0504536v1

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arXiv:math/0

504536v1

[math.A

P]26Apr2005

HIGH FREQUENCY ANALYSIS OF HELMHOLTZ EQUATIONS:CASE OF TWO POINT SOURCES

ELISE FOUASSIER

Abstract. We derive the high frequency limit of the Helmholtz equation with sourceterm when the source is the sum of two point sources. We study it in terms of Wignermeasures (quadratic observables). We prove that the Wigner measure associated with thesolution satisfies a Liouville equation with, as source term, the sum of the source termsthat 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 togetherwith 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 thefollowing Helmholtz equation

(1.1) i

u + u +n(x)2

2u = S(x), x R3

with

S(x) = S0 (x) + S1(x) = 13 S0x+ 13 S1x q1

where q1 is a point in R3 different from the origin.In the sequel, we assume that the refraction index n is constant, n(x) 1.

The equation (1.1) modelizes the propagation of a source wave in a medium withscaled refraction index n(x)2/2. There, the small positive parameter is related tothe frequency = 12 of u

. In this paper, we study the high frequency limit, i.e.the asymptotics 0. We assume that the regularizing parameter is positive,with 0 as 0. The positivity of ensures the existence and uniquenessof a solution u to the Helmholtz equation (1.1) in L2(R3) for any > 0.

The source term S models a source signal that is the sum of two source signalsconcentrating respectively close to the origin and close to the point q1 at the scale. The concentration profiles S0 and S1 are given functions. Since is also the

scale of the oscillations dictated by the Helmholtz operator + 12 , resonant inter-actions can occur between these oscillations and the oscillations due to the sourcesS0 and S

1. 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 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 iu prescribes a radiation condition atinfinity for u. One of the key difficulty in our problem is to follow this condition

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2 ELISE FOUASSIER

in the limiting process 0.

We study the high frequency limit in terms of Wigner measures (or semi-classicalmeasures). This is a mean to describe the propagation of quadratic quantities, likethe local energy density |u(x)|2, as 0. The Wigner measure (x, ) is theenergy carried by rays at the point x with frequency . These measures were intro-duced by Wigner [14] and then developed by P. Gerard [6] and P.-L. Lions and T.Paul [9] (see also the surveys [3] and [7]). They are relevant when a typical length is prescribed. They have already proven to be an efficient tool in the study ofhigh frequencies, see for instance [2], [4] for Helmholtz equations, P. Gerard, 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. Erdos, H.T. Yau [5]for an approach linked to statistical physics, and L. Miller [10] for a study in thecase with sharp interface.

Such problems of high frequency limit of Helmholtz equations have been studiedin Benamou, Castella, Katsaounis, Perthame [2] and Castella, Perthame, Run-borg [4]. In [2], the authors considered the case of one point source and a generalindex of refraction whereas in [4], they treated the case of a source concentratingclose 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 instance S0 only, with a constant index of re-fraction, it is proved in [2] that the corresponding Wigner measure 0 is the solutionto the Liouville equation

0+0(x, ) + x0(x, ) = Q0(x, ) = 1(4)2

(x)(||2 1)|S0()|2,the term 0+ meaning that is the outgoing solution given by

0(x, ) =

0

Q0(x + t,)dt.

In particular, the energy source created by S0 is supported at x = 0. Similarly,the energy source created by the source S1 is supported at x = q1. Thinking ofthe orthogonality property on Wigner measures, one can guess that the energysource generated by the sum S0 + S

1 is the sum of the two energy sources created

asymptotically by S0 and S1.

Indeed, we prove in this paper that the Wigner measure associated with thesequence (u) satisfies

(1.2) 0+(x, ) +

x(x, ) = Q0 + Q1,

where Q0 and Q1 are the source terms obtained in [2] in the case of one pointsource. 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). We also study

the limiting behaviour fo the rescaled solutions d12 u(x) and

d12 u(q1 + x).

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

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HIGH FREQUENCY ANALYSIS OF HELMHOLTZ EQUATIONS: CASE OF TWO POINT SOURCES3

on the study of the sequence (a) such that

ia

+ a + a = S1x q1

.Using the explicit formula for the Fourier tranform of a, we prove that a is uni-formly bounded in a suitable space and that a 0 as 0 weakly. We wouldlike to point out that our analysis, based on a study in Fourier space, strongly restson the assumption of a constant index of refraction.Second, our results on the Wigner measure then follow from the properties provedin [2]. They are essentially consequences of the uniform bounds on (u): we writethe equation satisfied by the Wigner transform associated with (u), and pass tothe limit 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 that is localized on the energy set {||2 = 1}, aproperty that was not exploited in [2].

