Boundary layers and glancing blow-up in nonlinear geometric optics

Ann. Scient. Éc. Norm. Sup.,4e série, t. 33, 2000, p. 383 à 432.

BOUNDARY LAYERS AND GLANCING BLOW-UP INNONLINEAR GEOMETRIC OPTICS

BY MARK WILLIAMS 1

ABSTRACT. – We construct rigorous geometric optics expansions of high order for semilinear hyperbolicboundary problems with oscillatory data. The errors approach zero inL∞ as the wavelengthε→ 0. Toachieve such errors it is necessary to incorporate profiles of theglancing, elliptic, and hyperbolic boundarylayers into the expansions. The analysis of the glancing boundary layer forces the introduction of a thirdscale 1/

√ε in addition to the usual oscillatory (1/ε) and spatial (1) scales. The evolution of the leading

part of the glancing profile is governed by a semilinear Schrödinger-type equation with nonhomogeneousboundary conditions. The description of the elliptic boundary layer involves complex phases and complextransport equations.

We also construct examples showing that when glancing modes of order at least 3 are present, the maximaltime of existenceTε of the exact solutionuε can approach 0 asε→ 0. The blow-up mechanism is differentfrom the types of focusing known to occur in free space. 2000 Éditions scientifiques et médicales ElsevierSAS

RÉSUMÉ. – On construit des développements de l’optique géométrique d’ordre supérieur pour desproblèmes aux limites hyperboliques semi-linéaires avec données oscillantes. Les erreurs tendent verszéro dansL∞ lorsque la longueur d’ondeε tend vers 0. Pour atteindre de telles erreurs il est nécessaired’incorporer les profils des couches limitesglancing, elliptiquesethyperboliquesdans les développements.L’analyse de la couche limiteglancingnous amène à introduire une troisième échelle 1/

√ε en plus des

échelles oscillantes (1/ε) et spatiales (1) habituelles. L’évolution de la partie dominante du profilglancingest contrôlée par une équation semi-linéaire de type Schrödinger avec conditions non-homogènes au bord.La description de la couche limite elliptique fait intervenir des phases complexes et des équations detransport complexes.

Nous construisons aussi des exemples montrant qu’en présence de modesglancingd’ordre au moins 3,le temps maximalTε d’existence de la solution exacteuε tend vers 0 lorsqueε → 0. Le mécanismed’explosion est différent des types de focalisations qui se manifestent en l’absence de frontières. 2000Éditions scientifiques et médicales Elsevier SAS

Part ISurvey of the main results

1. Introduction

As in [17] we construct geometric optics expansions for a class of Kreiss well-posed semilinear

boundary problems onRN+1+ = x= (x′,xN ) = (x0, . . . ,xN ): xN > 0

1 This research was supported in part by NSF grants DMS-9401248 and DMS-9706489.

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE. – 0012-9593/00/03/ 2000 Éditions scientifiques etmédicales Elsevier SAS. All rights reserved

384 M. WILLIAMS

Luε = f (uε, uε),

Buε|xN=0 = gε(x′),(1.1)

uε = 0 in x0 < 0,

and then show that the expansions are close to exact solutions for small wavelengthsε. HereLis anm ×m first-order operator, andgε(x′) = G(x′,x′ · ξ′/ε), whereG(x′,θ0) is smooth andperiodic inθ0. The main difference is that in this paper, we produce expansions for which theerrors approach zero inL∞, not merely inL2 as in [17]. (By “error” we mean the differencebetween the exact solution and the approximate solution given by the expansion.) To achieve thesmaller errors one must incorporate the profiles of theglancing, elliptic, and hyperbolic boundarylayersinto the expansions. The existence of the glancing and elliptic boundary layers, which aresmall inL2 but not inL∞, was already evident in [17], but there they were absorbed into theerror terms.

The analysis of the glancing boundary layer forces the introduction of a third scale (1/√ε),

in addition to the usual oscillatory (1/ε) and spatial (1) scales. If only the usual scales areused, trouble arises since the equation for the glancing profile is then governed by a vectorfield everywhere tangent to the boundary. The profile is thus uniquely determined by the initialcondition inx0 < 0, so one can’t impose a boundary condition. On the other hand, (generalized)eigenvectors associated to multiple real zeros of detL(ξ′,ξN ) = 0 are needed to make the exactsolution satisfy the boundary conditions (see Section 2). This inconsistency is one source of thelargeL∞ errors in [17].

As in [3,11,6,7] where different three-scale problems were considered (see Remark 8.9), theprofile equations exhibit second-order terms in the intermediate scale when the original system(1.1) is first-order. The second-order derivatives are transverse to the boundary, so one can imposea boundary condition. Indeed, we obtain semilinear Schrödinger-type profile equations withnonhomogeneousboundary conditions (8.34). Schrödinger profile equations of a different sortwere encountered in [3,11], while the second-order profile equations in [6,7] were of parabolic-hyperbolic type. In Section 2 a simple, explicit, linear example shows how the new scale andSchrödinger-type equations appear in connection with glancing modes.

The construction of the elliptic boundary layer also has some unusual features. Complexphases lead us to introduce periodic profiles that extend holomorphically into the upper complexhalf-plane. One must work with spaces of profiles that are invariant under complex conjugationand nonlinear functions (Remark 4.1). The transport equations associated to the complex phasesinvolve complex vector fields, but these equations need merely be solved to high enough order atthe boundaryxN = 0 (Proposition 9.4).

In contrast to [17], where a leading term expansion was constructed for solutionsuε to (1.1)with oscillatory datagε defined by almost-periodic profiles, here we work with periodic profilesand construct expansions of arbitrarily high order under an appropriate generically valid smalldivisor hypothesis (Definition 4.1). It turns out that complete expansions can be constructed evenwithout any hypotheses preventing rectification like the oddness hypotheses of [3].

The elliptic and glancing boundary layers are of width∼ ε and∼√ε, respectively, and appear

in the leading term of the expansions. Nonlinear interactions cause ahyperbolic boundary layerof width∼√ε to appear as well in the higher order terms (Definition 8.1). One of the main tasksof this paper is to understand precisely how the three layers interact and evolve. This informationis contained in the analysis of the profile equations. Much of it is summarized in Remark 8.4.

Once a sufficiently accurate approximate solution is constructed, the existence of a nearby ex-act solution follows from a Gues-type theorem for semilinear boundary problems (Theorem 6.2).

4e SÉRIE– TOME 33 – 2000 –N 3

BOUNDARY LAYERS AND GLANCING BLOW-UP 385

Glancing blow-up

Let p(ξ) = detL(ξ) be the scalar principal symbol of them×m first-order operatorL in (1.1).To anyξ′ = (ξ0, . . . ,ξN−1) ∈ RN \ 0 there is associated a combination of glancing, hyperbolic,and elliptic phases corresponding to the zerosξN of p(ξ′,ξN ) = 0 which are respectively multiplereal, simple real, and nonreal. The results of [17] were obtained under the assumption that therewere no real zerosξN of multiplicity higher than 2. In Section 12 we show the necessity ofthis assumption by constructing examples where the presence of real zeros of multiplicity 3results in a time of existenceTε→ 0 for the exact solutionuε asε→ 0. The examples involveconstant coefficient operators and linear phases so they show that, in contrast to the situation forhyperbolic equations in free space, coherence (see [8]) does not suffice to prevent blow-up. Themechanism of blow-up is completely different from the types of focusing in free space identifiedin [8] (see Remark 12.3). Theorem 4.2 of [17] shows that high-order nonreal zeros do not causeblow-up asε→ 0.

Remark1.1. – In [20] we construct rigorous geometric optics expansions for perturbations ofa stable planar shock produced by oscillations whose associated characteristic vector fields (3.6)all reflect strictly transversally off the shock (an oscillating free boundary). The results of thispaper are intended partly as a step toward understanding the more general situation where ellipticand glancing boundary layers form near the shock.

2. Appearance of the new scale in Schrödinger-type profile equations

Consider the 2× 2 example onR2+1+ = ( t,y,x): x> 0

Luε = 0,

Buε|x=0 = gε(t,y),(2.1)

uε = 0 in t < 0,

where

L= ∂t +

(1 00 −1

)∂x +

(0 11 0

)∂y,

andB is a 1× 2 constant, real matrix such that (L,B) satisfies the uniform Kreiss condition(Definition 3.2). We takegε(t,y) = b(t,y)eiβ·(t,y)/ε whereg ≡ 0 in t < 0 andβ = (τ ′,η′) with|τ ′|= |η′|.

The scalar symbol ofL is

p(τ ,η,ξ) = τ2−(ξ2 + η2

),(2.2)

so ξ = 0 is a double real root ofp(τ ′,η′,ξ) = 0. The associated glancing phase and glancingvector field are

φg(t,y,x) = (τ ′,η′, 0) · (t,y,x) and Xg = 2τ ′∂t − 2η′∂y(2.3)

respectively.Takingγ > 0 and ignoring powers of 2π we may write the exact solutionuε to (2.1) as

uε(t,y,x) =

∫eit(τ−iγ)+iyη+ix

√(τ−iγ)2−η2 r(τ − iγ,η)

Br(τ − iγ,η)

× b(τ − iγ − τ ′

ε,η− η′

ε

)dτ dη(2.4)

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE

386 M. WILLIAMS

where “√ ” always denotes the square root with positive part, andr is the eigenvector satisfying

L(τ − iγ,η,

√(τ − iγ)2− η2

)r(τ − iγ,η) = 0.(2.5)

For (τ ′,η′) fixed make the change of variable

τ = τ − τ ′

ε, η = η − η′

ε(2.6)

in (2.4), setX = (τ − iγ + τ ′/ε, η+ η′/ε), drop the tildes, and use|τ ′|= |η′| to obtain

uε(t,x,y) = ei(tτ′+yη′)/ε

∫ei(t,y)(τ−iγ,η)+i x√

ε

√2τ ′(τ−iγ)−2η′η+ε[(τ−iγ)2−η2] r(X)

Br(X)

× b(τ − iγ,η)dτ dη.(2.7)

(2.7) exhibits the new scalex/√ε. Observe that the glancing phaseφg appears on the far left,

and the first two terms under the square root give the symbol ofXg.Now fix (τ ′,η′) = (1, 1), and set

aε(χ, t,y)eiθ = eiθ∫ei(t,y)(τ−iγ,η)+iχ

√2(τ−iγ)−2η b(τ − iγ,η)

r(X)Br(X)

dτ dη.(2.8)

Consider the approximate solution

uε(t,y,x) = aε(χ, t,y)eiθ|χ= x√

ε, θ=

φgε

.(2.9)

By Taylor expanding the square root in (2.7) and using|eiw − eiz |6 |w− z|, it’s easy to show

∣∣e−iφgε (uε − uε)(t,y,x)∣∣Hsγ (t,y)

6Cx√ε|b|

Hs+(3/2)γ (t,y)

,(2.10)

where|v|Hsγ (t,y) ≡ |〈τ − iγ,η〉sv(τ − iγ,η)|L2(τ ,η).Finally, observe thataε(χ, t,y) satisfies a Schrödinger-type equation with nonhomogeneous

boundary conditions:

i∂2χaε +Xgaε = 0,

Baε|χ=0 = b(t,y),(2.11)

aε = 0 in t < 0.

Remark2.1. – Later we’ll encounter problems like (2.11) for thenth Fourier componentan(χ, t,y,x) of a periodic profile wherei∂2

χ is replaced byi∂2χ/n. Then-dependence is a source

of difficulty in the analysis. To obtain linear estimates suitable for Picard iteration, we are led tointroduce the cutoffsρ(χ〈D3/2

θ 〉) (8.35) and to make use of Wiener algebrasAθ(C(χ,Hs)) (7.7),(8.42).

3. Symbols, phases, the Kreiss condition, and regular boundary frequencies

We revert to the notationRN+1+ = x = (x′,xN ) = (x0,x′′) = (x0,y,xN ): xN > 0 of

Section 1. Denote the dual variables byξ = (ξ′,ξN ) = (ξ0,η,ξN ).

4e SÉRIE– TOME 33 – 2000 –N 3


A. In (1.1) L = L(∂) = ∂x0 +∑Nj=1Aj∂xN is an m × m first-order system, strictly

hyperbolic with respect tox0, and noncharacteristic with respect to the boundaryxN = 0. Thecorresponding symbols are

L(ξ) = ξ0I +

N∑j=1

Ajξj = ξ0 +A(ξ′′),

(3.1)p(ξ) = detL(ξ).

We assume theAj are real matrices.B in (1.1) is aµ×m real matrix whereµ is determinedfrom the Kreiss condition (Definition 3.2).

Define the matrixA(ξ′) by the equation

A−1N L(ξ) = ξN −A(ξ′).(3.2)

B. If ξ = (ξ′,ξN ) ∈ RN ×C, x ∈ RN+1+ we callξ (respectivelyξ · x) a characteristic mode

(respectivelycharacteristic phase) whenp(ξ) = 0, and we set

charL=

(ξ′,ξN ) ∈RN ×C: p(ξ) = 0.(3.3)

For ξ ∈ char L the mode (or associated phase) is calledglancing, hyperbolic, or ellipticdepending on whether the rootξN of p(ξ′,ξN ) = 0 is multiple real, simple real, or nonreal.

