that ζ ∈ H (if ζ ∈ E ∪ G then clearly Theorem 7.2 or 7.3 applies independently on Assumption 5.1). In this setting the stable (resp. unstable) eigenvalue X s := X s (τ , η) (resp. X u := X u (τ , η)) is given by:

X s := iξ = -i τ 2 -η 2 (resp. X u = -X s ) , (81)
from which we immediately deduce that the stable subspace E s (ζ) and the unstable subspace E u (ζ) are parametrized by:

E s (ζ) = vect (-ξ, τ + η) t and E u (ζ) = vect (ξ, τ + η) t .
We now study Assumption 5.1, in the setting of (79) the restriction of the operator

T ε (ζ) to E s (ζ) = E s h (ζ) is: T ε (ζ) -ξ τ + η = e 2 i ε ξ -ξ + α 0 (τ + η) ξ + α 0 (τ + η) • α 1 ξ + τ + η -α 1 ξ + τ + η -ξ τ + η . (82)
Consequently Assumption 5.1 is automatically satisfied for all boundary parameters α 0 , α 1 leading to strictly dissipative boundary conditions (for all ζ ∈ H) because in such a framework one can easily check that Porposition 8.1 applies with γ 0 = 0 so that T ε (ζ) is a contraction.

However, it is also interesting to note that in fact T ε (ζ) is a contraction for more boundary parameters than the ones leading to strictly dissipative boundary conditions. Indeed, it is not difficult to check that we have the following equivalence:

T ε (ζ) -ξ τ + η < -ξ τ + η ⇔ α 1 α 0 < 1,
so that Assumption 5.1 is satisfied for more parameters that the strictly dissipative ones. The aim of the next example is to give more details about this observation.

A modification of the wave equation

In this second example we consider the following modification of the classical wave equation:

           ∂ t u ε + A 1 ∂ 1 u ε + A 2 ∂ 2 u ε = 0 for (t, x) ∈ Ω, B 0 u ε |x2=0 = g ε on (t, x 1 ) ∈ ∂Ω 0 , B 1 u ε |x2=1 = 0 on (t, x 1 ) ∈ ∂Ω 1 , u ε |t≤0 = 0 for x ∈ Γ, (83)
where the coefficients A 1 , A 2 are given by:

A 1 :=   1 0 0 0 -1 0 0 0 a   , A 2 :=   0 1 0 1 0 0 0 0 -b   ,
for fixed parameters a ∈ R, b ∈ R * + . So the evolution equation of ( 83) is a wave equation (for the components u 1 and u 2 ) combined with an uncoupled transport phenomenon for the component u 3 . The source term g ε is of the form (80).

The boundary matrices in ( 83) are defined by (note that A 2 admits only one positive eigenvalue):

B 0 := 1 -α 0 -α 1 , B 1 := -1 1 0 0 1 -δ ,
where α 0 , α 1 , δ ∈ R. Consequently in (83) the coupling between u 1 , u 2 and u 3 is made in the boundary conditions.

As in Paragraph 9.1.1 in order to study Assumption 5.1 for (83) we are interested in the hyperbolic area of (83). The system is decoupled and the transport equation added on u 3 is hyperbolic whatever the frequency parameter is. So that we have the following decomposition of the boundary of the frequency space: Reiterating essentially the same computations as the ones performed in Paragraph 9.1.1, we can easily show that the stable subspace E s (ζ) and the unstable subspace E u (ζ) associated to (83) are given by: E s (ζ) := vect {e s } = vect (-ξ, τ + η, 0) t and E u (ζ) := vect {e u,1 , e u,2 } = vect (ξ, τ + η, 0) t , (0, 0, 1) t , where ξ is defined in (81).