The paper is organized as follows. In Section 2, we recall some definitions andstate our assumptions. Section 3 is devoted to the proof of uniform bounds on thesequence of solutions (u) and of the convergence of the rescaled solutions. Then,in Section 4, we establish the transport equation satisfied by the Wigner measure together with the radiation condition at infinity. In the appendix, we recall theproof 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 Hormander [1] for the study of Helmholtzequations. Then, we give our assumptions.

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

u() = (Fxu)() = 1(2)3

Rd

eixu(x)dx.

For u, v S(R3) and > 0, we define the Wigner transformW(u, v)(x, ) = (Fy)(u

x +

2y

v

x 2

y

),

W(u) = W(u, u).

In the sequel, we denote W = W(u).

If (u) is a bounded sequence in L2loc(Rd), it turns out that (see [6], [9]), up to

extracting a subsequence, the sequence (W(u)) converges weakly to a positiveRadon measure on the phase space TR3 = R3x R3 called Wigner measure (orsemiclassical measure) associated with (u):

(2.1) Cc (R6), lim0W(u), =

(x, )d

We recall that these measures can be obtained using pseudodifferential opera-tors. The Weyl semiclassical operator aW(x,Dx) (or Op

W (a)) is the continuous

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4 ELISE FOUASSIER

operator from S(Rd) to S(Rd) associated with the symbol a S(TRd) by Weylquantization rule

(2.2) (aW(x,Dx)u)(x) = 1(2)d

Rd

Rdy

ax + y2

, f(y)ei(xy)ddy.We have the following formula: for u, v S(Rd) and a S(Rd Rd),(2.3) W(u, v), aS,S = v, aW(x,Dx)uS,S ,where the duality brackets ., . are semi-linear with respect to the second argument.This formula is also valid for u, v lying in other spaces as we will see in Section 3.

2.2. Besov-like norms. In order to get uniform (in ) bounds on the sequence(u), we shall use the following Besov-like norms, introduced by Agmon and Horman-der [1]: for u, f L2loc(R3), we denote

uB = supj1

2j C(j)

|u|2dx1/2 ,fB =

j1

2j+1

C(j)

|f|2dx1/2

,

where C(j) denotes the ring {x R3/2j |x| < 2j+1} for j 0 and C(1) is theunit ball.

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

iv + v + v = fwhere > 0, then Agmon and Hormander [1] proved that there exists a constant

C independent of such thatvB CfB.

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

We denote for x R3, |x| =3

j=1 x2j and x = (1 + |x|2)1/2.

For all > 12 , we have

(2.4) uL2 := xuL2 C()uB.

We end this section by stating two properties of these spaces that will be usefulfor 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 in B on a linear manifoldof codimension 1.

Proposition 2.1. There exists a constant C such that for all f B, we haveR

f(x1, .)L2(R2)dx1 CfB.

The second property gives the stability of the space B by change of variables inFourier space.

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Proposition 2.2. Let 1, 2 be two open sets inR3, : 1 2 a C2 diffeo-morphism, C1c (R3). For all u B, we denote

T u = F1(u ).Then

T uB CC1bC2buB.2.3. Assumptions. We are now ready to state our assumptions. Our first assump-tion, borrowed from [2], concerns the regularizing parameter > 0.

(H1) for some > 0.This assumption is technical and is used to get a radiation condition at infinity inthe limit 0. Next, in order to get uniform bounds on u, we assume that thesource terms S0 and S1 belong to the natural Besov space that is needed to actuallysolve the Helmholtz equation (1.1).

(H2) S0B, S1B < .It turns out that, in order to compute the limit of the energy source, we shall needthe stronger assumption

(H3) xNS0 L2(R3) and xNS1 L2(R3) for some N > 12 + 3+1 .

3. Bounds on solutions to Helmholtz equations

In this section, we first establish uniform bounds on the sequence (u) that willimply estimates on the sequence of Wigner transforms (W). It turns out that weshall also need to compute the limit of the rescaled solutions w0 and w

1 defined

below in order to obtain the energy source in the equation satisfied by the Wignermeasure .Before stating our two results, let us define these rescaled solutions. Following [2]and [4], we denote

(3.1)

w0(x) = d12 u(x),

w1(x) = d12 u(q1 + x).

