High frequency analysis of Helmholtz equations: case of two point sources

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High frequency analysis of Helmholtz equations: case of two point sources

Elise Fouassier

To cite this version:

Elise Fouassier. High frequency analysis of Helmholtz equations: case of two point sources. SIAM

Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2006, 38 (2),

pp.617-636. �10.1137/050629999�. �hal-00004803�

ccsd-00004803, version 1 - 26 Apr 2005

CASE OF TWO POINT SOURCES

ELISE FOUASSIER

Abstract. We derive the high frequency limit of the Helmholtz equation with source term when the source is the sum of two point sources. We study it in terms of Wigner measures (quadratic observables). We prove that the Wigner measure associated with the solution satisfies a Liouville equation with, as source term, the sum of the source terms that would be created by each of the two point sources taken separately. The first step, and main difficulty, in our study is the obtention of uniform estimates on the solution.

Then, from these bounds, we derive the source term in the Liouville equation together with the radiation condition at infinity satisfied by the Wigner measure.

AMS subject classifications. 35Q60, 35J05, 81S30.

1. Introduction

In this article, we are interested in the analysis of the high frequency limit of the following Helmholtz equation

(1.1) −iαε

ε u^ε+ ∆u^ε+n(x)²

ε² u^ε=S^ε(x), x∈R³ with

S^ε(x) =S₀^ε(x) +S^ε₁(x) = 1 ε³S0

x ε

+ 1 ε³S1

x−q1

ε whereq1 is a point inR³ different from the origin.

In the sequel, we assume that the refraction indexnis constant,n(x)≡1.

The equation (1.1) modelizes the propagation of a source wave in a medium with scaled refraction indexn(x)²/ε². There, the small positive parameterεis related to the frequencyω= _2πε¹ ofu^ε. In this paper, we study the high frequency limit, i.e.

the asymptoticsε→0. We assume that the regularizing parameterαεis positive, withαε→0 as ε→0. The positivity of αε ensures the existence and uniqueness of a solutionu^εto the Helmholtz equation (1.1) inL²(R³) for anyε >0.

The source termS^εmodels a source signal that is the sum of two source signals concentrating respectively close to the origin and close to the pointq1at the scale ε. The concentration profiles S0 and S1 are given functions. Since ε is also the scale of the oscillations dictated by the Helmholtz operator ∆ +_ε¹2, resonant inter- actions can occur between these oscillations and the oscillations due to the sources S₀^ε and S₁^ε. On the other hand, since the two sources are concentrating close to two different points in R³, one can guess that they do not interact when ε →0.

These are the phenomena that the present paper aims at studying quantitatively.

We refer to Section 3 for the precise assumptions we need on the sources.

In some sense, the sign of the term−iαεεu^ε prescribes a radiation condition at infinity foru^ε. One of the key difficulty in our problem is to follow this condition

1

in the limiting processε→0.

We study the high frequency limit in terms of Wigner measures (or semi-classical measures). This is a mean to describe the propagation of quadratic quantities, like the local energy density |u^ε(x)|², as ε → 0. The Wigner measure µ(x, ξ) is the energy carried by rays at the pointxwith frequencyξ. These measures were intro- duced by Wigner [14] and then developed by P. G´erard [6] and P.-L. Lions and T.

Paul [9] (see also the surveys [3] and [7]). They are relevant when a typical length ε is prescribed. They have already proven to be an efficient tool in the study of high frequencies, see for instance [2], [4] for Helmholtz equations, P. G´erard, P.A.

Markowich, N.J. Mauser, F. Poupaud [7] for periodic media, G. Papanicolaou, L.

Ryzhik [11] for a formal analysis of general wave equations, L. Erd¨os, H.T. Yau [5]

for an approach linked to statistical physics, and L. Miller [10] for a study in the case with sharp interface.

Such problems of high frequency limit of Helmholtz equations have been studied in Benamou, Castella, Katsaounis, Perthame [2] and Castella, Perthame, Run- borg [4]. In [2], the authors considered the case of one point source and a general index of refraction whereas in [4], they treated the case of a source concentrating close to a general manifold with a constant refraction index. In the present paper, we borrow the methods used in both articles.

In the case of one point source, for instanceS^ε₀only, with a constant index of re- fraction, it is proved in [2] that the corresponding Wigner measureµ0is the solution to the Liouville equation

0⁺µ0(x, ξ) +ξ· ∇^xµ0(x, ξ) =Q0(x, ξ) = 1

(4π)²δ(x)δ(|ξ|²−1)|Sc0(ξ)|², the term 0⁺ meaning thatµis the outgoing solution given by

µ0(x, ξ) = Z 0

−∞

Q0(x+tξ, ξ)dt.

In particular, the energy source created by S₀^ε is supported at x = 0. Similarly, the energy source created by the source S₁^ε is supported at x =q1. Thinking of the orthogonality property on Wigner measures, one can guess that the energy source generated by the sumS₀^ε+S₁^εis the sum of the two energy sources created asymptotically byS₀^ε andS₁^ε.

Indeed, we prove in this paper that the Wigner measureµ associated with the sequence (u^ε) satisfies

(1.2) 0⁺µ(x, ξ) +ξ· ∇xµ(x, ξ) =Q0+Q1,

where Q0 and Q1 are the source terms obtained in [2] in the case of one point source. However, our proof does not rest on the mere orthogonality property.

Let us now give some details about our proof. Our strategy is borrowed from [2].