C. Supposeξ = (ξ0,ξ′′) ∈ charL ∩ (RN+1 \ 0). Strict hyperbolicity implies kerL(ξ) is onedimensional. Indeed, forξ′′ ∈RN \ 0,A(ξ′′) hasm distinct eigenvalues−ξi0(ξ′′) satisfying

ξ10(ξ′′)< ξ2

0(ξ′′)< · · ·< ξm0 (ξ′′)(3.4)

with associated right eigenvectorsrj (ξ′′), j = 1,. . . ,m.If ξ = (ξ

0,ξ′′) ∈ charL ∩ (RN+1 \ 0), ξ

0= ξi0(ξ′′) for somei and we setr(ξ) = ri(ξ

′′). Letπ(ξ) denote projection ofCm onto kerL(ξ) along rangeL(ξ). This is the same as the projectionontori(ξ

′′) with respect to the decomposition

Cm = spanr1(ξ′′)⊕ · · · ⊕ spanrm(ξ′′).(3.5)

Setpi(ξ) = ξ0− ξi0(ξ′′). ξi0(ξ′′) is real-analytic inξ′′ and the characteristic vector field associatedto ξ is given by

Xξ =∂pi∂ξ

(ξ) · ∂x(∂x = (∂x0, . . . ,∂xN )

).(3.6)

Xξ is calledglancing, outgoing,or incomingwhen ∂ξi0ξN

(ξ′′) is respectively= 0,< 0, or > 0.ξ (respectivelyξ · x) is then referred to as a glancing, outgoing, or incoming mode (respectivelyphase). The importance ofXξ is connected to the well-known fact [12] that

π(ξ)Akπ(ξ) =− ∂ξi0

∂ξk(ξ′′)π(ξ) and thus

(3.7)π(ξ)L(∂)π(ξ) =Xξπ(ξ).


388 M. WILLIAMS

D. When ξ = (ξ0,ξ′′) = (ξ′,ξ

N) ∈ charL and Im ξ

N6= 0, we don’t necessarily have a

decomposition like (3.5) or a well-defined vector fieldXξ as in (3.6). This leads to

DEFINITION 3.1. – The elliptic mode(ξ′,ξN

) is regularif(a) ξ

Nis a simple root ofp(ξ′,ξ

N) = 0.

(b) A(ξ′′) is diagonalizable.

Condition (b) means there is a decomposition

Cm =E1(ξ′′)⊕ · · · ⊕El(ξ′′), l6m,(3.8)

into eigenspaces ofA(ξ′′) associated to eigenvalues−ξ10(ξ′′), . . . ,−ξl0(ξ′′) ∈ C. ξ

0∈ R must

equal one of these, sayξ0

= ξ10(ξ′′). We may now defineπ(ξ) to be projection ofCm onto ker

L(ξ) along rangeL(ξ). This is just projection ontoE1 with respect to the decomposition (3.8).Since every eigenvector ofA(ξ′′) associated to−ξ

0is an eigenvector ofA(ξ′) associated to

ξN

and vice versa (the same is not true for generalized eigenvectors!), condition (a) in Definition3.1 and (3.8) implyξ

0is simple as a root ofp(ξ

0,ξ′′) = 0. Thus, forξ′′ = (η,ξN ) near (η,ξ

N)

in RN−1×C, ξ10(η,ξN ) is real-analytic inη and analytic inξN . Settingp1(ξ) = ξ0− ξ1

0(ξ′′) the(complex) characteristic vector field associated to the elliptic modeξ may now be defined by

Xξ =∂p1

∂ξ(ξ) · ∂x.(3.9)

Remark3.1. –(a) Since the matricesAj are real, if (ξ′,ξN ) is a regular elliptic mode, so is (ξ′,ξN ). For a

more general notion of regularity for elliptic modes see Definition 3.5.(b) (3.7) holds also for regular elliptic modesξ.

E. The uniform Kreiss condition and the spaces E+(ξ′). For ξ′ ∈RN andγ > 0 the stricthyperbolicity ofL implies that the eigenvaluesξjN (ξ0− iγ,η) of A(ξ0− iγ,η) (see (3.2)) havenonzero imaginary part. Denote byE+(ξ0− iγ,η) the direct sum of the generalized eigenspacesof A(ξ0− iγ,η) corresponding to theξjN with Im ξjN > 0:

E+(ξ0− iγ,η) =⊕

Im ξjN>0

ker[(ξjN −A(ξ0− iγ,η)

)mj ],(3.10)

wheremj is the multiplicity of ξjN . The dimensionµ =∑

Im ξjN>0mj of E+(ξ0 − iγ,η) is

independent of (ξ0,η) ∈RN , γ > 0. Let

X =

(ξ0,η,γ): (ξ0,η)∈RN , γ > 0, (ξ0,η,γ) 6= 0.

The spacesE+(ξ0 − iγ,η) form a C∞ subbundle of rankµ of the trivial Cm bundle overX ∩ γ > 0 which extends to a continuous subbundleE+ of rankµ overX [2]. For ξ′ ∈ RNE+(ξ′) denotes a fiber of this continuous extension.

DEFINITION 3.2. – The pair(L,B) satisfies theuniform Kreiss conditionif the restriction ofB to E+(ξ0− iγ,η) is an isomorphism for all(ξ0,η) ∈ RN , γ > 0 such that(ξ0,η,γ) 6= 0. ThisforcesB to be aµ×m matrix.

4e SÉRIE– TOME 33 – 2000 –N 3


F. Decomposition of E+(ξ′). For ξ′ ∈ RN \ 0 let ξiN (ξ′), i = 1,. . . ,M (ξ′) 6 m be thedistinct roots of det(ξN −A(ξ′)) = 0. Write the index setM(ξ′) = 1, . . . ,M (ξ′) as a disjointunion of subsetsG(ξ′),O(ξ′),P(ξ′),I(ξ′), andN (ξ′) corresponding to the modesβi(ξ′) ≡(ξ′,ξiN (ξ′)) that are respectively glancing, outgoing, such that ImξiN is positive, incoming, orsuch that ImξiN is negative.E+(ξ′) may now be written as a direct sum

E+(ξ′) =⊕

i∈G(ξ′)∪O(ξ′)∪P(ξ′)

E+(βi(ξ′)

).(3.11)

For i ∈ P(ξ′) E+(βi(ξ′)) is the generalized eigenspace associated to the nonreal eigenvalueξiN (ξ′). For i ∈O(ξ′) E+(βi(ξ′)) = kerL(βi(ξ′)), a one-dimensional space. Suppose finally thati ∈ G(ξ′), ξ′ = (ξ0,η), and thatmi > 1 is the multiplicity of the real rootξiN (ξ′). For γ > 0smallξiN (ξ0,η) splits intomi rootsξkN (ξ0− iγ,η), k = 1,. . . ,mi, with nonzero imaginary parts.Let µi(ξ′) be the number of these with positive imaginary part. Then dimE+(βi(ξ′)) = µi andE+(βi(ξ′)) is spanned by generalized eigenvectorsw such that [ξiN (ξ′) −A(ξ′)]µiw = 0. Thestrict hyperbolicity and noncharacteristic boundary assumptions imply [2]

µi =mi

2whenmi is even.

(3.12)

µi is eithermi + 1

2ormi − 1

2whenmi is odd.

DEFINITION 3.3. – Let P (βi(ξ′)) denote the projection ofE+(ξ′) onto E+(βi(ξ′)) withrespect to the decomposition(3.11).

Remark3.2. – Suppose dimE+(βi(ξ′)) = 2 for somei ∈ G(ξ′). Forξ′ nearξ′,E+(βi(ξ

′)) cansplit into a direct sum of two nearly parallel eigenspaces. The projections onto those eigenspacesblow up asξ′→ ξ′, which raises the question of whether such unbounded projections can lead toa shrinking time of existence,Tε→ 0, for the exact solutionuε to (1.1) asε→ 0. Example 2 ofSection 12 shows this to be the case. On the other hand Theorem 4.2 of [17] shows that ellipticmodes of high multiplicity do not causeTε→ 0.

To avoid glancing blow-up we shall later fixξ′ = (ξ0,η) ∈RN \ 0 such that

For all i ∈ G(ξ′),mi = 2 and thus dimE+(βi(ξ′)

)= 1.(3.13)

In this caseE+(βi(ξ′)) is the eigenspace ofA(η,ξiN (ξ′)) associated to the eigenvalue−ξ0.

DEFINITION 3.4. – We callξ′ = (ξ0,η) ∈RN \ 0 a regular boundary frequencyif ξ′ satisfies(3.13)and all the associated elliptic modesβi(ξ′), i ∈ P(ξ′)∪N (ξ′), are regular(Definition3.1).

The results of this paper will assume thatgε(x′) = G(x′,x′ · ξ′/ε) in (1.1) oscillates with afixed regular boundary frequencyξ′.

Whenξ′ is regular, for eachi= 1,. . . ,M (ξ′) the eigenspace ofA(η,ξiN (ξ′)) associated to−ξ0

is one-dimensional. Withr(ξ′,ξiN (ξ′)) denoting a corresponding basis vector, the decomposition(3.11) may now be written

E+(ξ′) =⊕

i∈G(ξ′)∪O(ξ′)∪P(ξ′)

spanr(ξ′,ξiN (ξ′)

).(3.14)


390 M. WILLIAMS

Example3.1. – Suppose

det(ξ0 +A(ξ′′)

)=(ξ2

0 − c1|ξ′′|2)(ξ2

0 − c2|ξ′′|2)(ξ2

0 − c3|ξ′′|2),

where 0< c1 < c2 < c3. Then everyξ′ ∈RN \ 0 is a regular boundary frequency. In particular, ifξ′ = (ξ0,η) satisfiesξ2

0 = c2|η|2, thenG(ξ′),O(ξ′),P(ξ′) are all nonempty.

The following simple proposition will be helpful in the construction of the elliptic boundarylayer.

PROPOSITION 3.1. – Assume(L,B) as in PartA satisfies the uniform Kreiss condition, andsupposeξ′ is a regular boundary frequency. ThencardP(ξ′) = cardN (ξ′) and the restriction

B :⊕

i∈G(ξ′)∪O(ξ′)∪N (ξ′)

spanr(η,ξiN (ξ′)

)→Cµ(3.15)

is an isomorphism. Ther(η,ξiN (ξ′)) can be chosen so that for eachi ∈ P(ξ′) there is ak ∈N (ξ′)such thatξiN (ξ′) = ξkN (ξ′) and r(η,ξiN (ξ′)) = r(η,ξkN (ξ′)). We denote byg :P(ξ′)→N (ξ′) themap that associatesk ∈N (ξ′) to i ∈P(ξ′).

Proof. –The statement follows immediately from the definitions, (3.14), and the fact that thematricesAj ,B are real. 2

Remark3.3. – Elliptic modes (ξ′,ξiN (ξ′)) of multiplicity > 1 do not cause blow-up, so one canattempt to construct geometric optics expansions foruε even when such modes are present. Weshall not attempt to deal here with boundary frequenciesξ′ for which the geometric multiplicityof ξiN (ξ′) as an eigenvalue ofA(ξ′) is less than the algebraic multiplicity (see [17], proof ofTheorem 4.7c). But in Remark (9.3) we describe how the results extend to the more general casewhere elliptic modes of possibly high multiplicity are required to be regular in the weaker senseof the following definition.

DEFINITION 3.5. – The elliptic mode(ξ′,ξN

) = (ξ0,η,ξ

N) is weakly regularif

(a) there are neighborhoodsω of (η,ξN

) in RN−1×C andO of (ξ0,η,ξ

N) in RN ×C such

that for each(η,ξN ) ∈ ω there is exactly one point(ξ0(η,ξN ),η,ξN ) ∈O ∩ charL,(b) the algebraic and geometric multiplicities ofξ

Nas an eigenvalue ofA(ξ′) are equal,

(c) A(η,ξN

) is diagonalizable.

G. In the analysis of the glancing boundary layer we shall need to work with WieneralgebrasAθ(C(χ,Hm(T ))) (7.7) which are invariant under the functionf (u,u) in (1.1). Thiswill be the case if we assumef (u,v) :Cm × Cm → Cm is entire withf (0, 0)= 0. One caneasily consider more general entire functionsf (x,u,v) =

∑(α,β)6=0 fα,β(x)uαvβ by imposing

appropriate restrictions on the coefficientsfα,β(x) (see [17], (2.7)), but we’ll refrain from doingso here.

4. Three-scale profile equations and small divisors

Using the notation and definitions of Section 3, PartF we fix a regular boundary frequencyξ′

and look for solutionsuε to (1.1) of the form

uε(x) =[a0(χ,x,θ) +

√εa1(χ,x,θ) + · · ·+ (

√ε)MaM (χ,x,θ)

]|χ=

xN√ε

, θ=φ(x)ε.(4.1)

where theM (ξ′)-tuple of phasesφ(x) and placeholderθ are defined as follows.