They respectively satisfy iw0 + w0 + w0 = S0(x) + S1x q1 ,iw1 + w1 + w1 = S0

x + q1

+ S1(x).

We are ready to state our results on u, w0 and w1.

Proposition 3.1. Assume S0, S1 B. Then, the solution u to the Helmholtzequation (1.1) satisfies the following bound

u

B

C(S0B + S1B),where C is a constant independent of.

Proposition 3.2. Letw0 and w1 be the rescaled solutions defined by (3.1). Then,

the sequences (w0) and(w1) are uniformly bounded in B

and they converge weakly- in B to the outgoing solutions w0 and w1 to the following Helmholtz equations

w0 + w0 = S0w1 + w1 = S1,

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6 ELISE FOUASSIER

i.e. w0 and w1 are given in Fourier space by

wj() = Sj()||2 1 + i0 = p.v. 1||2 1+ i(||2 1)Sj(), j = 0, 1.Remark. The Helmholtz equation w + w = S does not uniquely specify the

solution w. An extra condition is necessary, for instance the Sommerfeld radiationcondition. When the refraction index is constant equal to 1, this condition writes

(3.2) limr

1

r

Sr

wr

+ iw2d = 0.

Such a solution is called an outgoing solution.Alternatively, still assuming that the refraction index is constant, the outgoingsolution to the Helmholtz equation may be defined as the weak limit w of thesequence (w) such that

iw

+ w

+ w

= S(x).We point out that the two points of views are equivalent in the case of a constantindex 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 theproofs, our main difficulties are linked to the rays that are emitted by the source at0 towards the point q1 (and conversely). Hopefully, the interaction between thoserays is destructive and not constructive.

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

u

B

w

0B

,makes it sufficient to prove bounds on w0. Since w

0 is a solution to

iw0 + w0 + w0 = S0(x) + S1

x q1

we may decompose w0 = w0 + a, where w0 and a satisfy iw0 + w0 + w0 = S0(x),

ia + a + a = S1

x q1

.

First, we note that the bound w0B CS0B is established in Agmon-Horman-der [1] (see also Perthame-Vega [12]). Hence, the proof of Proposition 3.1 reducesto the proof of the following lemma.

Lemma 3.3. If a is the solution to

ia + a + a = S1

x q1

then a is uniformly (in ) bounded in B:

aB CS1B

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Proof. We want to prove that

v

B,

|a, v

| C

S1

B

v

B.

Using Parsevals equality, we write

(3.3) a, v =R3

eiq1 S1()v()

||2 + 1 i d.

To estimate this integral, we shall distinguish the values of close to or far fromtwo critical sets: the sphere {||2 = 1} (the set where the denominator in (3.3)vanishes when 0) and the line { collinear to x0} (the set where we cannotapply directly the stationary phase theorem to (3.3)).

More precisely, we first take a small parameter ]0, 1[, and we distinguish inthe integral (3.3), the contributions due to the values of such that |2 1| or

|2

1

| . Let

Cc (R) be a truncation function such that () = 0 for

|| 1. We denote () = ||21 . We accordingly decomposea, v =

R3

eiq1S1()v()()

||2 + 1 i d +R3

eiq1S1()v()(1 ())||2 + 1 i d

= I + II.

First, since the denominator is not singular on the support of, we easily boundthe first part with the L2 norms

|I| L

S1L2vL2 ,and using B L2, we obtain the desired bound(3.4)

|I

| C

S1

B

vB.

Let us now study the second part II where the denominator is singular. Up toa rotation, we may assume q1 = |q1|e1, where e1 is the first vector of the canonicalbase. We make the polar change of variables

=

r sin cos r sin sin r cos

.

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 dimension d 3.

Hence, q1 = |q1|r sin cos , and we get

II = ei |q1| r sin cos r2 + 1 i S1 v(1 )((r,,))r2 sin drddNow, we distinguish the contributions to the integral dd linked to the values closeto, or far from, the critical direction { = 2 , = 0} (which corresponds to the case{ collinear to q1}). To that purpose, let > 0 be a small parameter and denote

K =

(r,,)

1 r2 1 = 0,

2

= 0,

= 0

.