First, we prove uniform estimates on the sequence of solutions (u^ε). We also study the limiting behaviour fo the rescaled solutions ε^d−12 u^ε(εx) and ε^d−12 u^ε(q1+εx).

The obtention of these first two results is the key difficulty in our paper. It relies

on the study of the sequence (a^ε) such that

−iαεεa^ε+ ∆a^ε+a^ε=S1

x−q1

ε .

Using the explicit formula for the Fourier tranform ofa^ε, we prove thata^ε is uni- formly bounded in a suitable space and thata^ε →0 as ε→0 weakly. We would like to point out that our analysis, based on a study in Fourier space, strongly rests on the assumption of a constant index of refraction.

Second, our results on the Wigner measure then follow from the properties proved in [2]. They are essentially consequences of the uniform bounds on (u^ε): we write the equation satisfied by the Wigner transform associated with (u^ε), and pass to the limitε→0 in the various terms that appear in this equation. The only difficult (and new) term to handle is the source term.

Third, we prove an improved version of the radiation condition of [2]. Our argu- ment relies on the observation thatµ is localized on the energy set {|ξ|² = 1}, a property that was not exploited in [2].

The paper is organized as follows. In Section 2, we recall some definitions and state our assumptions. Section 3 is devoted to the proof of uniform bounds on the sequence of solutions (u^ε) and of the convergence of the rescaled solutions. Then, in Section 4, we establish the transport equation satisfied by the Wigner measure µtogether with the radiation condition at infinity. In the appendix, we recall the proof of some results established in [2] that we use in our paper.

2. Notations and assumptions

In this section, we recall the definitions of Wigner transforms and of the B, B^∗ norms introduced by Agmon and H¨ormander [1] for the study of Helmholtz equations. Then, we give our assumptions.

2.1. Wigner transform and Wigner measures. We use the following definition for the Fourier transform:

ˆ

u(ξ) = (Fx→ξu)(ξ) = 1 (2π)³

Z

R^d

e^−ix·ξu(x)dx.

Foru, v∈ S(R³) andε >0, we define the Wigner transform W^ε(u, v)(x, ξ) = (Fy→ξ)(u x+ε

2y

¯ v x−ε

2y ), W^ε(u) = W^ε(u, u).

In the sequel, we denoteW^ε=W^ε(u^ε).

If (u^ε) is a bounded sequence in L²_loc(R^d), it turns out that (see [6], [9]), up to extracting a subsequence, the sequence (W^ε(u^ε)) converges weakly to a positive Radon measureµon the phase spaceT^∗R³=R³

x×R³

ξ called Wigner measure (or semiclassical measure) associated with (u^ε):

(2.1) ∀ϕ∈ C^∞c (R⁶), lim

ε→0hW^ε(u^ε), ϕi= Z

ϕ(x, ξ)dµ

We recall that these measures can be obtained using pseudodifferential opera- tors. The Weyl semiclassical operatora^W(x, εDx) (orOp^W_ε (a)) is the continuous

operator fromS(R^d) toS^′(R^d) associated with the symbola∈ S^′(T^∗R^d) by Weyl quantization rule

(2.2) (a^W(x, εDx)u)(x) = 1 (2π)^d

Z

R^d_ξ

Z

R^dy

a x+y

2 , εξ

f(y)e^i(x−y)·ξdξdy.

We have the following formula: for u, v∈ S^′(R^d) anda∈ S(R^d×R^d), (2.3) hW^ε(u, v), ai^S′,S =hv, a¯ ^W(x, εDx)¯ui^S′,S,

where the duality bracketsh., .iare semi-linear with respect to the second argument.

This formula is also valid foru, vlying in other spaces as we will see in Section 3.

2.2. Besov-like norms. In order to get uniform (in ε) bounds on the sequence (u^ε), we shall use the following Besov-like norms, introduced by Agmon and H¨orman- der [1]: foru, f ∈L²_loc(R³), we denote

kuk^B∗ = sup

j≥−1

2^−j Z

C(j)|u|²dx

!1/2

,

kfkB = X

j≥−1

2^j+1 Z

C(j)|f|²dx1/2

,

whereC(j) denotes the ring{x∈R³/2^j ≤ |x|<2^j+1}forj ≥0 andC(−1) is the unit ball.

These norms are adapted to the study of Helmholtz operators. Indeed, if v is the solution to

−iαv+ ∆v+v=f

where α >0, then Agmon and H¨ormander [1] proved that there exists a constant C independent ofαsuch that

kvkB^∗ ≤CkfkB.

Perthame and Vega [12] generalised this result to Helmholtz equations with general indices of refraction.

We denote forx∈R³,|x|=qP3

j=1x²_j andhxi= (1 +|x|²)^1/2. For allδ > ¹₂, we have

(2.4) kukL²_−δ :=khxi^−δukL²≤C(δ)kukB^∗.

We end this section by stating two properties of these spaces that will be useful for our purpose (the reader can find the proofs in [1]). The first proposition states that, in some sense, we can define the trace of a function inBon a linear manifold of codimension 1.

Proposition 2.1. There exists a constant C such that for allf ∈B, we have Z

R

kf(x1, .)kL²(R²)dx1≤CkfkB.

The second property gives the stability of the spaceB by change of variables in Fourier space.

Proposition 2.2. Let Ω1, Ω2 be two open sets in R³, ψ : Ω1 → Ω2 a C² diffeo- morphism,χ∈ Cc¹(R³). For all u∈B, we denote

T u=F⁻¹ χ(ˆu◦ψ) . Then

kT ukB≤CkχkC¹_bkψkC²_bkukB.