4e SÉRIE– TOME 33 – 2000 –N 3


Notation 4.1. – For a fixed regular boundary frequencyξ′ andj ∈ G(ξ′) ∪ O(ξ′) ∪ I(ξ′) ∪P(ξ′)

(a) φj (x) = (ξ′,ξjN (ξ′)) · x= βj(ξ′) · x, ξjN as in Section 3, PartF.(b) rj = r(βj(ξ′)), a basis vector for kerL(βj(ξ′)),(c) πj = π(βj(ξ′)), projection onto kerL(βj(ξ′)) along rangeL(βj(ξ′)),(d) Xj =Xβj(ξ′), the characteristic vector field associated toβj(ξ′),(e) θj is the placeholder forφj/ε.

(f) For j ∈ N (ξ′), setφj(x) = (−ξ′,−ξkN (ξ′)) · x for k = g−1(j) (g as in Proposition 3.1),βj(ξ′) = dφj , rj = rk, πj = π(βj(ξ′)), andXj =Xβj(ξ′).

(g) Forj ∈N (ξ′) θj =−θk for k = g−1(j).(h) Setθ = (θj)j=1,...,M(ξ′).

For i ∈ G(ξ′) ∪ O(ξ′) ∪ I(ξ′) θi ∈ R, while for i ∈ P(ξ′) ∪ N (ξ′) we have Imθi > 0. Observethat fori ∈M(ξ′) \N (ξ′), φi restricts whenxN = 0 to the boundary phaseφ0(x)≡ ξ′ · x, whilefor i ∈N (ξ′), φi restricts to−φ0(x). We denote byθ0 the placeholder forφ0(x)/ε.

Each profileaj(χ,x,θ) is associated to a smooth periodic functionaj(χ,x, θ), θ ∈ RM(ξ′),whose Fourier expansion has the special form (dropping thej)

a(χ,x, θ) =∑

α∈Z(ξ′)

aα(χ,x)eiαθ,(4.2)

whereZ(ξ′)⊂ ZM(ξ′) is defined by

Z(ξ′) =α= (αi)i∈M(ξ′): αi ∈ Z if i ∈ G(ξ′)∪O(ξ′)∪ I(ξ′);

(4.3)αi ∈ Z+ if i ∈ P(ξ′)∪N (ξ′)

.

Here Z+ = 0, 1, 2,. . .. Note that since speca ⊂ Z(ξ′), a extends holomorphically in thevariables (θi)i∈P(ξ′)∪N (ξ′) to

(θi)i∈P(ξ′)∪N (ξ′): Im θi > 0.

Denoting the holomorphic extension also bya we define, forθ as in (4.1)(h),

a(χ,x,θ) = a(χ,x, θ)|θ=θ.(4.4)

PROPOSITION 4.1. – (a)For a(χ,x,θ) as in(4.4), there exists a functionc(χ,x, θ) of the form(4.2)such that

a(χ,x,θ) = c(χ,x, θ)|θ=θ.(b) Let f (u1, . . . ,un) : (Cm)n→Cm be entire and let

ai(χ,x,θ) = ai(χ,x, θ)|θ=θ, i= 1,. . . ,n,

for ai(χ,x, θ) as in(4.2). Then there exists a functionb(χ,x, θ) of the form(4.2)such that

f (a1, . . . ,an) = b(χ,x, θ)|θ=θ.

Proof. –(a) Write

a(χ,x,θ) =∑

α∈Z(ξ′)

aα(χ,x)eiαθ


392 M. WILLIAMS

and note that sinceθg(i) =−θi wheni ∈P(ξ′) for g as in Proposition 3.1, we have

iαθ = iβθ, for β ∈ Z(ξ′)

where

βi =−αi ∈ Z, i ∈ G(ξ′)∪O(ξ′)∪ I(ξ′),

βi = αg(i) ∈ Z+, i ∈ P(ξ′),

βg(i) = αi ∈ Z+, i ∈P(ξ′).

(b) Just takeb= f (a1, . . . , an). 2Remark4.1. – (a) In particular fora(χ,x,θ) as in (4.4) andf (u,v) :Cm×Cm→Cm entire,

we have

f(a(χ,x,θ),a(χ,x,θ)

)= b(χ,x, θ)|θ=θ(4.5)

for a functionb of the form (4.2).(b) In Section 9 we work with profilesa(χ,x,θ) of the form

a(χ,x,θ) =Mχa(χ,x,θ) +M∞a(x,θ),(4.6)

whereMχa is rapidly decreasing inχ and M∞a is independent ofχ. For f as in (4.5),f (a(χ,x,θ),a(χ,x,θ)) also has the form (4.6) (Proposition 5.1).

Notation 4.2. – For functionsa(χ,x,θ) = a(χ,x, θ)|θ=θ as in (4.4) we’ll often write∂θia(χ,x,θ) instead of [∂θi a(χ,x, θ)]|θ=θ.

Equations for the profilesa(χ,x,θ) are obtained by plugging (4.1) into (1.1), Taylor expandingnonlinear functions ofuε abouta0, and setting coefficients of different powers ofε equal to zero.With L(∂) andL(ξ) as in Section 3, PartA, set

P(∂θ) =∑

i∈M(ξ′)

L(βi(ξ′)

)∂θi .(4.7)

Let Ik (respectivelyBk) be the interior (respectively boundary) equations obtained by setting thecoefficient ofεk equal to zero. TheIk equations onxN > 0 are:

I−1: P(∂θ)a0 = 0,(4.8)

I− 12: P(∂θ)a1 +AN∂χa0 = 0,

I0: P(∂θ)a2 +AN∂χa1 +L(∂)a0 = f (a0),

I 12: P(∂θ)a3 +AN∂χa2 +L(∂)a1 = f ′(a0)a1,

I1: P(∂θ)a4 +AN∂χa3 +L(∂)a2 = f ′(a0)a2 + f ′′(a0)(a1,a1),

Ik, k > 1: P(∂θ)a2k+2 +AN∂χa2k+1 +L(∂)a2k = f ′(a0)a2k +F (a0, . . . ,a2k−1).

4e SÉRIE– TOME 33 – 2000 –N 3


Here we use the shorthandf (a0) for f (a0,a0) and do similarly for higher derivatives. Below welet s(θ0) denote theM (ξ′)-tuple such that

s(θ0)i = θ0 if i ∈M(ξ′) \ N (ξ′),

s(θ0)i =−θ0 if i∈N (ξ′).

The boundary equations onxN = 0 are:

B0: Ba0(χ,x,θ)|χ=0,xN=0,θ=s(θ0) =G(x′,θ0),(4.9)

B k2,k > 0: Bak(χ,x,θ)|χ=0,xN=0,θ=s(θ0) = 0.

Remark4.2. – If G(ξ′) = ∅, then all odd profiles turn out to vanish, all even profiles areindependent ofχ, and the equations (Ik,Bk) reduce to the usual 2-scale equations (see Section 9).

An important step in solving the (Ik,Bk) is solving equations like

P(∂θ)a(χ,x,θ) = F (χ,x,θ)

in algebras of functions of the form (4.4). This is generally impossible because of the occurrenceof small divisors. As in [10] we’ll impose a small divisor condition, but here the condition isslightly modified to take into account the elliptic boundary layer. Observe that fora(χ,x,θ) as in(4.4),

P(∂θ)a(χ,x,θ) = i∑

α∈Z(ξ′)

P(α)aα(χ,x)eiαθ,(4.10)

where

P(α) = L(α · dφ), φ(x) as in (4.1).(4.11)

Denote the characteristic set by

C =α ∈ ZM(ξ′): detP (α) = 0

.

DEFINITION 4.1. – The boundary frequencyξ′ (or phaseφ) satisfies thesmall divisorconditionif there existC > 0 andd ∈R such that

|dx(α · φ)|>C|α|d for all α ∈ ZM(ξ′) \ 0,(4.12)

|detP(α)|>C|α|d for all α ∈Z(ξ′) \ C.(4.13)

In what follows we shall always work with a fixed regular boundary frequencyξ′ chosen sothat

ξ′ satisfies the small divisor condition.(4.14)

Remark4.3. – (a) Property (4.12) implies that the component phases ofφ in (4.1) areQ-independent, a point that will be important in the solution of the profile equations. TheQ-independence would not follow ifZM(ξ′) \ 0 were replaced byZ(ξ′) \ 0 in (4.12). Situationswhere rational relations hold among just the real phases can be handled by choosing anappropriateadapted basis(Definition 8.2) for theQ-span of the real phases. This is done in [20]in the context of multidimensional shocks. It is possible to have relations involving a mixture of


394 M. WILLIAMS

real and nonreal phases. For example, an integer combination of glancing and elliptic phases canequal a different elliptic phase. We shall not treat such situations here.

(b) Suppose card (O(ξ′) ∪ I(ξ′)) = m for all ξ′ in some open subsetΩ ⊂ RN \ 0, so themodes associated to eachξ′ ∈Ω are all hyperbolic. Propositions implying the validity of (4.12),(4.13) for almost everyξ′ ∈Ω are given in Section 4 of [20]. Similarly, one can consider lowerdimensional submanifoldsΩ ⊂RN \0 such that a fixed number of glancing modes, and possiblyelliptic or hyperbolic modes as well, are associated to eachξ′ ∈ Ω. Examples where the smalldivisor properties hold for almost everyξ′ ∈ Ω are given in Section 11.

5. Function spaces

Here we define the spaces needed to state the main results.Fix a regular boundary frequencyξ′ ∈RN \ 0 and letM (ξ′) be as in Section 3, PartF. Let

ΩT =x ∈RN+1;−∞< x0 < T , xN > 0

, ΩT ,θ =ΩT ×TM(ξ′),

and set

bΩT =x′ ∈RN : −∞< x0 <T

, bΩT ,θ0 = bΩT ×T1.

We regard the functionsa(χ,x, θ) in (4.2) as functions onR+ ×ΩT ×TM(ξ′).

5.1. Spaces for profiles

(a) C∞χ (T ,ξ′) = a(χ,x, θ) ∈ C∞(R+ × ΩT × TM(ξ′)): a has the special form (4.2) and∂α

(χ,x,θ)a is bounded for eachα.

(b) C∞(T ,ξ′) = a(x, θ) ∈ C∞(ΩT × TM(ξ′)): a has the form (4.2) and∂α(x,θ)

a is boundedfor eachα..

(c) Γ∞χ (T ,ξ′) = a(χ,x, θ) ∈ C∞χ (T ,ξ′): χk∂α(χ,x,θ)

a is bounded for each pair (α,k), k > 0 .

(d) Γ∞χ (T ,ξ′) = a(χ,x,θ): a= a|θ=θ , with a ∈ Γ∞χ (T ,ξ′) andθ as in (4.1)(h).

(e) C∞(T ,ξ′) = a(x,θ): a= a|θ=θ , with a ∈ C∞(T ,ξ′) andθ as in (4.1)(h).(f) Pχ(T ,ξ′) = a(χ,x,θ): a= b+ c, whereb(χ,x,θ)∈ Γ∞χ (T ,ξ′) andc ∈C∞(T ,ξ′).(g) C∞(T ) = a(x′,θ0) ∈C∞(bΩT ,θ0): ∂

α(x′,θ0)a is bounded for eachα.

In (5.1)(d)a has been holomorphically extended as in (4.4).

PROPOSITION 5.1. – Supposef (u1, . . . ,un) : (Cm)n→Cm is entire. Leta1, . . . ,an ∈ Pχ(T ,ξ′).Thenf (a1, . . . ,an) ∈ Pχ(T ,ξ′). In particular, if f (u,v) is entire anda ∈ Pχ(T ,ξ′), thenf (a,a) ∈Pχ(T ,ξ′).

Proof. –Let ai = bi + ci, wherebi ∈ Γ∞χ (T ,ξ′) andci ∈C∞(T ,ξ′). Write

f (a1, . . . ,an) =[f (b1 + c1, . . . ,bn + cn)− f (c1, . . . ,cn)

]+ f (c1, . . . ,cn),

rewrite the first term on the right using Taylor’s theorem, and apply Proposition 4.1.25.2. Spaces onΩT

LetHk(ΩT ),Wm,∞(ΩT ) be the usual Sobolev spaces. Form ∈ 0, 1, 2,. . ., ρ > 0, ε ∈ (0, 1]let

(a) Bmρ (T ) =uε(x) ∈Hm(ΩT ): |∂αuε|L2(ΩT ) 6 ρε−|α| for |α|6m, ε ∈ (0, 1]

.

4e SÉRIE– TOME 33 – 2000 –N 3


Bmρ (T ) is the analogous space onbΩT .

(b) Dmρ (T ) =uε(x) ∈Wm,∞(ΩT ): |∂αuε|L∞(ΩT ) 6 ρε−|α| for |α|6m, ε ∈ (0, 1]

.

Dmρ (T ) is the analogous space onbΩT .