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Then K is a compact set. Let k Cc be such that (1 )k(, ) is a localizationfunction on K. We write

II = ei |q1| r sin cos r2 + 1 i (S1 v)((r,,))(1 (r))k(, )r2 sin drdd+

ei

|q1|

r sin cos

r2 + 1 i (S1 v)((r,,))(1 (r))(1 k(, ))r2 sin drdd

II = II I + IV.

To estimate the contribution II I, we apply the stationary phase method. Wedenote = 2 . The phase function is gr(, ) = r cos cos so

gr

= r sin cos = 0 at (, ) = (0, 0),gr

= r cos sin = 0 at (, ) = (0, 0),

and the Hessian at the point (0, 0) is

D2gr(0, 0) =

r 00 r

,

which is invertible at any point in K. Since K is a compact set, we can apply theMorse lemma: there exists a finite covering (j)j=1,n (n N) of K such that oneach set j , there exists a C change of variables (, ) (j , j) such that

gr(, ) = r r2j2

r 2j

2.

Moreover, we can write (1 )k =n

j=1 j where j Cc and supp(j) j .Then, we make the changes of variables j =

r2

j , j =

r2

j . Finally, we

decompose j = 1j

2j . Thus, we obtain, for the contribution II I

, the formula

(3.5) II I =n

j=1

ei x1 (r+2j +2j )r + 1 + i

T1j S1(r, j ,

j)T2j v(r, j , j)drdjdj ,

where

T1j S1 := F

(1jS1) (r, (j , j), (j , j)),T2j v := F

r + 1 + ir2 + 1 i (

2j v) (r, (j , j), (j , j))

2r

dd(r,,)

d(, )d(j , j)

.As a first step, using Proposition 2.2, we directly get T1j S1 B with

T1j S1B CS1B.As a second step, we study T2j v. Since for r close to 1, r + 1 + ir2 + 1 i

1,we recover, from Proposition 2.2,

T2j v B and T2j vB CvB.

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Now, we apply Parsevals equality with respect to the r variable in the formula ( 3.5)

II I

=

n

j=1

ei|q1|

(2j +2j )

r + 1 + i

T1j S1(. |

q1| , . , .)T2j vdrdjdj

=

ei

|q1|

(2j +2j )1{>0}e

(i)tFr(

T1j S1(. |q1|

, . , .))( t, j , j)

Fr(T2j v)(, j , j)dtddjdj .where 1{>0} denotes the characteristic function of the set { > 0}.Hence, we obtain

|II I| n

j=1

Fr(

T1j S1(.

|q1|

, . , .))())L2d

Fr(T2j v)()L2d|II I|

nj=1

T1j S1(

|q1|

)L2d

T2j v()L2d

|II I| n

j=1

T1j S1()L2d

T2j v()L2d

|II I| C

nj=1

T1j S1BT2j vB

|II I| CS1BvB,which is the desired estimate.

We are left with the part IV, which corresponds to the directions that arenot collinear to q1. We denote K the support of (1 )(1k) which is a compactset. In K, we can choose as new independent variables

1 = q1 , 2 = ||2 1.More precisely, since

d(1, 2)

d=

q12

is of maximal rank 2, there exists a finite covering (j)j=1,m (m N) of K suchthat in j , we can make the change of variables . As before, we denotej =

3j

4j some localization functions on

j such that (1 )(1 k) =

mj=1

j .

Thus, for j = 1, . . . , m,ei

q1

||2 + 1 + iS1vjd = ei21 + i (S1vj)(())

dd

d.If we denote

T3j S1 := F1((3jS1) ),T4j v := F1

(4j v) )

dd

,

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and ifF1 denotes the Fourier transform with respect to the 1 variable, Parsevalsequality with respect to 1 gives

ei q1 ||2 + 1 + iS1vjd = (2)d {t>0}et(F11 (T3j S1))(x1 t)(F11 (T4j v))(x1)ei2/dtdx1d2d3

CS1BvB.Summing over j, we obtain

|IV| CS1BvB,which ends the proof of the bound

|a, v| CS1BvB.

3.2. Proof of Proposition 3.2. We prove the result for the sequence (w0), theconvergence of the sequence (w1) can be obtained similarly. As we did in the proof

of Proposition 3.1, we write w0 = w0 + a. Since w0 is the solution to a Helmholtzequation with constant index of refraction and fixed source, it converges weakly-to the outgoing solution w0 to w +w = S0. Hence, it suffices to show the followingresult.