2.3. Assumptions. We are now ready to state our assumptions. Our first assump- tion, borrowed from [2], concerns the regularizing parameterαε>0.

(H1)αε≥ε^γ for someγ >0.

This assumption is technical and is used to get a radiation condition at infinity in the limitε→0. Next, in order to get uniform bounds onu^ε, we assume that the source termsS0andS1belong to the natural Besov space that is needed to actually solve the Helmholtz equation (1.1).

(H2)kS0k^B,kS1k^B<∞.

It turns out that, in order to compute the limit of the energy source, we shall need the stronger assumption

(H3)hxi^NS0∈L²(R³) andhxi^NS1∈L²(R³) for someN > ¹₂+_γ+1^3γ .

3. Bounds on solutions to Helmholtz equations

In this section, we first establish uniform bounds on the sequence (u^ε) that will imply estimates on the sequence of Wigner transforms (W^ε). It turns out that we shall also need to compute the limit of the rescaled solutions w^ε₀ and w^ε₁ defined below in order to obtain the energy source in the equation satisfied by the Wigner measureµ.

Before stating our two results, let us define these rescaled solutions. Following [2]

and [4], we denote (3.1)

( w₀^ε(x) = ε^d−21u^ε(εx), w₁^ε(x) = ε^d−12 u^ε(q1+εx).

They respectively satisfy

−iαεεw^ε₀+ ∆w^ε₀+w^ε₀ = S0(x) +S1 x−^q_ε¹ ,

−iαεεw^ε₁+ ∆w^ε₁+w^ε₁ = S0 x+^q_ε¹

+S1(x).

We are ready to state our results onu^ε,w₀^εandw^ε₁.

Proposition 3.1. Assume S0, S1 ∈ B. Then, the solution u^ε to the Helmholtz equation (1.1) satisfies the following bound

ku^εkB^∗ ≤C(kS0kB+kS1kB), whereC is a constant independent ofε.

Proposition 3.2. Letw^ε₀ andw₁^ε be the rescaled solutions defined by (3.1). Then, the sequences(w^ε₀)and(w₁^ε)are uniformly bounded inB^∗and they converge weakly-

∗ inB^∗ to the outgoing solutionsw0 andw1 to the following Helmholtz equations ∆w0+w0=S0

∆w1+w1=S1,

i.e. w0 andw1 are given in Fourier space by c

wj(ξ) = −Scj(ξ)

|ξ|²−1 +i0 =−

p.v. 1

|ξ|²−1

+iπδ(|ξ|²−1)

Scj(ξ), j= 0,1.

Remark. The Helmholtz equation ∆w+w=S does not uniquely specify the solutionw. An extra condition is necessary, for instance the Sommerfeld radiation condition. When the refraction index is constant equal to 1, this condition writes

(3.2) lim

r→∞

1 r

Z

Sr

∂w

∂r +iw²dσ= 0.

Such a solution is called an outgoing solution.

Alternatively, still assuming that the refraction index is constant, the outgoing solution to the Helmholtz equation may be defined as the weak limit w of the sequence (w^δ) such that

−iδw^δ+ ∆w^δ+w^δ =S(x).

We point out that the two points of views are equivalent in the case of a constant index of refraction (which is not true for a general index of refraction).

We prove the two propositions in the following two sections. As we will see in the proofs, our main difficulties are linked to the rays that are emitted by the source at 0 towards the point q1 (and conversely). Hopefully, the interaction between those rays is ”destructive” and not constructive.

3.1. Proof of Proposition 3.1. In the sequel,C will denote any constant inde- pendent ofε.

The scaling invariance

ku^εk^B∗ ≤ kw^ε₀k^B∗,

makes it sufficient to prove bounds onw₀^ε. Sincew^ε₀ is a solution to

−iαεεw^ε₀+ ∆w^ε₀+w^ε₀=S0(x) +S1 x−q1

ε we may decomposew₀^ε=wf^ε₀+a^ε, wherewf^ε₀ anda^εsatisfy

−iαεεwf^ε₀+ ∆wf^ε₀+wf^ε₀ = S0(x),

−iαεεa^ε+ ∆a^ε+a^ε = S1 x−^q_ε¹ .

First, we note that the boundkwf^ε₀k^B∗ ≤CkS0k^B is established in Agmon-H¨orman- der [1] (see also Perthame-Vega [12]). Hence, the proof of Proposition 3.1 reduces to the proof of the following lemma.

Lemma 3.3. If a^εis the solution to

−iαεεa^ε+ ∆a^ε+a^ε=S1 x−q1

ε thena^ε is uniformly (in ε) bounded inB^⋆:

ka^εkB^∗ ≤CkS1kB

Proof. We want to prove that

∀v∈B, |ha^ε, vi| ≤CkS1k^Bkvk^B. Using Parseval’s equality, we write

(3.3) ha^ε, vi=

Z

R³

e^−iq1·ξε Sc1(ξ)¯v(ξ)ˆ

−|ξ|²+ 1−iεαεdξ.

To estimate this integral, we shall distinguish the values of ξ close to or far from two critical sets: the sphere {|ξ|² = 1} (the set where the denominator in (3.3) vanishes when ε → 0) and the line {ξ collinear to x0} (the set where we cannot apply directly the stationary phase theorem to (3.3)).