Notation 5.1. – The subscript “0” attached to any of the function spaces defined in this paper,e.g.,Pχ,0(T ,ξ′), will be used to denote the subspace of functions vanishing inx0 < 0.

DEFINITION 5.1. – For functions a(χ,x,θ) ∈ Pχ(T ,ξ′) we define projectionsM∞a =limχ→∞ a ∈C∞(T ,ξ′) andMχa= (1−M∞)a ∈ Γ∞χ (T ,ξ′).

Remark5.1. – (a)a(χ,x,θ) ∈ Pχ(T ,ξ′) impliesa(x/√ε,x,φ(x)/ε) ∈ Dmρ (T ) for all m, for

someρ= ρ(m)> 0.(b) Spaces likeBmρ (T ) andDmρ (T ) were used in [4].

Notation 5.2. – (a) For functionsa(χ,x, θ) ∈ C∞χ (T ,ξ′) we seta(χ,x) equal to the mean ofa, anda∗ = a− a. Fora(χ,x,θ) = a|θ=θ, we seta(χ,x) = a anda∗ = a− a.

(b) If B is any space of periodic functions,B∗ will denote the subspace of functions withmean 0.

6. Main results

We shall construct exact solutionsuε of (1.1) inHm1(ΩT ) wherem1 > (N + 1)/2 by firstconstructing an approximate solutionuε of the form (4.1) withM terms where (M − 1)/2>m1,and then proving a general Gues-type theorem for boundary problems to obtain a unique exactsolution uε nearby. The smooth profilesaj in (4.1) will be constructed to lie inPχ,0(T ,ξ′)(Definition 5.1(f)).

The analysis of the profile equations produces detailed information about the interaction andevolution of the boundary layers. Some of that information is summarized in Remark 8.4.

THEOREM 6.1. – Fix a regular boundary frequencyξ′ ∈RN \0 (Definition3.4)satisfying thesmall divisor condition(Definition4.1).Consider the problem(1.1)where(L,B) as in Section3A satisfies the uniform Kreiss condition(Definition 3.2), f :Cm × Cm → Cm is entire withf (0, 0)= 0, andgε(x′) = G(x′,x′ · ξ′/ε), whereG(x′,θ0) ∈ C∞(T ) has compact support inx′

andG≡ 0 in x0 < 0 (soG ∈ C∞0 (T )). Choose(M ,m1) such that(M − 1)/2>m1 > (N + 1)/2.There existT0, 0< T06 T , and profilesaj(χ,x,θ)∈ Pχ,0(T0,ξ′), for j = 1,. . . ,M such thatuεas in(4.1)satisfies

Luε = f(uε, uε

)+ ε(M−1)/2Rε,

Buε|xN=0 = gε + ε(M−1)/2rε,(6.1)

uε = 0 in x0 < 0,

whereRε,rε lie in Dm1ρ,0(T0)∩Bm1

ρ,0(T0),Dm1ρ,0(T0)∩ Bm1

ρ,0(T0) respectively for someρ > 0.

Remark6.1. – (a) In fact for allm> 0,Rε,rε lie in Dmρ(m),0(T0)∩ Bmρ(m),0(T0),Dmρ(m),0(T0)∩Bmρ(m),0(T0) respectively for someρ(m)> 0.

(b) Corner compatibility conditions in (6.1) hold to infinite order sinceG ∈ C∞0 (T ) andf (0, 0)= 0.


396 M. WILLIAMS

Exact solutions near approximate ones

The following general Gues-type theorem also applies to boundary problems not involvingoscillatory data.

THEOREM 6.2. – Let (L,B) andf be as in Theorem6.1and chooseM1>m1 > (N + 1)/2.Supposeuε ∈Dm1

ρ,0(T ) satisfies

(a) Luε = f(uε, uε

)+ εM1Rε,

Buε|xN=0 = gε + εM1rε,

uε = 0 in x0 < 0,

whereRε ∈ Bm1ρ,0(T ), rε ∈ Bm1

ρ,0(T ) for someρ > 0.Then there existε0 > 0 andσ > 0 such that for0< ε< ε0 the problem

(b) Luε = f (uε,uε),

Buε|xN=0 = gε,

uε = 0 in x0 < 0,

has a unique exact solutionuε ∈ uε + εM1Bm1σ,0(T ).

THEOREM 6.3 (Exact oscillatory solutions). –Let (L,B), f , gε, (M ,m1), T0, and the profilesaj be just as in Theorem6.1. There existε0 > 0 andσ > 0 such that for0< ε< ε0, the problem(1.1)has a unique exact solution inHm1(ΩT0) given byuε = uε + ε(M−1)/2Bm1

σ,0(T ) where

uε = a0

(xN√ε

,x,φ

ε

)+ · · ·+

(√ε)M

aM

(xN√ε

,x,φ

ε

).(6.2)

Proof. –We haveuε ∈ Dm1ρ,0(T0), so (6.1) implies the hypotheses of Theorem 6.2 are satisfied

with M1 = (M − 1)/2. An application of that theorem concludes the proof.2Remark6.2. – (a) A standard continuation principle and uniqueness for (1.1) imply that the

solutionuε in Theorem 6.3 is in factC∞.(b) Theorems 6.1, 6.2, and 6.3 remain true exactly as stated whenf (uε,uε) in (1.1) is replaced

by certain more general forcing terms. See Remark 8.5.(c) Cases where there are nontrivial rational relations (henceresonances) among the

components ofφ, but which involve only the real phases, can be handled similarly usingadaptedbases. See Section 8.2.

(d) The treatment of boundary frequenciesξ′ for which the corresponding elliptic modes areof possibly high multiplicity, but still weakly regular in the sense of Definition 3.5, is describedin Remark 9.3.

7. Spaces for constructing the glancing boundary layer

The leading term of the glancing boundary layer will be written as a sum of two pieces,b0 + c0

(8.35), (8.36). Different spaces are used in the Picard iterations forb0 andc0. Recall the notationx= (x0,x′′) = (x0,y,xN ) = (x′,xN ) for x ∈ΩT .

4e SÉRIE– TOME 33 – 2000 –N 3


7.1. b0-spaces

Form ∈ 0, 1, 2,. . ., T > 0, γ > 0 let

Hm(T ) =

u(x): |u|m,T =

∑|α|6m

|∂αu|L2(ΩT ) <∞.(7.1)

Hmγ =

u(x): |u|m,γ =

∑|α|6m

γm−|α||e−γx0∂αu|L2(RN+1

+ )<∞

.(7.2)

Hmγ (T ) =

u(x): |u|m,γ,T =

∑|α|6m

γm−|α||e−γx0∂αu|L2(ΩT ) <∞.(7.3)

C(χ,Hm(T )

)=u(χ,x): u is a continuous, bounded function of(7.4)

χ> 0 with values inHm(T )

,

|u|χ,m,T = supχ>0

∣∣u(χ, ·)∣∣m,T

.

Ck(χ,Hm(T )

)=u(χ,x): ∂lχu ∈C(χ,Hm(T )), l6 k

.(7.5)

The spaces and normsC(χ,Hmγ ), |u|χ,m,γ andC

(χ,Hm

γ (T )), |u|χ,m,γ,T(7.6)

are defined similarly.

Wiener algebrasAθ(B). It was observed in [9] that Wiener algebras are useful for provingestimates on profiles by mode-by-mode analysis. For a Banach spaceB Aθ(B) is the space ofperiodicB-valued functions ofθ ∈Rp with absolutely summable Fourier coefficients. Here andelsewhere we suppress the “p” in the notation. ThusV ∈Aθ(B) if and only if

V =∑n∈Zp

Vneinθ with |V |Aθ (B) = |V |θ;B ≡

∑n

|V |B <∞.(7.7)

If k > 1 we define

Akθ (B) =V : ∂αθ V ∈Aθ(B) for |α|6 k

with

|V |Akθ

(B) = |V |k,θ;B ≡∑n

〈n〉k|Vn|B.(7.8)

A simple application of the triangle inequality shows that ifB is a Banach algebra satisfying|uv|B 6C|u|B|v|B for someC > 0, then for the sameC

|UV |θ;B 6C|U |θ;B|V |θ;B.(7.9)

In particular (7.9) holds forB = Hm(T ) or C(χ,Hm(T )) when m > (N + 1)/2. Thecorresponding|U |θ;B norms are written|U |θ;m,T , |U |θ;χ,m,T respectively.

Supposeu,v ∈C(χ,Hm(T ))∩L∞(R+×ΩT ) for somem> 0. We have the weighted Moser-type inequality [15]

|uv|χ,m,γ,T 6C(|u|χ,m,γ,T |v|L∞ + |u|L∞ |v|χ,m,γ,T

).(7.10)


398 M. WILLIAMS

(7.10) implies, with the obvious notation,

|UV |θ;χ,m,γ,T 6C(|U |θ;χ,m,γ,T |V |θ;L∞ + |U |θ;L∞ |V |θ;χ,m,γ,T

).(7.11)

Supposef :Cm ×Cm→Cm is entire withf (0, 0)= 0. If B is a Banach algebra ofC-valuedfunctions, straightforward use of the triangle inequality shows the mapu→ f (u,u) sendsB toitself and is bounded on bounded sets inB. Similarly, (7.9) implies the map

U → f (U ,U) sendsAθ(B) to itself and is bounded on bounded sets inAθ(B).(7.12)

ForU ∈Aθ(C(χ,Hm(T )))∩L∞(R+ ×ΩT )),m> 0, (7.11) implies by the same argument∣∣f (U )∣∣θ;χ,m,γ,T

6C(|U |θ;L∞)

(|U |θ;χ,m,γ,T

).(7.13)

7.2. c0-spaces

With a= a+ a∗ as in Notation 5.2, set

L2∗ =a(χ,x′′,θ)∈ L2

(R+ ×R

N

+ ×T1): a= 0

.(7.14)

Let ∂−1θ be the mapL2∗ → L2∗ which assigns toa(χ,x′′,θ) its periodic primitive. Then

D−1θ = i∂−1

θ is bounded and self-adjoint onL2∗. The operatorD2χD−1θ , which will appear later

in Schrödinger-type profile equations, is an unbounded self-adjoint operator onL2∗ with domain

D∗ =a(χ,x′′,θ) ∈L2∗: a is anL2 function of (x′′,θ)

with values inH2(R+)∩H10(R+)

.(7.15)

Form> 0 let

Γm =

a(χ,x′′,θ)∈ L2(R+ ×R

N

+ ×Tp): |a|Γm =

∑|α,k|6m

∣∣χk∂α(χ,x′′,θ)a∣∣L2 <∞

,(7.16)

Γm,∗ =a(χ,x′′,θ): χk∂lχa ∈D∗, |k, l|6m

. (Hereθ ∈ T1).(7.17)

Spaces likeΓm were used in [3]. The following Proposition is an immediate consequence ofLemma 4.2 of [3].

PROPOSITION 7.1 (Moser inequality forΓm spaces). – (a)There existsC such that fora,b∈ Γm ∩L∞

|ab|Γm 6C(|a|Γm |b|L∞ + |a|L∞ |b|Γm

).(7.18)

(b) Supposef isC∞ anda,b∈ Γm satisfy for someR> 0

|b|L∞ 6R and∣∣χk∂α(χ,x′′,θ)a

∣∣L∞6R for all |α,k|6m.(7.19)

Then there is a constantC(m,R,f ) such that∣∣f (a+ b,a+ b)− f (a,a)∣∣Γm6C|b|Γm .(7.20)

In particular, if f (0, 0)= 0 we have∣∣f (b,b)∣∣Γm6C

(|b|L∞

)|b|Γm .(7.21)

4e SÉRIE– TOME 33 – 2000 –N 3


Form> 0,T > 0 let

C(T ,Γm) =a(χ,x,θ): a is a bounded, continuous function of

x0 ∈ (−∞,T ] with values inΓm

,

|a|T ,Γm = supx0∈(−∞,T ]

|a(·,x0, ·)|Γm .(7.22)

SpacesCk(T ,Γm) are defined in the obvious way.

Remark7.1. – Because of the initial condition in (1.1), functions with nonconstant depen-dence onx0 in the following sections will always vanish identically inx0 < 0.

8. Discussion of the proofs

8.1. Discussion of Theorem 6.1

We shall analyze the profile equations (4.8), (4.9) using the projectionsMχ, M∞ fromDefinition 5.1, as well as operatorsE which project onto kerP(∂θ) and approximate inversesQ(∂θ) such thatQP = PQ= I −E.

Recall the characteristic set

C =α ∈ ZM(ξ′): detL(α · dφ) = 0

,(8.1)

and forj = 1,. . . ,M (ξ′) let

Cj =α ∈ ZM(ξ′): α · φ is in theR-span ofφj

.(8.2)

We haveC =⋃M(ξ′)j=1 Cj and in view of the first small divisor condition (4.12),

Cj =

(0,. . . , 0,αj, 0,. . . , . . . , 0)∈ ZM(ξ′): αj ∈ Z.

Forα ∈ ZM(ξ′) defineπα :Cm→Cm by

(a) πα = 0 if α /∈ C, πα = Id if α= 0.(8.3)

(b) If α ∈ Cj \ 0, πα = πj .