Lemma 3.4. If a B is the solution toia + a + a = S1

x q1

then a 0 in B.Proof. The proof of this result requires two steps (using a density argument):

(1) for v B, we have the bound a, v CS1BvB(2) ifS1 and v are smooth, then a, v 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

a, v =R3

eiq1S1()v()

||2 + 1 i d.

We are thus left with the study of

(3.6) R() = R3

eiq1 ()

||2 + 1 i d

where = S1v belongs to S(R3).As in the proof of Lemma 3.3, we distinguish the contributions of various values

of . We shall use exactly the same partition, according to the values of close to,or far from, the sphere || = 1 and collinear or not to q1. We shall use the same

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notations for the various truncation functions.We first separate the contributions of such that |2 1| and |2 1|

R() = R3

eiq1 ()()

||2 + 1 i d +R3

eiq1 ()(1 ())||2 + 1 i d

= I + II.

In the support of , since the denominator is not singular, we can apply thenon stationary phase method.Since q1 = 0, we may assume q11 = 0 and we have

I =

iq11

R3

eiq1 1

()()

||2 + 1 i

d

=

iq11

R3

eiq1

1(()())

||2 + 1 i 2()()1

(||2 + 1 i)2

.

Hence, we obtain the bound

|I| |q11 |R3

1|1()| +

2

2|1|

d.

Since 1() and 1 belongs to S, we have, as 0,I 0.

Let us now study the second term II. We use the same changes of variables asin Section 3.1. It leads to the following formula

II =n

j=1

ei

q1

(r2j 2j )

r + 1 + i j(r, j , j) (r, j , j)drdjdj ,where

j(r, j , j) = j (r, (j , j), (j , j)) 2(r + 1 + i)r(r2 + 1 i) d(, )d(j , j) ,(r, j , j) = (r, (j , j), (j , j)),are still smooth functions that are bounded independently from .Using Parsevals inequality with respect to the variables (j ,

j) for each integral,

we obtain the bound

|II| Cn

j=1

ei|q1|

rei(2

j+2

j )

r + 1 + i Fj,j (j )drdjdj .

To obtain the convergence of II, it remains to study an integral of the followingtype

|r1|

ei

|q1|

r

w(r)r + 1 + i dr, where w S.

This is done in the following lemma.

Lemma 3.5. w S, (0, 1), we have|r|

ei|q1|

rw(r)

r + i dr = iw(0) + O0().

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12 ELISE FOUASSIER

Using this lemma, we readily get the estimate

(3.7) |II I| C1 (0, 1),which proves that II I 0 as 0.

There remains to give theProof of Lemma 3.5. We write

ei|q1|

rw(r)

r + i dr =

ei|q1|

rw(r)

r2 + ()2(r i)dr

= i

ei|q1|

rw(r)

r2 + ()2dr +

ei|q1|

r rw(r)

r2 + ()2dr

= I+ II .

We have

I = i

ei|q1|yw(y)

y2 + 1 dy iw(0),and

II =

ei

|q1|

rw(r) w(0) rr2 + ()2

dr +

w(0)r

r2 + ()2dr .

The last term vanishes because the integrand is odd. Moreover, using the smooth-ness of w, we easily obtain that for all (0, 1),ei |q1| rw(r) w(0) Cr

Thus,

ei

|q1|

rw(r) w(0)

r

r2 + ()2dr

C

|r|1drand the result is proved.

We are left with the study of IV. We use the same change of variables as inSection 3.1.

IV =mj=1

ei

q1

||2 + 1 + i ()j()d

=

mj=1

ei

2

1 + i (j)(()) d

d

d= i

m

j=1

ei2

1 + i 1(j)(())

d

d

d.

The integral obviously converges with respect to all the variables except 1. It re-mains to prove the convergence with respect to the 1 variable, i.e. the convergenceof

()

1 + i d1,where

= 1

(j)(())

dd

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is smooth and compactly supported with respect to . It is a consequence of thefact that the distribution (x + i0)1 is well-defined on R by

1x + i0

= v.p.( 1x

) i(x).

We conclude that IV 0 and a, v 0 as 0.

4. Transport equation and radiation condition on

In this section, we state and prove our results on the Wigner measure associatedwith (u). Since we established the uniform bounds on (u) and the convergenceof (w0), (w

1), these results now essentially follows from the results proved in [2].