More precisely, we first take a small parameter δ∈]0,1[, and we distinguish in the integral (3.3), the contributions due to the values of ξ such that|ξ²−1| ≥ δ or |ξ²−1| ≤ δ. Let χ ∈ Cc^∞(R) be a truncation function such that χ(λ) = 0 for

|λ| ≥1. We denoteχδ(ξ) =χ ^|ξ|

2−1 δ

. We accordingly decompose

ha^ε, vi = Z

R³

e^−iq1ε·ξSc1(ξ)¯ˆv(ξ)χδ(ξ)

−|ξ|²+ 1−iεαε

dξ+ Z

R³

e^−iq1ε·ξcS1(ξ)¯ˆv(ξ)(1−χδ(ξ))

−|ξ|²+ 1−iεαε

dξ

= I^ε+II^ε.

First, since the denominator is not singular on the support ofχδ, we easily bound the first part with theL² norms

|I^ε| ≤ kχk^L∞

δ kSc1kL²kˆvkL², and usingB ֒→L², we obtain the desired bound (3.4) |I^ε| ≤CkS1kBkvkB.

Let us now study the second partII^εwhere the denominator is singular. Up to a rotation, we may assumeq1=|q1|e1, wheree1 is the first vector of the canonical base. We make the polar change of variables

ξ=





rsinθcosϕ rsinθsinϕ rcosθ

.

Remark. In order to make the calculations easier, we write this paper in di- mension equal to 3, but the proof would be similar in any dimensiond≥3.

Hence,q1·ξ=|q1|rsinθcosϕ, and we get II^ε=

Z e^{−i|q1|ε rsinθcosϕ}

−r²+ 1−iεαε

cS1v(1¯ˆ −χδ)

(ξ(r, θ, ϕ))r²sinθdrdθdϕ

Now, we distinguish the contributions to the integraldθdφlinked to the values close to, or far from, the critical direction{θ= ^π₂, ϕ= 0}(which corresponds to the case {ξcollinear toq1}). To that purpose, letη >0 be a small parameter and denote

K=

(r, θ, φ)

1−χr²−1 δ

6= 0, χ θ−^π2

η

6

= 0, χϕ η

6= 0

.

ThenKis a compact set. Letk∈ Cc^∞be such that (1−χδ)k(θ, ϕ) is a localization function onK. We write

II^ε =

Z e^{−i|q1|ε rsinθcosϕ}

−r²+ 1−iεαε

(cS1v)(ξ(r, θ, ϕ))(1¯ˆ −χδ(r))k(θ, ϕ)r²sinθdrdθdϕ +

Z e^{−i|q1|ε rsinθcosϕ}

−r²+ 1−iεαε

(cS1v)(ξ(r, θ, ϕ))(1¯ˆ −χδ(r))(1−k(θ, ϕ))r²sinθdrdθdϕ II^ε = III^ε+IV^ε.

To estimate the contributionIII^ε, we apply the stationary phase method. We denoteα=θ−^π2. The phase function isgr(α, ϕ) =rcosαcosϕso

∂gr

∂α = −rsinαcosϕ = 0 at (α, ϕ) = (0,0),

∂gr

∂ϕ = −rcosαsinϕ = 0 at (α, ϕ) = (0,0), and the Hessian at the point (0,0) is

D²gr(0,0) =

−r 0 0 −r

,

which is invertible at any point inK. SinceK is a compact set, we can apply the Morse lemma: there exists a finite covering (Ωj)j=1,n (n ∈N) ofK such that on each set Ωj, there exists aC^∞ change of variables (α, ϕ)7→(αj, ϕj) such that

gr(α, ϕ) =r−rα²_j 2 −rϕ²_j

2 . Moreover, we can write (1−χδ)k=Pn

j=1χj whereχj ∈ Cc^∞ and supp(χj)⊂Ωj. Then, we make the changes of variables α^′_j = p_r

2αj, ϕ^′_j = p_r

2ϕj. Finally, we decomposeχj=χ¹_jχ²_j. Thus, we obtain, for the contributionIII^ε, the formula (3.5) III^ε=

Xn j=1

Z e^{ix1ε (−r+α′j2+ϕ′j2)}

−r+ 1 +iεαε

T[_j¹S1(r, α^′_j, ϕ^′_j)Td_j²v(r, α^′_j, ϕ^′_j)drdα^′_jdϕ^′_j, where

T_j¹S1 := F

(χ¹_jSc1)◦ξ(r, α(αj, ϕj), ϕ(αj, ϕj)) , T_j²v := F

−r+ 1 +iεαε

−r²+ 1−iεαε

(χ²_jˆv)◦ξ(r, α(αj, ϕj), ϕ(αj, ϕj))

×2 r dξ

d(r, α, ϕ)

d(α, ϕ) d(αj, ϕj)

.

As a first step, using Proposition 2.2, we directly getT_j¹S1∈B with kT_j¹S1kB≤CkS1kB.

As a second step, we studyT_j²v. Since forrclose to 1,

−r+ 1 +iεαε

−r²+ 1−iεαε

≤1, we recover, from Proposition 2.2,

T_j²v∈B and kT_j²vk^B ≤Ckvk^B.