It is also easy to show that the boundary condition on ∂Γ 0 is strictly dissipative if and only if α 0 < 0 and α 2 1 + 2α 0 b < 0. This condition satisfies the uniform Kreiss-Lopatinskii if and only if α 0 < 0, independently on α 1 . With such a choice of α 0 , the inverse given by the uniform Kreiss-Lopatinskii condition is given by:

φ 0 (ζ) : C → E s (ζ) φ 0 (ζ)x := -x ξ + α 0 (τ + η) e s .
The boundary condition on ∂Γ 1 satisfies the uniform Kreiss-Lopatinskii condition for all δ = 0 and is strictly dissipative if and only if we have δ 2 > b 2 . For δ = 0 the inverse given by the uniform Kreiss-Lopatinskii condition is: 2ξ(τ + η) δ(-ξ + τ + η)

• α 1 ξ + α 0 (τ + η) e s :=( 1 + 2 )e s .

(84)

Let us first remark that if in B 0 one chooses α 0 < 0 and α 1 = 0 (so that the boundary on ∂Γ 0 is strictly dissipative) then from ( 84

Conclusion and comments

In this article we show that to construct the geometric optics expansion associated to a hyperbolic boundary value problem defined in a strip a new inversibility condition has to be imposed (see Assumption 5.1). This condition involves the traces of the hyperbolic components of the geometric optics expansion and is automatically satisfied if during a full circuit of reflection the coefficients of reflection ensure that the energy descreases.

As a consequence, this condition meets the intuition that if after a full reflection the boundary conditions are such that the energy inscreases then as the full reflection are periodically repeated in time the associated ansatz should have an exponential growth in time (with some rate depending on the time needed to perform a full reflection).

Moreover the examples described in Paragraphs 9.1.1-9.1.2 seem to indicate that Assumption 5.1 is trivially (in the sense that T (ζ) is a contraction) satisfied for all strictly dissipative boundary conditions (for which the (lower exponential) strong well posedness of ( 19) is known to hold).

A point of interest is that in the expansions described so far the inversibility condition used to construct the WKB expansions does not involve the elliptic or the glancing parts of the ansatz. This point meets the intuition that these parts of the ansatz are linked to boundary layers so that they can not propagate the information from one side to the other and consequently they should behave like they do in the half space geometry.

However, in the author's opinion, this observation has an important counterpart. More precisely, in [Benoit, ] the author obtains a full characterization of lower exponentially strongly well posed problems (see Definition 7.1) in terms of new inversibility conditions involving the traces of the solution of each side of the strip. Nevertheless compared to the inversibility condition Assumption 5.1 one of the inversibility conditions used in [Benoit, ] differs by the following:

• firstly, as it is not at the microlocalized level, it has to hold uniformly in terms of the frequency parameter ζ ∈ Ξ \ Ξ 0 .

• Secondly this condition has to be imposed on the full stable subspace E s (ζ) and not only on the hyperbolic part of this space that is E s h (ζ) (note that by Hersh's lemma [Hersh, 1963] this space is empty for ζ ∈ Ξ \ Ξ 0 ).

The main issue with the characterization used in [Benoit, ] is its uniformity in terms of ζ ∈ Ξ \ Ξ 0 which seems really difficult to check in practice. To overcome this difficulty the natural strategy is to have a look to the boundary frequencies ζ ∈ Ξ 0 to obtain the uniform bound by compactness arguments (it is the classical method of [Kreiss, 1970]).

First let us remark that the extension of the condition made in [Benoit, ] to hyperbolic frequency ζ ∈ Ξ 0 is nothing but Assumption 5.1 for hyperbolic frequencies. Consequently Assumption 5.1 is a microlocalized version of the condition ensuring the lower exponential strong well posedness. This phenomenon already appeared for the geometric optics expansions of boundary value problems in the half space. So we believe that it is interesting to notice that such a situation also occurs in more complex geometries.