Next defineE :Pχ(T ,ξ′)→ Pχ(T ,ξ′) (notation as in (f) of Section 5.1) by the following actionon monomials.

E(Uα(χ,x)eiαθ

)= (παUα)eiαθ, α ∈Z(ξ′).(8.4)

Note thatE=E0 +∑M(ξ′)j=1 Ej where

Ej(Uαe

iαθ)

= (παUα)eiαθ if α ∈ Cj \ 0, Ej(Uαe

iαθ)

= 0 otherwise;(8.5)

E0(Uαe

iαθ)

= Uα if α= 0, E0(Uαe

iαθ)

= 0 otherwise.

By an argument of [10] the small divisor property (4.14) implies the existence of operators

Q(∂θ) :Pχ(T ,ξ′)→ Pχ(T ,ξ′)


400 M. WILLIAMS

satisfying

(a) P(∂θ)Q=QP = I −E,

(b) rangeE= kerP = kerQ,(8.6)

(c) kerE= rangeP = rangeQ.

Indeed, setQ(∂θ)(Uαeiαθ) =−iQ(α)Uαeiαθ where

(a) Q(α) = L−1(α · dφ) if α /∈ C,(b) Q(0) = 0,(8.7)

(c) Q(α) =∑k 6=p

1

ξp0 (ξ′′)− ξk0 (ξ′′)πk(ξ′′) if α · dφ=

(ξp0 (ξ′′),ξ′′

),

where−ξk0 (ξ′′), k = 1,. . . , l, are the eigenvalues ofA(ξ′′), and πk the corresponding spectralprojections. (8.6)(a) follows from

P(α)Q(α) =Q(α)P(α) = 1− πp(ξ′′) = 1− πα(8.8)

with P(α) as in (4.11).

Decomposition of the profile equationsIn order to make use of the projectionsMχ, M∞ (Definition 5.1) to break up the profile

equations into manageable pieces, we’ll assume for the moment that solutionsak(χ,x,θ) ∈Pχ,0(T0,ξ′) to (4.8), (4.9) do exist for someT0 > 0. That assumption is verified to be true inSection 9.

Apply Q to the profile equationI−1 to obtain

(I −E)a0 = 0.(8.9)

For allk let us use

E=E0 +

M(ξ′)∑j=1

Ej ; I =Mχ +M∞

to write

(Eak)(χ,x,θ) = ak(χ,x) +

M(ξ′)∑j=1

σk,j (χ,x,θj)rj ,

(8.10)ak =M∞ak +Mχak.

ApplyingE0, Ej , andQ to I− 12

we find, respectively,

(a) AN∂χa0 = 0 soa0 =M∞a0,

(b) ∂χσ0,j = 0 for j ∈M(ξ′) \ G(ξ′),(8.11)

(c) (I −E)a1 =−QAN∂χa0.

In (8.11)(b) we’ve used (3.7). Note that forj ∈ G(ξ′), EjANEj = 0.Next applyE0,M∞E0, Ej , andM∞Ej to I0 to obtain, respectively,

4e SÉRIE– TOME 33 – 2000 –N 3


(a) AN∂χa1 +L(∂)a0 = f (a0),(8.12)

(b) L(∂)a0 =M∞f (a0),

(c) EjAN∂χa1 +Xjσ0,jrj = Ejf (a0) (recall (3.7))

(d) XjM∞σ0,jrj =M∞Ejf (a0).

The first observation is that (8.12)(b),(d) constitute a coupled semilinear system fora0,M∞σ0,j , j ∈ G(ξ′)∪O(ξ′)∪ I(ξ′) since for anyk ∈ G(ξ′)∪O(ξ′)∪ I(ξ′) the functions

M∞f (a0), M∞Ekf (a0) depend only ona0,M∞σ0,j ,

j ∈ G(ξ′)∪O(ξ′)∪ I(ξ′)(8.13)

(see Proposition 9.1). They are independent of

σ0,j , j ∈ P(ξ′)∪N (ξ′) and Mχσ0,j , j ∈ G(ξ′).

To find the corresponding boundary conditions use the boundary equationB0 to write (withs(θ0) as in (4.9))

(a) Ba0|xN=0 =G(x′),(8.14)

(b) B

(M(ξ′)∑j=1

σ0,jrj

)∣∣χ=0, xN=0, θ=s(θ0)

=G∗(x′,θ0).

Determination of boundary data for a∗0We now show how the Kreiss condition and Proposition 3.1 are used to obtain the boundary

values of

a∗0 =∑

j∈M(ξ′)\G(ξ′)

σ0,j(x,θj)rj +∑

j∈G(ξ′)

σ0,j(χ,x,θj)rj

from (8.14)(b). Lettingσ0,j,n, G∗n denote thenth Fourier coefficients ofσ0,j (respectivelyG∗),we obtain the following equations from (8.14)(b). Forn > 0,χ= 0,xN = 0,

B

( ∑j∈G∪O∪P

σ0,j,nrj

)=G∗n −B

∑j∈I

σ0,j,nrj ,(8.15)

while for n < 0,χ= 0, xN = 0

B

( ∑j∈G∪O

σ0,j,nrj +∑j∈N

σ0,j,−nrj

)=G∗n −B

(∑j∈I

σ0,j,nrj

)(8.16)

(recalls(θ0)j =−θ0 for j ∈N sinceφj |xN=0 =−φ0 for j ∈N ).The Kreiss condition implies Brj j∈G∪O∪P is a basis forCµ, so the Fourier coefficients on

the left in (8.15) can be expressed as a linear function of the coefficients on the right. Similarly,Proposition 3.1 implies Brj j∈G∪O∪N is a basis ofCµ, so the Fourier coefficients on the leftin (8.16) can be expressed as a linear function of the coefficients on the right.

If b is a periodic function ofθ0 with b= 0, letb± denote the pieces with positive (respectivelynegative) spectra. The above discussion implies in particular the existence of constant matrices


402 M. WILLIAMS

M± such that(M∞σ

±0,j |xN=0, θj=θ0, j ∈O(ξ′)

)=M±

(G∗,±,σ±0,j|xN=0, θj=θ0, j ∈ I(ξ′)

).(8.17)

In the solution of the semilinear system (8.12)(b),(d) the boundary values ofM∞σ0,j ,j ∈ G(ξ′) ∪ I(ξ′) are determined by transport. Kreiss-type estimates (Lemma 9.2) for thelinearized system are readily obtained since (L,B) satisfies the Kreiss condition and theXj

are just vector fields.After this semilinear system is solved by iteration, the values of

σ0,j |χ=0, xN=0, j ∈ G(ξ′) and M∞σ0,j |xN=0, j ∈ P(ξ′)∪N (ξ′)

are then read off from (8.15) and (8.16) as explained in the above discussion. Forj ∈ G(ξ′) set

Mχσ0,j |χ=0, xN=0 = (σ0,j −M∞σ0,j)|χ=0, xN=0.

To complete the determination ofa0 it remains to find

σ0,j =M∞σ0,j , j ∈ P(ξ′)∪N (ξ′) and Mχσ0,j , j ∈ G(ξ′).

The elliptic modes are determined by solving the coupled system (8.12)(d) to sufficiently highorder atxN = 0 (see Proposition 9.4) with the boundary data already determined.

Remark8.1. – Fork ∈ P(ξ′) ∪ N (ξ′) M∞Ekf (a0) depends ona0, M∞σ0,j , j ∈ G(ξ′) ∪O(ξ′) ∪ I(ξ′), which have already been determined, as well asM∞σ0,j , j ∈ P(ξ′) ∪ N (ξ′).M∞Ekf (a0) is independent ofMχσ0,j , j ∈ G(ξ′) (Proposition 9.3).

The leading part of the elliptic boundary layer is given by∑j∈P(ξ′)∪N (ξ′)

σ0,j(x,θj)rj |θj=φjε

.(8.18)

Leading part of the glancing boundary layerTo determineMχσ0,j , j ∈ G(ξ′) first applyMχ to (8.12)(c) and then rewrite it using (8.11)

andEjANEj = 0 (for j ∈ G(ξ′)) as

EjANQAN∂2χ(Mχσ0,j)rj +XjMχσ0,jrj =MχEjf (a0).(8.19)

WriteMχσ0,j(χ,x,θj) =∑nUn(χ,x)einθj and observe that

Q(∂θ)Uneinθj =−iQ(nαj)Uneinθj =−iQ(αj)D

−1θj

(Une

inθj)

(8.20)

whereαj = (0,. . . , 0, 1, 0,. . ., 0) with 1 in thejth slot. The following Lemma proved in Section 9implies that the first term in (8.19) is also given by ascalaroperator.

LEMMA 8.1. – For j ∈ G(ξ′) let (ξ′,ξjN (ξ′)) = (ξk0 (ξ′′),ξ′′), with ξk0 as in(3.4). We have

πjANQ(αj)ANπj =12πj∂2ξk0∂ξ2N

(ξ′′)πj ≡1cjπj .(8.21)

cj ∈R \ 0 since(ξ′,ξjN (ξ′)) is a glancing mode of order2.

4e SÉRIE– TOME 33 – 2000 –N 3


For j ∈ G(ξ′), cj as in (8.21) define the scalar operator

Lj =−iD2χD−1θj

cj+Xj .(8.22)

(8.19) becomes

Lj(Mχσ0,j)rj =MχEjf (a0), j ∈ G(ξ′).(8.23)

(8.23) is a coupled system for theMχσ0,j , j ∈ G(ξ′) with boundary data determined by (8.14)(b)as described above, whose solution we’ll discuss shortly.

Remark8.2. – (a) The nonlinear termMχEjf (a0) in (8.23) depends only on the knownfunctions

a0, σ0,k =M∞σ0,k, k ∈O(ξ′)∪ I(ξ′), M∞σ0,k, k ∈ G(ξ′),

and the unknownsMχσ0,k, k ∈ G(ξ′) (Proposition 9.5).(b) The value ofMχσ0,j , j ∈ G atχ= 0 is independent ofxN . TheMχσ0,j inherit nontrivial

xN dependence inχ > 0 from the forcing term.

The leading part of the glancing boundary layer is given by∑j∈G(ξ′)

Mχσ0,j(χ,x,θj)rj |χ=

xN√ε

, θj=φjε

.(8.24)

Determination of a1,a2, . . . ,aMWith a0 ∈ Pχ(T0,ξ′) thereby constructed, we read off (I −E)a1 from (8.11)(c).∂χa1 ∈ Γ∞χ (T0,ξ′) is determined by (8.12)(a) and we recoverMχa1 ∈ Γ∞χ (T0,ξ′) using the

following obvious Lemma.

LEMMA 8.2. – Supposeb(χ,x,θ)∈ Γ∞χ (T0,ξ′) and set

B(χ,x,θ) =−∞∫χ

b(χ′,x,θ)dχ′.

ThenB ∈ Γ∞χ (T0,ξ′) and∂χB = b.

To determineMχσ1,j , j ∈ O(ξ′) ∪ I(ξ′) ∪ P(ξ′) ∪ N (ξ′) apply Mχ to (8.12)(c) and use(8.11)(c) to get

EjAN∂χEa1 +EjAN∂χ(I −E)a1 +XjMχσ0,jrj =MχEjf (a0).(8.25)

For thesej, EjANEj = djEj for somedj ∈C \ 0 by (3.7), so∂χMχσ1,j ∈ Γ∞χ (T0,ξ′) can beread off from (8.25). Application of Lemma (8.2) yields the functionsMχσ1,j ∈ Γ∞χ (T0,ξ′).

Next applyE0,M∞E0, Ej , andM∞Ej to I 12

to obtain respectively:

(a) AN∂χa2 +L(∂)a1 = f ′(a0)a1,(8.26)

(b) L(∂)M∞a1 =M∞f′(a0)a1,


404 M. WILLIAMS

(c) EjAN∂χa2 +Xjσ1,jrj = Ejf ′(a0)a1 +EjF ,

(d) XjM∞σ1,jrj =M∞Ejf ′(a0)a1 +M∞EjF ,

whereF here and henceforth represents an already determined element ofPχ(T0,ξ′).Parallel to an earlier argument we observe that (8.26)(b),(d) constitute a linear system for

M∞a1, M∞σ1,j , j ∈ G(ξ′)∪O(ξ′)∪ I(ξ′) sinceM∞f ′(a0)a1 andM∞Ejf ′(a0)a1 depend onlyonM∞a1,M∞σ1,k, k ∈ G(ξ′)∪O(ξ′)∪ I(ξ′), and the known functiona0.

To find the boundary data for this system, rewriteB 12

in (4.9) as

(a) BM∞a1|xN=0 =−BMχa1|χ=0, xN=0,

(b) BEa1 =B

M(ξ′)∑j=1

(M∞σ1,j +Mχσ1,j)rj =−B(I −E)a1,(8.27)

onxN = 0, χ= 0, θ = s(θ0).