Now, we apply Parseval’s equality with respect to the r variable in the formula (3.5) III^ε =

Xn j=1

Z e^{i|q1|ε (α′2j+ϕ′2j)}

−r+ 1 +iεαε

\ T_j¹S1(.−|q1|

ε , ., .)Td_j²vdrdα^′_jdϕ^′_j

= Z

e^{i|q1|ε (α′j2+ϕ′j2)}1{ρ>0}e^{−(εαε−i)t}F^r→ρ(

\ T_j¹S1(.−|q1|

ε , ., .))(ρ−t, α^′_j, ϕ^′_j)

×Fr→ρ(Td_j²v)(ρ, α^′_j, ϕ^′_j)dtdρdα^′_jdϕ^′_j. where1{ρ>0} denotes the characteristic function of the set {ρ >0}. Hence, we obtain

|III^ε| ≤ Xn j=1

Z

kF^r→ρ(

\ T_j¹S1(.−|q1|

ε , ., .))(ρ))kL²dρ

!

× Z

kF^r→ρ(Td_j²v)(ρ)kL²dρ

|III^ε| ≤ Xn j=1

Z kT_j¹S1(ρ−|q1|

ε )kL²dρ Z

kT_j²v(ρ)kL²dρ

|III^ε| ≤ Xn j=1

Z kT_j¹S1(ρ)kL²dρ Z

kT_j²v(ρ)kL²dρ

|III^ε| ≤ C Xn j=1

kT_j¹S1kBkT_j²vkB

|III^ε| ≤ CkS1k^Bkvk^B, which is the desired estimate.

We are left with the part IV^ε, which corresponds to the directions ξ that are not collinear toq1. We denoteK^′the support of (1−χδ)(1−k) which is a compact set. InK^′, we can choose as new independent variables

η1=−q1·ξ, η2=|ξ|²−1.

More precisely, since

d(η1, η2)

dξ =

−q1

2ξ

is of maximal rank 2, there exists a finite covering (Ω^′_j)j=1,m (m∈N) of K^′ such that in Ω^′_j, we can make the change of variables ξ 7→ η. As before, we denote χ^′_j =χ³_jχ⁴_j some localization functions on Ω^′_j such that (1−χδ)(1−k) =Pm

j=1χ^′_j. Thus, forj= 1, . . . , m,

Z e^−iq1ε·ξ

−|ξ|²+ 1 +iεαε

Sc1vχˆ ^′_jdξ=

Z e^iη2ε

−η1+iεαε

(cS1vχˆ ^′_j)(ξ(η))dξ dη dη.

If we denote

T_j³S1 := F⁻¹((χ³_jSc1)◦ξ), T_j⁴v := F⁻¹ (χ⁴_jv)ˆ ◦ξ)dξ

dη ,

and ifF¹denotes the Fourier transform with respect to theη1variable, Parseval’s equality with respect toη1 gives

Z e^−iq1ε·ξ

−|ξ|²+ 1 +iεαε

cS1vχˆ ^′_jdξ

= (2π)^d Z

χ{t>0}e^−εαεt(F1⁻¹(T[_j³S1))(x1−t)

×(F1⁻¹(Td_j⁴v))(x1)e^iη2/εdtdx1dη2dη3

≤ CkS1k^Bkvk^B. Summing overj, we obtain

|IV^ε| ≤CkS1kBkvkB, which ends the proof of the bound

|ha^ε, vi| ≤CkS1kBkvkB.

3.2. Proof of Proposition 3.2. We prove the result for the sequence (w₀^ε), the convergence of the sequence (w^ε₁) can be obtained similarly. As we did in the proof of Proposition 3.1, we writew^ε₀=wf₀^ε+a^ε. Sincewf₀^εis the solution to a Helmholtz equation with constant index of refraction and fixed source, it converges weakly-∗ to the outgoing solutionw0to ∆w+w=S0. Hence, it suffices to show the following result.

Lemma 3.4. If a^ε∈B^⋆ is the solution to

−iαεεa^ε+ ∆a^ε+a^ε=S1 x−q1

ε

thena^ε→0 inB^⋆.

Proof. The proof of this result requires two steps (using a density argument):

(1) for v∈B, we have the boundha^ε, vi≤CkS1k^Bkvk^B (2) if S1and vare smooth, thenha^ε, vi →0.

The first point is exactly the result in Lemma 3.3. It remains to prove the conver- gence in the smooth case (the second point above).

We write

ha^ε, vi= Z

R³

e^−iq1ε·ξcS1(ξ)¯v(ξ)ˆ

−|ξ|²+ 1−iεαεdξ.

We are thus left with the study of

(3.6) Rε(ψ) =

Z

R³

e^−iq1ε·ξψ(ξ)

−|ξ|²+ 1−iεαε

dξ whereψ=Sc1vbbelongs toS(R³).

As in the proof of Lemma 3.3, we distinguish the contributions of various values ofξ. We shall use exactly the same partition, according to the values ofξclose to, or far from, the sphere |ξ| = 1 and collinear or not to q1. We shall use the same

notations for the various truncation functions.

We first separate the contributions ofξ such that|ξ²−1| ≤δand|ξ²−1| ≥δ Rε(ψ) =

Z

R³

e^−iq1ε·ξψ(ξ)χδ(ξ)

−|ξ|²+ 1−iεαε

dξ+ Z

R³

e^−iq1ε·ξψ(ξ)(1−χδ(ξ))

−|ξ|²+ 1−iεαε

dξ

= I^ε+II^ε.

In the support of χδ, since the denominator is not singular, we can apply the non stationary phase method.