We first prove bounds on the sequence of Wigner transforms (W) that allow us todefine a Wigner measure associated to (u). Then, we get the transport equationsatisfied by together with the radiation condition at infinity, which uniquely

determines .

4.1. Results.

Theorem 4.1. Let S0, S1 B and > 0. The sequence (W) is bounded inthe Banach space X and up to extracting a subsequence, it converges weak- to apositive and locally bounded measure such that

(4.1) supR>0

1

R

|x| 0,

(4.5) x 12u(x)L2 CuB CfB,

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14 ELISE FOUASSIER

hence, for any function satisfying (4.2), we have

|W(u), |

R6

|u|(x + 2 y)|u|(x 2 y)x + 2 y

1

2+0x 2 y

1

2+0

x + 2

y 12+0x 2

y 12+0||(x, y)dxdy

Cf2BR3

supxR3

|x| + |y|1+0|(x, y)|dy.

So (W(u)) is bounded in X, > 0. We deduce that, up to extracting a subse-quence, (W(u)) converges weak- to a nonnegative measure satisfying(4.6) |, | Cf2B

R3

supxR3

|x| + |y|1+0|(x, y)|dy.

We refer for instance to Lions, Paul [9] for the proof of the nonnegativity of .The bound (4.1) is obtained using the following family of functions

R

(x, y) =

1

3/2 e|y|2/ 1

R (x R)and letting 0, R .

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

4.3.1. Transport equation satisfied by W. W satisfies the following equation

(4.7) W + xW = i

2Im W(f, u) := Q.

This equation can be obtained writing first the equation satisfied by

v(x, y) = u

x +

2y

u

x 2

y

.

From the equality

y xv = 2

u(x +

2y)u(x

2y) u(x

2y)u(x +

2y)

,

we deduce

v + iy xv + i

2

n2(x +

2y) n2(x

2y)

v = (x, y),

where

(x, y) := i2S(x +

2y)u(x

2y) S(x

2y)u(x +

2y).

After a Fourier transform, we obtain the equation (4.7).Then we write the dual form of this equation. Let S(R6), we have(4.8) W, W, x = Q, .By the definition of the Wigner measure , we get

W, 0 and W, x , x.

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Hence we are left with the study of the source term Q, .4.3.2. Convergence of the source term. In order to compute the limit of the source

term in (4.7), we develop

Q, = i2

ImW(S0, u), + W(S1 , u),

.

Thus, the result is contained in the following proposition.

Proposition 4.3. The sequences

W(S0 , u)

and

W(S1 , u)

are bounded in

S(R6) and for all S(R6), we have

lim0

W(S0 , u), S,S =1

(2)3

R3

w0()S0()(0, )d,(4.9)lim0

W(S1 , u), S,S =1

(2)3

R3

w1()

S1()(q1, )d,(4.10)

where w0 and w1 are defined in Proposition 3.2.

Using Proposition 4.3, we readily get

lim0

Q, = i2(2)3

Im

R3

w0()S0()(0, )d + R3

w1()S1()(q1, )d=

1

(4)2

R3

|S0()|2(2 1)(0, )d+

R3

|S1()|2(2 1)(q1, )d,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 onlyconsider the first one in our proof. Let S(TRd) and (x, y) = F1y((x, )),then we have

W(S0 , u), S,S =

S0

x +

2y

u

x 2

y

(x, y)dxdy

=

S0(x)w0(x + y)((x +

y

2), y)dxdy.

Hence, using that S(R2d), we getW(S0 , u), S,S CxN|S0(x)| |w0(x + y)|x + y x + yxNyk dxdy

C

xNS0

L2

w0

B R3y supxR3 x + y

xNykdy

for any k 0 and > 1/2, upon using the Cauchy-Schwarz inequality in x.Then, we distinguish the cases |x| |y| and |x| |y| : the term stemming fromthe first case gives a contribution which is bounded by C

dy

ykand the second

contribution is bounded by C

dyyk

. So, upon choosing k large enough, we obtain

that|W(S0 , u), S,S | CxNS0L2wB .

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16 ELISE FOUASSIER

Now, in order to compute the limit (4.9), we write

W(S0, u

),

= S0(x)w0 (x + y)x +y

2, y (0, y)dxdy+

w0(x)S0(x y)(0, y)dxdy

= I + II.

Reasonning as above, we readily get that lim0 I = 0. For the second term, wehave

II =

w0(x)

S0 (0, .)