The functions

Mχa1, Mχσ1,j , j ∈O(ξ′)∪ I(ξ′)∪P(ξ′)∪N (ξ′)

and (I−E)a1 are already known. The procedure followed earlier yields boundary values first for

M∞σ1,j , j ∈ G(ξ′)∪O(ξ′)∪ I(ξ′),

and then for the remaining pieces ofa1, namely

Mχσ1,j , j ∈ G(ξ′) and M∞σ1,j , j ∈P(ξ′)∪N (ξ′).

(8.26)(d) gives a linear system for theM∞σ1,j , j ∈ P(ξ′) ∪ N (ξ′) which is solved tosufficiently high order atxN = 0 with the boundary data just determined. For thesej,M∞Ejf ′(a0)a1 depends on the known functions

a0, M∞a1, M∞σ1,k, k ∈ G(ξ′)∪O(ξ′)∪ I(ξ′)

and the unknownsM∞σ1,k, k ∈ P(ξ′)∪N (ξ′).It remains to find theMχσ1,j , j ∈ G(ξ′). First applyQ to I0 to get

(I −E)a2 =−QAN∂χa1 +F .(8.28)

ApplyingMχ to (8.26)(c) and using (8.28) yields

EjAN∂χEa2 +EjAN∂χ(I −E)a2 +XjMχσ1,jrj =MχEf ′(a0)a1 +MχEjF .(8.29)

For j ∈ G(ξ′), (8.29) may be rewritten using (8.28),EjANEj = 0, and Lemma 8.1 as

Lj (Mχσ1,j)rj =MχEjf ′(a0)a1 +MχEjF .(8.30)

The linear system (8.30) is solved in Section 9 with the boundary data determined above.

Remark8.3. – The construction ofa2,a3, . . . ,aM−2 is exactly parallel to the construction ofa1. Note that for anyk, makingIk hold (to sufficiently high order atxN = 0) requires completedetermination ofa0, . . . ,a2k, but only determination of

(I −E)a2k+1, Mχa2k+1, Mχσ2k+1,j , j ∈O(ξ′)∪ I(ξ′)∪P(ξ′)∪N (ξ′),

4e SÉRIE– TOME 33 – 2000 –N 3


and (I − E)a2k+2. To achieve (6.1) we need to solve (I0,B0), . . . , (IM−22

,BM−22

) so aM−1,aMneed only be partially determined.

DEFINITION 8.1. – For anyK theglancing, elliptic, and hyperbolic boundary layers of orderK are, respectively,

K∑i=0

(√ε)i ∑j∈G(ξ′)

Mχσi,j(χ,x,θj)rj |χ=xN√ε

, θj=φjε

,(8.31)

∑j∈P(ξ′)∪N (ξ′)

σ0,j(x,θj)rj |θj=φjε

+

K∑i=1

(√ε)i ∑j∈P(ξ′)∪N (ξ′)

σi,j (χ,x,θj)rj |χ=xN√ε

, θj=φjε

,

(8.32)and

K∑i=1

(√ε)i

∑j∈O(ξ′)∪I(ξ′)

Mχσi,j(χ,x,θj)rj |χ=xN√ε

, θj=φjε

(8.33)

with σi,j as in(8.10).

Remark8.4. – (a)

a0(χ,x,θ) = a0(x) +∑

j∈G(ξ′)

σ0,j(χ,x,θj)rj +∑

j∈M(ξ′)\G(ξ′)

σ0,j(x,θj)rj .

Fork > 1,Eak(χ,x,θ) = ak(χ,x) +∑

j∈M(ξ′) σk,j(χ,x,θj)rj .(b) For anyk > 0 the pieces ofak are determined in the following order:

(1) (I −E)ak,

(2) Mχak, Mχσk,j , j ∈M(ξ′) \ G(ξ′),

(3) M∞ak, M∞σk,j , j ∈ G(ξ′)∪O(ξ′)∪ I(ξ′),

(4) M∞σk,j , j ∈ P(ξ′)∪N (ξ′),

(5) Mχσk,j , j ∈ G(ξ′).

(c) The ordering in (b) reflects the boundary and interior coupling (via the forcing term) of thedifferent pieces ofak.

The term in (1) is determined directly from knowledge of the previous profiles. The terms in (2)can be read off from knowledge of (1) and the previous profiles. The terms in (3) are determinedby a system whose forcing term depends only on the terms in (3), (1) and previous profiles, whilethe forcing term for (4) depends on the terms in (4), (3), (1) and previous profiles. Finally, theforcing term for (5) depends on the terms in (5), (3),Mχak andMχσk,j , j ∈O(ξ′)∪I(ξ′) from(2), (1), and previous profiles.

These systems are semilinear whenk = 0 and linear fork > 1. The time of existenceT0 isdetermined by the systems for (3) and (5) whenk = 0.

The coupling on the boundary is somewhat different. For example, the boundary data forthe terms in (1), (2) is needed to determine that for the terms in (3), (4). On the other handwhile there is interior coupling among the terms in (4), there is no boundary coupling in thesense that for a givenj ∈ P(ξ′) ∪ N (ξ′), M∞σk,j |xN=0 can be read off from an equation like


406 M. WILLIAMS

(8.27)(b) independently ofM∞σk,i|xN=0, i 6= j. The same applies to the terms of (5). However,M∞σk,j |xN=0, j ∈ G(ξ′) is needed to determineMχσk,j |χ=0, j ∈ G(ξ′).

(d) For any fixedk there is no interior coupling or boundary coupling (in the sense used in (c))between the elliptic and glancing boundary layer terms given by (4) and (5) respectively. Theevolution of these terms is influenced by the glancing and hyperbolic pieces in (3), but as notedin (c) the influence is not mutual.

(e) In particular (k = 0) there is no interior or boundary coupling between the leading partsof the elliptic(8.18)and glancing(8.24)boundary layers.

Remark8.5. – The same analysis handles the case whenf (uε,uε) in (1.1) is replaced, forexample, byf (x,uε,uε) +F (xN/

√ε,x,φ/ε), providedF (χ,x,θ)∈ Pχ,0(T ,ξ′) andf (x,u,v) is

smooth inx, entire in (u,v), vanishes inx0 < 0, and satisfiesf (x, 0, 0)= 0. Remark 8.4 continuesto apply. Note that whenF (χ,x,θ)≡ 0, we haveM∞σk,j ≡ 0, j ∈ G(ξ′), for all k.

Construction of the glancing boundary layerWe must solve the system (8.23) with specified data atχ= 0. Sets0(χ,x,θ) = (s0,j(χ,x,θj)) =

(Mχσ0,j), j ∈ G(ξ′). Remark 8.2(a) implies this system has the form, withLj as in (8.22),

Lj(s0,j)rj = Ejh(x,ω,s0), j ∈ G(ξ′), ω = (θk), k ∈ G ∪O ∪ I,

s0,j|χ=0 = gj(x′,θj),(8.34)

s0,j = 0 in x0 < 0,

where theh,gj are smooth inx,ω,θj, vanish inx0 < 0, and by finite propagation speed havecompact support inx′ on [0,T ]. h is entire in (s0,s0), h(x,ω, 0)= 0, andh has compact supportin xN on [0,T ] if it is not independent ofxN (see Proposition 9.5).

Choose a cutoff functionρ(r) ∈C∞0 (R+), suppρ⊂ [0, 1],ρ≡ 1 nearr = 0. We will constructs0 in the forms0 = b0 + c0 (see Remark 8.7) whereb0,c0 satisfy respectively

Lj(b0,j)rj = ρ(χ〈D3/2

θj〉)Ejh(x,ω,b0), j ∈ G(ξ′),

b0,j|χ=0 = gj ,(8.35)

b0,j = 0 in x0 < 0,

Lj(c0,j)rj = Ejh(x,ω,b0 + c0)− ρ(χ〈Dθj〉3/2

)Ejh(x,ω,b0),

c0,j |χ=0 = 0,(8.36)

c0,j = 0 in x0 < 0.

Hereρ(χ〈D3/2θj〉) acts on periodic functions ofθj as the multiplierρ(χ〈n〉3/2).

These two systems are solved by separate Picard iterations corresponding in the obvious wayto the following scalar (hence noj) linear problems:

Lb0 = ρ(χ〈Dθ〉3/2)h(χ,x,θ), h ∈ Γ∞χ,0(T ,ξ′), θ ∈ T1,

b0|χ=0 = g(x′,θ), g ∈ C∞0 (T ),(8.37)

b0 = 0 in x0 < 0,

Lc0 = k(χ,x,θ), k ∈ Γ∞χ,0(T ,ξ′),

c0|χ=0 = 0,(8.38)

c0 = 0 in x0 < 0.

4e SÉRIE– TOME 33 – 2000 –N 3


Remark8.6. – k vanishes to infinite order atχ = 0, andh,k,g have compact support inx′ on [0,T ], with h,k having compact support inxN if not independent ofxN . We also leth ∈ Γ∞χ,0(T + 1,ξ′), g ∈ C∞0 (T + 1) denote extensions ofh,g vanishing inx0 > T + 1/2 withh= k = g = 0.

(8.37) is solved for each Fourier coefficient. The problem for thenth coefficientb0,n(χ,x) is,with c,X corresponding tocj ,Xj in (8.22),(

i∂2χ

cn+X

)b0,n = ρ

(χ〈n〉3/2)hn,

b0,n|χ=0 = gn(x′),(8.39)

b0,n = 0 in x0 < 0.

Let X(ξ0 − iγ,ξ′′) denote the symbol of1iX . For γ > 0 the partial Fourier transform of thesolution is

b∧0,n(χ,ξ0− iγ,ξ′′,xN ) =

χ∫0

ei(χ−χ′)√cnX(ξ0−iγ,ξ′′)

( ∞∫χ′

e−i(χ′−χ′′)

√cnX(ξ0−iγ,ξ′′)icnρ

(χ′′〈n〉3/2

)h∧n(χ′′,ξ0− iγ,ξ′′,xN )dχ′′

)dχ′

+ eiχ√cnX(ξ0−iγ,ξ′′)gn(ξ0− iγ,ξ′′),(8.40)

where√ denotes the square root with positive imaginary part. (8.40) leads to the estimates(notation as in Section 7)

|b0,n|C(χ,Hmγ (T )) 6C√γ〈n〉 |hn|C(χ,Hmγ (T )) + 〈gn〉Hmγ (T ),(8.41)

|b|A1θ

(C(χ,Hmγ (T ))) 6C√γ|h|Aθ (C(χ,Hmγ (T ))) + 〈g〉A1

θ(Hmγ (T )).(8.42)

Iteration together with a continuation principle proved using the Moser-type inequalityfor Wiener algebras (7.13) yields a solutionb0 ∈ Γ∞χ,0(T0,ξ′) to (8.35) for someT0 > 0(Proposition 9.6).

Remark8.7. – (a) (8.42) is immediate from (8.41).(b) The factor 1/〈n〉 in (8.41) is there because suppρ(χ〈n〉3/2) ⊂ [0, 1/〈n〉3/2]. Without the

cutoff 1/〈n〉 would be replaced by〈n〉, and the resulting estimate forb0 would not be suitablefor Picard iteration. Having 1 in place of 1/〈n〉 would make iteration possible, but we need thegain of oneθ derivative in (8.42) to deduce infinite regularity inθ of the solution to (8.35).

We can write down the solution to (8.38) using the unitary group corresponding to the self-adjoint operatorD2

χD−1θ /cj with domainD∗ ⊂ L2∗ (7.15). Integrating along characteristics of

the glancing vector fieldX = ∂x0 + v · ∂x′′ , we obtain

c0(χ,x,θ) =

x0∫0

ei(x0−s)

D2χD−1θ

cj k(χ,x+ (s− x0)(1,v, 0),θ

)ds(8.43)


408 M. WILLIAMS

which yields the estimate

|c0(·,x0, ·)|L2 6x0∫

0

|k(·,s, ·)|L2 ds.(8.44)

To getΓm estimates,m>N/2 we use the following commutator result, proved in Section 9.

PROPOSITION 8.1. – Supposek(χ,x,θ)∈L2∗ is such thatχk∂lχk ∈D∗, |k, l|6 1. Then

[χ,e

itD2χD−1θ

cj

]k = 2te

itD2χD−1θ

cjDχD

−1θ

cjk.(8.45)

This readily gives the key estimate

∣∣c0(·,x0, ·)∣∣Γm6

x0∫0

|k(·,s, ·)|Γm ds for k ∈ Γm,∗ (7.17).(8.46)

Iteration together with a continuation principle obtained from the Moser-type inequality forΓm

spaces (7.21) produces a solutionc0 ∈ Γ∞χ,0(T0,ξ′) to (8.36) for a possibly smallerT0 > 0.The equations for higher order glancing profiles are similar but linear (recall (8.30)). They are

solved in Section 9.4.

8.2. Real resonances and adapted bases

A resonanceoccurs when there is a relation∑i∈M(ξ′)

αiφi = 0,

where at least 3 of theαi ∈ Z are nonzero. Supposeξ′ is such that all resonances involve onlythe real phases. That is,∑

i∈M(ξ′)

αiφi = 0⇒ αi = 0, i ∈P(ξ′)∪N (ξ′).(8.47)

This situation can be handled by constructing anadapted basisB(ξ′) for the Q-span ofφj j∈G(ξ′)∪O(ξ′)∪I(ξ′).