Sinceq16= 0, we may assumeq¹₁6= 0 and we have I^ε = ε

iq₁¹ Z

R³

e^−iq1ε·ξ∂ξ1

ψ(ξ)χδ(ξ)

−|ξ|²+ 1−iεαε

dξ

= ε

iq₁¹ Z

R³

e^−iq1ε·ξ

∂ξ1(ψ(ξ)χδ(ξ))

−|ξ|²+ 1−iεαε− 2ψ(ξ)χδ(ξ)ξ1

(−|ξ|²+ 1−iεαε)²

. Hence, we obtain the bound

|I^ε| ≤ ε

|q¹₁| Z

R³

1

δ|∂ξ1(χψ)|+ 2

δ²|ξ1χψ| dξ.

Since∂ξ1(χψ) andξ1χψ belongs toS, we have, asε→0, I^ε→0.

Let us now study the second termII^ε. We use the same changes of variables as in Section 3.1. It leads to the following formula

II^ε= Xn j=1

Z e^{−iq1ε(r−α′j2−ϕ′j2)}

−r+ 1 +iεαε fχj(r, α^′_j, ϕ^′_j)ψ(r, αe ^′_j, ϕ^′_j)drdα^′_jdϕ^′_j, where

f

χj(r, α^′_j, ϕ^′_j) = χj◦ξ(r, α(α^′_j, ϕ^′_j), ϕ(α^′_j, ϕ^′_j))2(−r+ 1 +iεαε) r(−r²+ 1−iεαε)

d(α, ϕ) d(αj, ϕj)

, ψ(r, αe ^′_j, ϕ^′_j) = ψ◦ξ(r, α(α^′_j, ϕ^′_j), ϕ(α^′_j, ϕ^′_j)),

are still smooth functions that are bounded independently fromε.

Using Parseval’s inequality with respect to the variables (α^′_j, ϕ^′_j) for each integral, we obtain the bound

|II^ε| ≤Cε Xn j=1

Z e^{−i|q1|ε r}e^{−iε(λ2j+µ2j)}

−r+ 1 +iεαε F^λj,µj(fχjψ)drdλe jdµj

.

To obtain the convergence ofII^ε, it remains to study an integral of the following

type Z

|r−1|≤δ

e^{−i|q1|ε r}w(r)

−r+ 1 +iεαε

dr, wherew∈ S. This is done in the following lemma.

Lemma 3.5. ∀w∈ S, ∀θ∈ (0,1), we have Z

|r|≤δ

e^{−i|q1|ε r}w(r)

−r+iεαε

dr=−iπw(0) +Oε→0(ε^−θ).

Using this lemma, we readily get the estimate

(3.7) |III^ε| ≤Cε^1−θ ∀θ∈ (0,1), which proves thatIII^ε→0 asε→0.

There remains to give the Proof of Lemma 3.5. We write

Z δ

−δ

e^{−i|q1|ε r}w(r)

−r+iεαε

dr = Z δ

−δ

e^{−i|q1|ε r}w(r)

r²+ (εαε)²(r−iεαε)dr

= −iεαε

Z δ

−δ

e^{−i|q1|ε r}w(r) r²+ (εαε)²dr+

Z δ

−δ

e^{−i|q1|ε r} rw(r) r²+ (εαε)²dr

= I+II.

We have

I=−i Z _εαε^δ

−_εαε^δ

e^{−i|q1|αεy}w(εαεy)

y²+ 1 dy→ −iπw(0), and

II= Z δ

−δ

e^{−i|q1|ε r}w(r)−w(0) r

r²+ (εαε)²dr+ Z δ

−δ

w(0) r

r²+ (εαε)²dr . The last term vanishes because the integrand is odd. Moreover, using the smooth- ness ofw, we easily obtain that for allθ∈ (0,1),

e^{−i|q1|ε r}w(r)−w(0)≤Cθ

r ε

^θ Thus,

Z δ

−δ

e^{−i|q1|ε r}w(r)−w(0) r r²+ (εαε)²dr

≤ C

ε^θ Z δ

−δ|r|^θ−1dr and the result is proved.

We are left with the study ofIV^ε. We use the same change of variables as in Section 3.1.

IV^ε = Xm j=1

Z e^−iq1ε·ξ

−|ξ|²+ 1 +iεαε

ψ(ξ)χj(ξ)dξ

= Xm j=1

Z e^iη2ε

−η1+iεαε

(ψχj)(ξ(η))dξ dη dη

= iε Xm j=1

Z e^iη2ε

−η1+iεαε∂η1

(ψχj)(ξ(η))dξ dη

dη.

The integral obviously converges with respect to all the variables exceptη1. It re- mains to prove the convergence with respect to theη1variable, i.e. the convergence

of Z φ(η)

−η1+iεαεdη1, where

φ=∂η1

(ψχj)(ξ(η))dξ dη

is smooth and compactly supported with respect to η. It is a consequence of the fact that the distribution (x+i0)⁻¹is well-defined onRby

1

x+i0 =v.p.(1

x)−iπδ(x).

We conclude thatIV^ε→0 andha^ε, vi →0 asε→0.

4. Transport equation and radiation condition on µ

In this section, we state and prove our results on the Wigner measure associated with (u^ε). Since we established the uniform bounds on (u^ε) and the convergence of (w₀^ε), (w^ε₁), these results now essentially follows from the results proved in [2].

We first prove bounds on the sequence of Wigner transforms (W^ε) that allow us to define a Wigner measureµassociated to (u^ε). Then, we get the transport equation satisfied by µ together with the radiation condition at infinity, which uniquely determinesµ.