Secondly as pointed in [Benoit, ], the condition ensuring the lower exponential strong well posedness can not hold for glancing modes. So the fact that Assumption 5.1 only holds on E s h (ζ) seems to indicate that in fact in the extension to Ξ 0 only the hyperbolic part of the solution should be considered. So probably the extension of the characterization in [Benoit, ] up to Ξ 0 does not require any inversibility property on glancing modes. Meaning that it may be possible to extend the symmetrizor construction of [Benoit, ] up to Ξ 0 (except at glancing modes) to recover the uniformity of the bound. We expect to have further results about this conjecture in some forthcoming publications.

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 9.1.1 The wave equation in two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 9.1.2 A modification of the wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 9.2 Conclusion and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Figure 1 :

 1 Figure 1: Apperance of the source term induced phases.

Figure 3 :

 3 Figure 3: Phases in the WKB expansion and selfinteraction.

  (I -T (ζ)) and (I -T (ζ)) on the stable subspace E s (ζ) for (I -T (ζ)) and on ker B 0 for (I -T (ζ)), uniformly in terms of the frequency parameter ζ ∈ Ξ. That is we have that there exist C, C > 0 such that for all ζ ∈ Ξ ∀u ∈ E s (ζ), |u| ≤C|(I -T (ζ))u|, (71) ∀v ∈ ker B 0 , |v| ≤ C|(I -T (ζ))v|,

  EH = (τ, η) ∈ R 2 \ |η| > |τ | , H = (τ, η) ∈ R 2 \ |τ | > |η| , G = (τ, η) ∈ R 2 \ |η| = |τ | and E = ∅,consequently in the following we will assume that |τ | > |η| to be in the hyperbolic area. 9

  φ 1 (ζ) : C 2 → E u (ζ) φ 1 (ζ) :=With these expressions in hand it is easy to show that the operator T ε (ζ) applied to e s reads:T ε (ζ)e s = e 2 iξ ε ξ + τ + η -ξ + τ + η • -ξ + α 0 (τ + η) ξ + α 0 (τ + η)

Figure 4 :

 4 Figure 4: The set (in blue) of (x, y) ∈ R 2 such that y+x -y+x

This assumption is clearly not necessary at all. But we can easily show that if it is not satisfied, then the WKB expansion for (1) is the same as the one for the same problem in the half space {x d ≥ 0} and this case is of little interest.

Let us stress that it is not the case in the corner geometry, see[Benoit, 2016], for which the dispersion relation changes at each rebound. This explains why in the strip geometry, the phase generation process is not as rich as in the quarter space.

Remark that in (22) we take an arbitrary number of correctors for the non-glancing modes while we restrict the expansion to only one corrector for the glancing one. This choice is motivated by[Williams, 1996]-[Williams, 2000] and will be explain in Paragraphs

5.2 and 6.3.

The scaling ε -1/2 for the size of the boundary layers comes from[Williams, 1996] and is explained in Paragraph

6.3. It permits to construct a corrector for glancing modes such that the error in the interior in O(ε 1/4 ) in L 2 γ (Ω). Note that this is the sharpest possible error rate.

Note that compared to the boundary layer for glancing modes the size of the boundary layer for elliptic modes can be made independent on ε.satisfying the cascade of equations (23),(26),(27),(28) written for n = 0. satisfying the cascade of equations (23),(26),(27),(28) written for n = 0.

In fact in this setting as we only construct a first order corrector in (22) it is in fact sufficient to take g ∈ H 2 (∂Ω 0 ) to ensure that the u ε g,1,k ∈ H 1 γ (Ω) for all γ > γ 0 in such a way that the previous discussion makes sense.

Such γ 0 always exists because min k∈I , ∈O α k,l > 0

In this example, the mixed area is of little interest because A (ζ) has two elliptic roots and only one hyperbolic root so that the selfinteraction phenomenon can not occur.

Note that in Figure9.1.2 we make a crude estimate in the sense that we do not take into account the oscillating factors and the dependency of ξ < 0 with respect to (τ , η).

The author acknowledge financial support from the ANR research project NABUCO.