SupposeB(ξ′) = ψk(x) k=1,...,b is a basis for theQ-span of φj j∈G∪O∪I . With ψ =(ψ1, . . . ,ψb) andj ∈ G ∪O ∪ I let

Dj =α ∈ Zb: α ·ψ is in theR-span ofφj

,(8.48)

and letα∗j be the element of minimal length inDj such thatα∗j · ψ is a positive multiple ofφj .Observe that every element ofDj is some integer multiple ofα∗j .

DEFINITION 8.2. – A basis B(ξ′) = ψk k=1,...,b for the Q-span of φj j∈G∪O∪I inC∞(RN+1

x ) is adaptedif:(a) for eachk, ψk|xN=0 =mkφ0, mk ∈ Z;(b) for eachj ∈ G(ξ′)∪O(ξ′)∪ I(ξ′), α∗j ·ψ = φj .

4e SÉRIE– TOME 33 – 2000 –N 3


Remark8.8. – (a) Iff (θ) is a periodic function ofθ ∈ Rb with spectrum contained inDj ,there is a uniquely determined periodic functionf∗(ωj) of ωj ∈R such that

f (θ) = f∗|ωj=α∗j ·θ.

To see this write

f (θ) =∑α∈Dj

fαeiαθ =

∑l∈Z

flα∗jeilα

∗j θ.

(b) No matter what relations exist among the φj j∈G∪O∪I , an adapted basis can always bechosen. The construction and use of adapted bases is explained in detail in [20]. The definitionthere is slightly more complicated since the shock is a free oscillating boundary.

If we now take

φ(x) =((ψk)k=1,...,b, (φj)j∈P(ξ′)∪N (ξ′)

)and assumeφ(x) satisfies a small divisor property similar to Definition 4.1, a construction justlike that outlined above produces approximate solutionsuε of the form (6.2).

8.3. Discussion of Theorem 6.2

Consider the linearized problem corresponding to (1.1):

Lu= f ,

Bu|xN=0 = g,(8.49)

u= 0 in x0 < 0,

wheref ∈ L2(ΩT ), g ∈ L2(bΩT ) and both vanish inx0 < 0. The starting point is the classicinequality for systems (L,B) satisfying the uniform Kreiss condition:

There existC > 0,λ0 > 0 such that forλ> λ0,

|u|0,µ,λ +1√λ〈u〉0,µ,λ 6C

(|f |0,µ,λ

λ+〈g〉0,µ,λ√

λ

).(8.50)

Here we use Gues’s weighted norms

|w|m,µ,λ =∑

06|α|6mµm−|α||e−λx0∂αw|L2(ΩT ),(8.51)

and denote boundary norms by〈〉m,µ,λ.As in [4] the idea is to take advantage of the factorεM1 in hypothesis (a) of Theorem 6.2 to

find an exact solution

uε = uε +wε

as in (b) of the theorem.wε is constructed by the iteration scheme (some epsilons are suppressed):

Lwn+1 = f (u+wn)− f (u)− εM1Rε,

Bwn+1|xN=0 =−εM1rε,(8.52)

wn+1 = 0 in x0 < 0.


410 M. WILLIAMS

The key estimate is proved in Section 10 by differentiating (8.52) and applying (8.50). Herewe set

‖u‖m,µ,λ = |u|m,µ,λ +1√λ〈u〉m,µ,λ.(8.53)

PROPOSITION 8.2. – Let m1 > (N + 1)/2 and ρ be as in Theorem6.2. Supposewn ∈Hm1

0 (ΩT ) satisfies

|wn|L∞(ΩT ) 6K.(8.54)

There existC(K,ρ)> 0, λ0(K,ρ)> 0 and a positive functionφ(λ) such that forλ> λ0(K,ρ)

‖wn+1‖m1, λ/ε, λ 6C(K,ρ)

λ‖wn‖m1, λ/ε, λ + εM1−m1φ(λ).(8.55)

(8.55) easily implies the existence ofλ1(K,ρ) > λ0(K,ρ) and ε1(λ) such that forλ > λ1,ε ∈ (0,ε1] we have

(a) ‖wn‖m1, λ/ε, λ 6 2εM1−m1φ(λ) for all n,(8.56)

(b) |wn|L∞(ΩT ) 6K for all n.

Convergence of the iterates is proved by a similar argument.

Remark8.9. – (a) In [6,7] Gues considers semilinear Dirichlet problems for viscousperturbationsH + εE of a first-order symmetric hyperbolic operatorH (E is second-order,elliptic). He shows that the solutionuε converges inHs, s < 1/2, to the solutionu0 of a maximaldissipative boundary problem forH. The obstruction to convergence fors > 1/2 is a boundarylayer of sizeε. When the boundary is charac teristic, the layer has another piece of size

√ε. The

profile equations in [6,7] are of parabolic-hyperbolic type.(b) [3] and [11] construct rigorous geometric optics expansions valid for times of order 1/ε for

nonlinear hyperbolic initial value problems in free space. The profilesa(εx,x,β · x/ε) involve athird (slow) scale (ε). Some of the profile equations are of Schrödinger type and have the form

∂a

∂T− P (∂Y ,∂y,∂θ)a= f (a),(8.57)

where a = a(T ,Y ,y,θ), x = (t,y), X = (T ,Y ) = (εt,εy). Here P is anti-self-adjoint andsecond-order only in the∂y derivatives.

These equations differ from our profile equations in 2 respects. The second-order derivativesdo not occur in variablesX = εx corresponding to the new scale. More significantly, thosederivatives do not involve the time variable, so the solution to the linearized initial value problemfor (8.57) is given by the operatoreTP . Because our profile equations involve second-orderderivatives inχ = xN/

√ε and nonhomogeneous boundary conditions onxN = 0, we were

forced to do mode-by-mode analysis involving the cutoffsρ(χ〈n〉3/2) and the use of Wieneralgebras.

4e SÉRIE– TOME 33 – 2000 –N 3


Part IIThe proofs

9. Theorem 6.1: Construction of the approximate solution

9.1. Determination ofa0, M∞σ0,j , j ∈ G(ξ′)∪O(ξ′)∪ I(ξ′)

To solve the system (8.12)(b,d) with boundary conditions (8.14)(a) and (8.17), we’ll need thefollowing spaces of profiles.

DEFINITION 9.1. – LetΩT ,θ =ΩT ×Tp, bΩT ,θ0 = bΩT ×T1. For m ∈ 0, 1, 2,. . . , T > 0,σ > 0 let

(a) Hm(T ) = U (x,θ): |U |m,T =∑|α|6m |∂α(x,θ)U |L2(ΩT ,θ ) <∞;

(b) Hm(T ) = u(x′,θ0): 〈u〉m,T =∑|α|6m〈∂α(x′,θ0)u〉L2(bΩT ,θ0) <∞;

(c) Hmσ (T ) = U (x,θ): |U |σ,m,T =∑|α|6m σ

m−|α||e−σx0∂α(x,θ)U |L2(ΩT ,θ ) <∞ . Hmσ (T ) isdefined similarly onbΩT ,θ0.

The following proposition clarifies an earlier assertion about the nonlinear terms in (8.12)(b,d).

Notation 9.1. – Supposeu(χ,x,θ)∈ Pχ(T ,ξ′) satisfiesEu= u, so

u= u(χ,x) +

M(ξ′)∑j=1

σj(χ,x,θj)rj

for someu, σj . Forf as in Theorem 6.1 we sometimes write

f (u,u) = f(Mχu;M∞u; (Mχσj), j ∈M(ξ′); (M∞σj ), j ∈M(ξ′)

)= f(Mχu,M∞u,MχuG(ξ′),M∞uG(ξ′), . . . ,MχuP∪N ,M∞uP∪N

),(9.1)

whereM(ξ′) = G(ξ′)∪O(ξ′)∪ I(ξ′)∪P(ξ′)∪N (ξ′).

PROPOSITION 9.1. – Supposea ∈ Pχ(T ,ξ′) satisfies(8.9)and(8.11)(b). Then forj ∈ G(ξ′)∪O(ξ′)∪ I(ξ′)

(a) M∞f (a) =E0f (0,a, 0,M∞aG , 0,aO∪I, 0),

(b) M∞Ejf (a) = Ejf (0,a, 0,M∞aG , 0,aO∪I, 0).

The proof makes use of the following Lemma, which shows that elliptic modes are nevercancelled by multiplication with a nonzero element ofPχ(T ,ξ′).

LEMMA 9.1. – Let a,b ∈ Pχ(T ,ξ′) and supposeEja= a for somej ∈ P(ξ′) ∪ N (ξ′). Thenfor k = 0, andk ∈ G(ξ′)∪O(ξ′)∪ I(ξ′) we haveEk(a · b) = 0.

Proof. –Write out the Fourier series ofa and b, observe that ifa 6= 0, the spectrum ofacontains only positive integers while specb ⊂Z(ξ′), and use the definition ofEk. 2

Proof of Proposition 9.1. –Note that

a=M∞a, aO∪I∪P∪N =M∞aO∪I∪P∪N .

Now


412 M. WILLIAMS

f (a)− f (0,a, 0,M∞aG , 0,aO∪I, 0)= f (0,a,MχaG ,M∞aG , 0,aO∪I, 0,aP∪N )

− f (0,a, 0,M∞aG , 0,aO∪I, 0)

= (MχaG)g1 + (aP∪N )g2

=F1 +F2(9.2)

for g1,g2 coming from Taylor’s formula. We haveM∞F1 = 0, sinceMχaG is rapidly decreasingin χ and g1 ∈ Pχ(T ,ξ′) by Proposition 5.1. Also,g2 ∈ Pχ(T ,ξ′) so EjF2 = 0 for j = 0 orj ∈ G(ξ′) ∪ O(ξ′) ∪ I(ξ′) by Lemma 9.1. (a) and (b) now follow since theEj commute withM∞. 2

Proposition 9.1 leads us to consider the system

L(∂)a0 = E0f (0,a0, 0,M∞a0,G , 0,M∞a0,O∪I, 0),

XjM∞σ0,jrj = Ejf (0,a0, 0,M∞a0,G , 0,M∞a0,O∪I, 0), j ∈ G ∪O ∪ I,

Ba0|xN=0 =G(x′),(M∞σ

±0,j |xN=0, θj=θ0, j ∈O

)=M±

(G∗,±,M∞σ

±0,j |xN=0, θj=θ0, j ∈ I

).(9.3)

PROPOSITION 9.2. – The semilinear system(9.3)has solutionsa0, M∞σ0,j , j ∈ G ∪O ∪ Ibelonging toC∞0 (T0,ξ′) for someT0 > 0 and with compact support inx.

Proof. –Convergence of the usual iteration scheme will follow from an a priori estimate forthe following linear system for the unknownsu, σj(x,θj), j ∈ G ∪O ∪ I:

(a) L(∂)u= f (x),(9.4)

(b) Xjσj (x,θj) = fj(x,θj), j ∈ G ∪O ∪ I,

(c) Bu|xN=0 =G(x′),

(d)(σ±j |xN=0, θj=θ0, j ∈O

)=M±

(G∗,±,σ±j |xN=0, θj=θ0, j ∈ I

),

whereM± is defined as in (8.17), all functions vanish inx0 < 0, and the data (f ,fj),G is smoothwith compact support inΩT , bΩT respectively.

Set

W (x,θ) = u+∑

j∈G∪O∪Iσj (x,θj)rj

and

F (x,θ) = f +∑

j∈G∪O∪Ifj(x,θj)rj .

LEMMA 9.2. – Letm ∈ 0, 1, 2,. . . . There existσ0 > 0, C > 0 such that forσ > σ0,

|W |σ,m,T +1√σ〈W 〉σ,m,T 6C

(1σ|F |σ,m,T +

1√σ〈G〉σ,m,T

).(9.5)

Proof. –Write Xp = ∂x0 +∑N−1

j=1 ap,j∂xj + ap,N∂xN , whereap,N > 0 (respectively< 0,= 0) if Xp is outgoing (respectively incoming, glancing). Assume now thatXp is outgoing,Xq incoming, andXr glancing.e−σx0W satisfies a system just like (9.4) except∂x0 in L(∂) andXj is replaced by (∂x0 + σ), andF ,G are multiplied bye−σx0.

4e SÉRIE– TOME 33 – 2000 –N 3


Multiply the σp equation in this new system bye−σx0σp(x,θp), integrate overΩT × T1, anddo similarly for theσq andσr equations to obtain:

(a) σ|σp|2σ,0,T 6 ap,N 〈σp〉2σ,0,T + |fp|σ,0,T |σp|σ,0,T ,

(b) σ|σq|2σ,0,T + |aq,N |〈σq〉2σ,0,T 6 |fq|σ,0,T |σq|σ,0,T ,(9.6)

(c) σ|σr |2σ,0,T 6 |fr|σ,0,T |σr |σ,0,T .