4.1. Results.

Theorem 4.1. Let S0, S1 ∈ B and λ > 0. The sequence (W^ε) is bounded in the Banach spaceX_λ^⋆ and up to extracting a subsequence, it converges weak-⋆ to a positive and locally bounded measureµsuch that

(4.1) sup

R>0

1 R

Z

|x|<R

Z

ξ∈R³

µ(x, ξ)dxdξ ≤C(kS0kB+kS1kB)².

The Banach spaceX_λ^∗ is defined as the dual space of the setXλof functionsϕ(x, ξ)ˆ such that ϕ(x, y) :=F^ξ→y( ˆϕ(x, ξ))satisfies

(4.2)

Z

R^d

sup

x∈R^d

(1 +|x|+|y|)^1+λ|ϕ(x, y)|dy <∞.

Theorem 4.2. Assume (H1), (H2), (H3). Then the Wigner measureµasssociated with(u^ε)satisfies the following transport equation

(4.3) ξ· ∇xµ= 1 (4π)²

δ(x)|cS0(ξ)|²+δ(x−q1)|Sc1(ξ)|²

δ(|ξ|²−1) :=Q(x).

Moreover, µ is the outgoing solution to the equation (4.3) in the following sense:

for all test functionR∈ Cc^∞(R⁶), if we denote g(x, ξ) =R∞

0 R(x−ξt, ξ)dt, then (4.4)

Z

R⁶

R(x, ξ)dµ(x, ξ) =− Z

R⁶

Q(x, ξ)g(x, ξ)dxdξ.

Remark. Here the support of the test functionR contains 0, contrary to [2].

4.2. Proof of Theorem 4.1. This theorem, that is proved in [2], is a consequence of the uniform estimate on the sequence (u^ε) in the spaceB^∗obtained in Proposition 3.1. We observe that for anyλ >0,

(4.5) khxi^−12−λu^ε(x)kL² ≤Cku^εk^B∗ ≤Ckfk^B,

hence, for any functionϕsatisfying (4.2), we have

|hW^ε(u^ε),ϕˆi|

≤ Z

R⁶

|u^ε|(x+^ε₂y)|u^ε|(x−^ε2y) hx+^ε₂yi¹²⁺⁰hx−₂^εyi¹²⁺⁰hx+ε

2yi¹²⁺⁰hx−ε

2yi¹²⁺⁰|ϕ|(x, y)dxdy

≤Ckfk²B

Z

R³

sup

x∈R³h|x|+|y|i¹⁺⁰|ϕ(x, y)|dy.

So (W^ε(u^ε)) is bounded in X_λ^∗,λ >0. We deduce that, up to extracting a subse- quence, (W^ε(u^ε)) converges weak-∗to a nonnegative measureµsatisfying

(4.6) |hµ,ϕˆi| ≤Ckfk²B

Z

R³

sup

x∈R³

h|x|+|y|i¹⁺⁰|ϕ(x, y)|dy.

We refer for instance to Lions, Paul [9] for the proof of the nonnegativity ofµ.

The bound (4.1) is obtained using the following family of functions ϕ^R_µ(x, y) = 1

µ^3/2e^−|y|2/µ1

Rχ(hxi ≤R)

and lettingµ→0,R→ ∞.

4.3. Proof of the transport equation 4.3. This section is devoted to the proof of the transport equation satisfied byµ. We first write the transport equation satisfied by W^ε in a dual form. Then, we study the convergence of the source term (the convergence of the other terms is obvious). Finally, choosing an appropriate test function in the limiting process, we get the radiation condition at infinity satisfied by µ. Proving first a localization property, we improve the radiation condition proved in [2].

4.3.1. Transport equation satisfied by W^ε. W^ε satisfies the following equation (4.7) αεW^ε+ξ· ∇xW^ε= iε

2Im W^ε(f^ε, u^ε) :=Q^ε. This equation can be obtained writing first the equation satisfied by

v^ε(x, y) =u^ε x+ε 2y

u^ε x−ε 2y

. From the equality

∇y· ∇xv^ε= ε 2

∆u^ε(x+ε

2y)u^ε(x−ε

2y)−∆u^ε(x−ε

2y)u^ε(x+ε 2y)

, we deduce

αεv^ε+i∇^y· ∇^xv^ε+ i 2ε

n²(x+ε

2y)−n²(x−ε 2y)

v^ε=σε(x, y), where

σε(x, y) := iε 2

S^ε(x+ε

2y)u^ε(x−ε

2y)−S^ε(x−ε

2y)u^ε(x+ε 2y)

. After a Fourier transform, we obtain the equation (4.7).

Then we write the dual form of this equation. Letψ∈ S(R⁶), we have (4.8) αεhW^ε, ψi − hW^ε, ξ· ∇^xψi=hQ^ε, ψi.

By the definition of the Wigner measureµ, we get

αεhW^ε, ψi →0 and hW^ε, ξ· ∇^xψi → hµ, ξ· ∇^xψi.

Hence we are left with the study of the source termhQ^ε, ψi.

4.3.2. Convergence of the source term. In order to compute the limit of the source term in (4.7), we develop

hQ^ε, ψi=iε 2Im

hW^ε(S₀^ε, u^ε), ψi+hW^ε(S₁^ε, u^ε), ψi . Thus, the result is contained in the following proposition.