(x)dx,

hence, since w0 converges weakly- in B, it suffices to prove that S0(0, .) belongsto B. We denote = (0, .). We have, for > 1/2,

S0 B CS0 L2

= C

x |S0 (x)|2dx

CL1 x |S0|2 ||(x)dxwhere we used the Cauchy-Schwarz inequality. Hence, we get

S0 B CxNS0L2R3y

supxR3

x + yxNyk dy,

for any k. As before, this integral converges. Thus, we have established that

S0 (0, .) belongs to B, which implies thatII

S0(x)w0(x + y)(0, y)dxdy.

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

4.4.1. Support of . In order to prove the radiation condition without restrictionon the test function R (as assumed in [2]), we first prove a localization property onthe Wigner measure . This property is well-known when u satisfies a Helmholtzequation without source term. It is still valid here thanks to the scaling of S.

Proposition 4.4. Under the hypotheses (H1), (H2), (H3), the Wigner measure satisfies

supp() {(x, ) R6/ ||2 = 1}.Proof. Let Cc (R6) and = W(x,Dx). Let us denote H = 2 1.Since u satisfies the Helmholtz equation (1.1), we have

(4.11) iu

+ H

u

= 2

S

.Moreover, H is a pseudodifferential operator with symbol ||2 1. By pseudodif-ferential calculus, H = OpW ((x, )(||2 1)) + O() so, using the definition ofthe measure , we get that

lim0

(Hu, u) = lim0

(OpW ((x, )(||2 1))u, u)

=

(x, )(||2 1))d.

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Using the equation (4.11), we write

(Hu, u) = 2(S, u) i(u, u) = 2(W(S, u), ) i(u, u).On the first hand, Proposition 4.3 gives that lim0 2(W(S, u), ) = 0. On theother hand, (u, u) is bounded so lim0 (

u, u) = 0. Therefore, for any Cc (R6), we have

(||2 n2(x))d = 0, so supp() {||2 = 1}.

4.4.2. Proof of the condition (4.4). Using the previous localization property, inorder to prove the radiation condition (4.4), one may only use test functions R Cc (R6) such that supp(R) R6\{ = 0}.Let R be such a test function. We associate with R the solution g to

g + xg = R(x, ).By duality, we have

Q, g = W, R,so that it sufffices to establish the following two convergences:

(4.12) lim0

Q, g = Q, g,

(4.13) lim0

W, R = f, R,where Q et g are defined in Theorem 4.2.As before, since R X for any > 0, the limit (4.13) follows from the weak-convergence of W in X.On the other hand,

Q, g = ImR6

S0(x)w0(x + y)g([x + y2 ], y)dxdy(4.14)

+

ImR6 S1(x)w

1(x + y)g

(q1 + [x +y

2], y)dxdy,

so Q, g is the sum of two terms of the same type. Such a term has been studiedin [2], where the following result is proved.

Proposition 4.5. Assume (w) is bounded in B and that(w) converges weakly-in B to w0. Assume S0 satisfy (H3). Let R Cc (R6) be such that supp(R) R6\{ = 0}. Let g be the solution to

g + xg = R(x, )and g(x, ) =

0

R(x + t,)dt. Then, we have

lim0

R6

S0(x)w0(x + y)g([x + y2 ], y)dxdy = 1(2)3

R3

S0()w0()g(0, )d.Proof. The proof of this result is written in the appendix.

Using the proposition above together with Proposition 3.2, we get that

lim0

Q, g

= Im

1

(2)3

R3

S0()w0()g(0, )d + 1(2)3

R3

S1()w1()g(q1, )d=

1

(4)2

R3

|S0()|2(2 1)g(0, )d + R3

|S1()|2(2 1)g(q1, )d .

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18 ELISE FOUASSIER

Thus, the radiation condition (4.4) is proved.

Appendix A. Proof of Proposition 4.5

In the sequel, we denote

G = lim0

R6

S0(x)w0(x + y)g([x + y2 ], y)dxdy.

A.1. Bounds on G. In order to study G, we need a preliminary result on thetest function g.

Lemma A.1. LetR Cc (R6 \ { = 0}). We denote g the solution to(A.1) g + .xg = R(x, ).

It is given by the explicit formula

(A.2) g(x, ) =

0

exp(||1s) 1||R(x

||s, )ds.