Proposition 4.3. The sequences εW^ε(S₀^ε, u^ε)

and εW^ε(S₁^ε, u^ε)

are bounded in S^′(R⁶)and for allψ∈ S(R⁶), we have

ε→0limεhW^ε(S₀^ε, u^ε), ψi^S′,S = 1 (2π)³

Z

R³

c

w0(ξ)cS0(ξ)ψ(0, ξ)dξ, (4.9)

ε→0limεhW^ε(S₁^ε, u^ε), ψiS^′,S = 1 (2π)³

Z

R³wc1(ξ)cS1(ξ)ψ(q1, ξ)dξ, (4.10)

wherew0 andw1 are defined in Proposition 3.2.

Using Proposition 4.3, we readily get

ε→0limhQ^ε, ψi = i 2(2π)³Im

Z

R³wc0(ξ)cS0(ξ)ψ(0, ξ)dξ+ Z

R³wc1(ξ)cS1(ξ)ψ(q1, ξ)dξ

= 1

(4π)² Z

R³|cS0(ξ)|²δ(ξ²−1)ψ(0, ξ)dξ +

Z

R³|cS1(ξ)|²δ(ξ²−1)ψ(q1, ξ)dξ

, which is the result in Theorem 4.2.

Let us now prove Proposition 4.3.

Proof of Proposition 4.3. The two terms to study being of the same type, we only consider the first one in our proof. Letψ∈ S(T^∗R^d) andϕ(x, y) =Fy→ξ⁻¹ (ψ(x, ξ)), then we have

εhW^ε(S₀^ε, u^ε), ψiS^′,S = ε Z

S₀^ε x+ε 2y

u^ε x− ε 2y

ϕ(x, y)dxdy

= Z

S0(x)w^ε₀(x+y)ϕ(ε(x+y

2), y)dxdy.

Hence, using thatψ∈ S(R^2d), we get εhW^ε(S₀^ε, u^ε), ψi^S′,S ≤ C

Z

hxi^N|S0(x)||w^ε₀(x+y)| hx+yi^β

hx+yi^β hxi^Nhyi^kdxdy

≤ Ckhxi^NS0kL²kw^ε₀k^B∗ Z

R³y

sup

x∈R³

hx+yi^β hxi^Nhyi^kdy for anyk≥0 andβ >1/2, upon using the Cauchy-Schwarz inequality inx.

Then, we distinguish the cases |x| ≤ |y| and |x| ≥ |y| : the term stemming from the first case gives a contribution which is bounded byCR dy

hyi^k−β and the second contribution is bounded byCR dy

hyi^k. So, upon choosingklarge enough, we obtain that

|εhW^ε(S₀^ε, u^ε), ψi^S′,S| ≤Ckhxi^NS0kL²kw^εk^B∗.

Now, in order to compute the limit (4.9), we write εhW^ε(S₀^ε, u^ε), ψi =

Z

S0(x)w₀^ε(x+y)

ϕ ε

x+y 2

, y

−ϕ(0, y)

dxdy +

Z

w^ε₀(x)S0(x−y)ϕ(0, y)dxdy

= Iε+IIε.

Reasonning as above, we readily get that limε→0Iε= 0. For the second term, we have

IIε= Z

w^ε₀(x) S0∗ϕ(0, .) (x)dx,

hence, sincew^ε₀converges weakly-∗inB^∗, it suffices to prove thatS0∗ϕ(0, .) belongs toB. We denoteφ=ϕ(0, .). We have, forβ >1/2,

kS0∗φk^B ≤ CkS0∗φkL²_β =C Z

hxi^β|S0∗φ(x)|²dx

≤ CkφkL¹

Z

hxi^β|S0|²∗ |φ|(x)dx where we used the Cauchy-Schwarz inequality. Hence, we get

kS0∗φk^B ≤Ckhxi^NS0kL²

Z

R³y

sup

x∈R³

hx+yi^β hxi^Nhyi^kdy,

for any k. As before, this integral converges. Thus, we have established that S0∗ψ(0, .) belongs tob B, which implies that

IIε→ Z

S0(x)w0(x+y)ψ(0, y)dxdy.b

4.4. Proof of the radiation condition (4.4). It remains to prove thatµsatisfies the weak radiation condition (4.4).

4.4.1. Support of µ. In order to prove the radiation condition without restriction on the test functionR(as assumed in [2]), we first prove a localization property on the Wigner measureµ. This property is well-known whenu^ε satisfies a Helmholtz equation without source term. It is still valid here thanks to the scaling ofS^ε. Proposition 4.4. Under the hypotheses (H1), (H2), (H3), the Wigner measureµ satisfies

supp(µ)⊂ {(x, ξ)∈R⁶/ |ξ|²= 1}.

Proof. Let φ ∈ Cc^∞(R⁶) and φ^ε = φ^W(x, εDx). Let us denote H^ε = −ε²∆−1.

Sinceu^εsatisfies the Helmholtz equation (1.1), we have (4.11) iαεεu^ε+H^εu^ε=ε²S^ε.

Moreover,H^εis a pseudodifferential operator with symbol|ξ|²−1. By pseudodif- ferential calculus,φ^εH^ε=Op^W_ε (φ(x, ξ)(|ξ|²−1)) +O(ε) so, using the definition of the measureµ, we get that

ε→0lim(φ^εH^εu^ε, u^ε) = lim

ε→0(Op^W_ε (φ(x, ξ)(|ξ|²−1))u^ε, u^ε)

= Z

φ(x, ξ)(|ξ|²−1))dµ.