Then we have the estimate

(A.3) M 0, |g(x, y)| CxM MyM ,where . . denotes the infimum of two numbers, and C is a constant depending onM and R.

Proof. Let a be a multiindex such that |a| M. We denote = || . We write

yag(x, y) = Fy0

(i)ae||1s 1||R(x s,)ds

=

R3

dei.y+s=0

ds

b,c,d,e,f

C(a,b,c,d,e,f)e||

1s(i)b(||1s)

(||1s)f(i)c(s)(ix)d(i)e

1

||R

(x s,).(A.4)

Using that R C0 (R6\{|| = 0}) so

there exists r0, A, B > 0 such that supp(R) {|x| r0} {A || B}, R and its derivatives belong to L1(R6).

|(i)

b(

|

|1s)

| Cs,

|

| A.

|(i)c(s)| Cs, || A. in the integral above, s [|x| r0, |x| + r0], so for |x| large enough, we can usethe equivalence s |x| x (where we dente for a, b > 0, a b if c1, c2 >0/ c1a < b < c2a), we get in (A.4),

|yag(x, y)| CxM exp(B

x).

The desired estimate follows.

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Using this lemma, we estimate

|G

| R6 w0(x + y)S0(x)g((x +y

2, y)dxdy

CR6

|w0(x + y)|x + y 12+0 x + y

1

2+0xN1 |S0(x)|xN1

(x + y2 )M MyM

Cw0BxN1S0(x)L2

R3

supxR3

x + y 12+0xN1 (|x| + |y|)M M

yM dy.

Now, we prove that the integral

I =R3

supxR3

x + y 12+0xN1 (|x| + |y|)M M

yM dy

is bounded uniformly with respect to .

Let us define the following three subsets inR6

(A.5)A = {(x, y) R6/|x| |y|}, B = {|x| |y|, |10y| 1},C = {|x| |y|, |10y| 1},

where 10 means 1 with > 0 sufficiently small.If (x, y) A, then

x + y 12+0xN1 (|x| + |y|)M M

yM CxN1+

1

2+0xM M

yM.

Now, we distinguish the relative size of x and : if x , we have

x 12+0N1

xM M

x 12+0N1M (12+0N1+M). if x , we have x 1 x, hence we get

x 12+0N1xM M

x 12+0N1M M(+1)(12+0N1).

Now, we choose 1/2 + 0 such that N1 3+1 +

12 , we can choose M > 3 such that this contribution is uniformly

bounded with respect to .If (x, y) B, then |y| 1 so we obtain

x + y 12+0xN1 (|x| + |y|)M

M

yM CyM+1

2+0.

Thus, the corresponding contribution to I is bounded by CR3

yM+ 12+0dy whichis convergent.If (x, y) C, then

x + y 12+0xN1 (|x| + |y|)M M

yM Cy1

2+0 yM M

yM .

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20 ELISE FOUASSIER

Since |y| 1+0, if we denote z = y, we have |z| , for some > 0, whichimplies that

z

2 z.Thus, we have

I C3R3

zM+ 12+0(zM M)dz

CM(1)4R3

zM+ 12 +0(zM M)dz

where we used the hypothesis (H1) .We distinguish two cases, according to the relative size ofz and . We have

zz 12+0 z

M MzM dz

M

z

z 12+0Mdz

C( 3

2+0)

,and

zz 12+0 z

M MzM dz

z

z 12+0dz

C( 32+0).Hence, we get

I CM(1)4( 32+0).To conclude, we choose M >

4+ 32

1 , which gives that I is uniformly bounded withrespect to .

A.2. Convergence of G. We decompose G in the following way

G = R6

w0(x + y)S0(x)g((x + y2 ), y) g(0, y)dxdy+

R6

w0(x + y)S0(x)g(0, y) g(0, y)dxdy

+

R6

w0(x + y)S0(x)g(0, y)dxdy= I + II + II I

Using the same method as in Section A.1, we prove that I, II 0. Then, wemay write

II I =

R6

w0(x)

S0

g(0, .)

(x)dx.

Moreover, we established in the proof of Proposition 4.3 that if is rapidly decreas-

ing at infinity, then S0 B. Hence, since (w0) converges weakly- in B to w0,we get

lim0

II I =

R6

w0(x)

S0 g(0, .)(x)dx,which ends the proof.

Acknowledgement. I would like to thank my advisor Francois Castella for havingguided this work.

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