Use (9.4)(d) in (9.6)(a) to writeσp|xN=0 in terms of (G∗, σj ,j ∈ I). Foru we have the Kreissestimate

|u|σ,0,T +1√σ〈u〉σ,0,T 6C

(1σ|f |σ,0,T +

1√σ〈G〉σ,0,T

).(9.7)

Summing over all modes and absorbing terms in the usual way by takingσ large, we obtainthe estimate (9.5) for| |σ,0,T norms. Differentiating (9.4) and applying theL2 estimate yields(9.5). 2

ForM > (N + 2)/2, f (a0) as in (8.12), andW as above, since

|UV |m,T 6C(|U |L∞ |V |m,T + |U |m,T |V |L∞

),(9.8)

∣∣f (W )∣∣m,T6C

(|W |L∞

)|W |m,T , and(9.9) ∣∣Ejf (W )

∣∣m,T6C

∣∣f (W )∣∣m,T

, j = 0 or j ∈ G ∪O ∪ I,(9.10)

a standard iteration argument and continuation principle based on (9.5) yield solutionsa0,M∞σ0,j , j ∈ G∪O∪I to the semilinear system belonging toH∞0 for someT0 > 0. Their supportin x is compact by finite propagation speed. This concludes the proof of Proposition 9.2.29.2. Determination ofσ0,j , j ∈ P(ξ′)∪N (ξ′)

The next Proposition is proved just like Proposition 9.1.

PROPOSITION 9.3. – Supposea ∈ Pχ(T ,ξ′) satisfies(8.9)and(8.11)(b). Then forj ∈ P(ξ′)∪N (ξ′),

M∞Ejf (a) = Ejf (0,a, 0,M∞aG , 0,aO∪I, 0,aP∪N ).

This proposition and (8.12)(d) lead us to consider the system

(a) Xjσ0,jrj = Ejf (0,a0,M∞a0,G , 0,a0,O∪I, 0,a0,P∪N )≡Fj , j ∈ P ∪N ,(9.11)

(b) σ0,j |xN=0, θj=s(θ0)j = gj(x′,θ0),

where thegj ∈ C∞0 (T0) are obtained from (8.14)(b) using the Kreiss condition, ands(θ0) is as in(4.9).

PROPOSITION 9.4. – Let T0 be as in Proposition9.2. There exist functionsσ0,j , R0,j(x,θj),j ∈P(ξ′)∪N (ξ′) all belonging toC∞0 (T0,ξ′) with compact support inx, such that

Xjσ0,jrj =Fj + x[(M−1)/2]N R0,j(x,θj),

(9.12)σ0,j |xN=0, θj=s(θ0)j = gj(x′,θ0).

Here [(M − 1)/2] is the smallest integer> (M − 1)/2.


414 M. WILLIAMS

Proof. –The coefficient of∂xN in the complex vector fieldXj is nonzero. Requiring (9.11)(a)to hold just atxN = 0 determines∂xNσ0,j |xN=0, j ∈ P ∪ N . Similarly, requiring (9.11)(a) tohold to order [(M − 1)/2]− 1 atxN = 0 determines∂lxNσ0,j |xN=0, l6 [(M − 1)/2]. Choosingelements ofC∞0 (T0,ξ′) with compact support inx with these traces atxN = 0 determines theR0,j . 29.3. Determination ofMχσ0,j , j ∈ G(ξ′)

Although the next proof uses an argument from [3], we include it here to clarify howLj as in(8.22) arises from a glancing mode (ξ′,ξjN (ξ′)).

Proof of Lemma 8.1

Proof. –Forαj as in (8.21)

αj · dφ=(ξ′,ξjN (ξ′)

)=(ξk0 (ξ′′),ξ′′

)for somek. We’ll write Q(αj) =Q(ξk0 ,ξ′′). As in (3.7)

L(ξk0 ,ξ′′)π(ξk0 ,ξ′′

)= 0

implies

π(ξk0 (ξ′′),ξ′′

)( ∂ξk0∂ξN

+AN

)π = 0.(9.13)

Differentiate (9.13) to get

∂ξNπ

(∂ξk0∂ξN

+AN

)π+ π

∂2ξk0∂ξ2N

π + π

(∂ξk0∂ξN

+AN

)∂ξNπ = 0.(9.14)

Differentiate (I − π) = L(ξk0 ,ξ′′)Q(ξk0 ,ξ′′) (respectivelyI − π =QL) and multiply on the left(respectively right) byπ to get

−π · ∂ξNπ = π

(∂ξk0∂ξN

+AN

)Q,(9.15)

respectively

−∂ξNπ · π =Q(∂ξk0∂ξN

+AN

)π.(9.16)

Multiply (9.14) on the left and right byπ and use (9.15), (9.16) to find

−π(∂ξk0∂ξN

+AN

)Q(∂ξk0∂ξN

+AN

)π+ π

∂2ξk0∂ξ2N

π

−π(∂ξk0∂ξN

+AN

)Q(∂ξk0∂ξN

+AN

)π = 0.(9.17)

(ξk0 ,ξ′′) is glancing so∂ξk0 /∂ξN = 0 and (9.17) implies (8.21).2The same argument as for Proposition 9.1 gives

4e SÉRIE– TOME 33 – 2000 –N 3


PROPOSITION 9.5. – Supposea ∈ Pχ(T ,ξ′) satisfies(8.9)and(8.11)(b). Then forj ∈ G(ξ′)

MχEjf (a) = Ej[f (0,a,MχaG ,M∞aG , 0,aO∪I, 0)− f (0,a, 0,M∞aG , 0,aO∪I, 0)

].

This proposition and (8.23) lead us to consider

Lj (Mχσ0,jrj ) = Ej[f (0,a0,Mχa0,G ,M∞a0,G , 0,a0,O∪I, 0)

− f (0,a0, 0,M∞a0,G , 0,a0,O∪I, 0)], j ∈ G,(9.18)

Mχσ0,j |χ=0, θj=θ0 = gj(x′,θ0),

where thegj ∈ C∞0 (T ) are obtained from (8.14)(b) as before.The system (9.18) is of the form (8.26) and is solved by constructingb0 as in (8.35) and then

c0 as in (8.36).

PROPOSITION 9.6. – There existT0 > 0 and a solutionb0 ∈ Γ∞χ,0(T0,ξ′) to the semilinearsystem(8.35).

Proof. –1. The first step is to establish the estimate (8.41) on thenth Fourier coefficientb0,n(χ,x) and (8.42). Defineζ+

n ,βn by rewriting (8.40) as

b∧0,n(χ,ξ0− iγ,ξ′′,xN ) =

χ∫0

ei(χ−χ′)ζ+n βn(χ′, ·)dχ′+ eiχζ

+

g∧n =B∧n,1 +B∧n,2.(9.19)

Integrate by parts twice in

B∧n,1 =

χ∫0

∂χ′ (ei(χ−χ′)ζ+n )

−iζ+n

βn(χ′, ·)dχ′

to get

2B∧n,1(χ, ·) =ei(χ−χ

′)ζ+n

−iζ+n

βn(χ′, ·)|χ0 +

χ∫0

ei(χ−χ′)ζ+n

iζ+n

(−icnρ

(χ′〈n〉3/2

))h∧n(χ′, ·)dχ′

=i

ζ+n

[ ∞∫χ

e−i(χ−χ′)ζ+n icnρ

(χ′〈n〉3/2

)h∧n(χ′, ·)dχ′

− eiχζ+n

∞∫0

eiχ′ζ+n icnρ

(χ′〈n〉3/2

)h∧n(χ′, ·)dχ′

+

χ∫0

ei(χ−χ′)ζ+n icnρ

(χ′〈n〉3/2

)h∧n(χ′, ·)dχ′

].(9.20)

Now ζ+ =√cnX(ξ0− iγ,ξ′′), |Im X|= γ, and

suppχ′

ρ(χ′〈n〉3/2

)⊂[0,〈n〉−3/2

],

so we read off from (9.19), (9.20) the estimate


416 M. WILLIAMS

|b0,n|C(χ, Hmγ ) 6C√γ〈n〉 |hn|C(χ, Hmγ ) + 〈gn〉Hmγ ,(9.21)

for C independent ofn,Hmγ as in (7.2). (9.21) implies thatb0,n must vanish inx0 < T if hn and

gn do, so we may replace theHmγ norms in (9.21) byHm

γ (T ) norms (7.3). This gives (8.41) andsumming overn we obtain

|b0(χ,x,θ)|A1θ(C(χ, Hmγ (T ))) 6

C√γ|h|Aθ (C(χ, Hmγ (T ))) + 〈g〉A1

θ(Hmγ (T )).(9.22)

2. There exist constantsC1,C2 independent ofγ > 0,T > 0 such that

C1|u(x)|Hmγ (T ) 6∑|α|6m

|e−γx0∂αu|L2(ΩT ) 6C2|u|Hmγ (T ).(9.23)

Indeed, the second inequality is obvious. WhenΩT =RN+1+ the first inequality follows from the

Plancherel theorem and the fact that

(Dt − iγ)k(e−γtu

)= e−γtDk

t u.

Reduce to this case using the fact that foru ∈Hmγ (T ) there exist Seeley extensionsu ∈Hm

γ suchthat

|u|m,γ 6C|u|m,γ,T .(9.24)

3. Fix m> (N + 1)/2. Using (9.23) we may setγ = 1/T in (9.22) to obtain

|b0|A1θ

(C(χ, Hm(T ))) 6C√T |h|Aθ(C(χ, Hm(T ))) + 〈g〉A1

θ(Hm(T )).(9.25)

Since

|Ejh(x,ω,b0)|Aθ (C(χ, Hm(T ))) 6C|h|Aθ (C(χ, Hm(T ))),

we may use (7.9), (7.12) and Picard iteration to find a solutionb0(χ,x,θ) ∈ A1θ(C(χ,Hm(T0))),

for someT0 > 0. A continuation principle based on (7.13) yieldsb0 ∈ A1θ(C(χ,H∞(T0))) in the

usual way.4. Differentiate (8.35) to obtain (with obvious notation)

L∂θb0 = ρEj[∂h

∂ω(x,ω,b0) +

∂h

∂b0

∂b0

∂θ

] (ω = (θk),k ∈ G ∪O ∪ I

),

∂θb0|χ=0 = ∂θg,(9.26)

∂θb0 = 0 in x0 < 0.

Apply (9.25) and use (7.9), (7.12) again to deduce

∂θb0 ∈A1θ

(C(χ,H∞(T0))

).

Repeating the argument we obtain

b0 ∈A∞θ(C(χ,H∞(T0))

).

Eq. (8.35) then gives

b0 ∈A∞θ(C∞(χ,H∞(T0))

).(9.27)

4e SÉRIE– TOME 33 – 2000 –N 3


5. To seeb0 ∈ Γ∞χ,0(T0,ξ′) let b(k)0 denote thekth iterate in the Picard scheme and write

b(k)0 (χ,x,θ) =B(k)

1 +B(k)2 , b0 =B1 +B2(9.28)

where theB(k)i correspond in the obvious way to the termsBn,i of (9.19). Observe thatB(k)

2 isthe same for allk. SinceG ∈ C∞0 (T ) and

Im ζ+n = Im

√cnX(ξ0− iγ,ξ′′)>C

√|n|√γ for γ large,(9.29)

it follows easily thatB2 =B(k)2 ∈ Γ∞χ,0(T0,ξ′).

For eachn, Bn,1 in (9.19) is constant inχ for χ > 1 since suppχρ(χ〈n〉3/2) ⊂ [0, 〈n〉−3/2].We have

LBn,1 =

(i∂2χ

cn+X

)Bn,1 = ρ

(χ〈n〉3/2

)hn = 0 in χ> 1.(9.30)

Thus,XBn,1 = 0 in χ> 1. SinceX is glancing andBn,1 = 0 in x0 < 0 we deduceBn,1≡ 0 inχ> 1. ThusB1 ∈ Γ∞χ,0(T0,ξ′) and we concludeb0 ∈ Γ∞χ,0(T0,ξ′). 2

The followingL2 estimate for linear Schrödinger-type equations withhomogeneousboundaryconditions will be used in the construction ofc0 (8.36) and also in the solution of the higherprofile equations.H1(x′,L2∗(χ,xN ,θ)) denotes the space ofH1 functions ofx′ with values inthe space ofL2 functions of (χ,xN ,θ) with mean 0.

PROPOSITION 9.7. – Let a(χ,x,θ) = (aj(χ,x,θj),j ∈ G(ξ′)) belong toL20((−∞,T ] : D∗) ∩

H1(x′,L2∗(χ,xN ,θ)) and satisfy

Ljaj = Ej(ba) + dj , j ∈ G(ξ′),(9.31)

whereb(χ,x,θ)∈ Pχ,0(T ,ξ′), dj (χ,x,θ)∈ Γ∞χ,0(T ,ξ′). Then for06 x06 T

|a(·,x0, ·)|2L2(χ,x′′,θ) 6x0∫

0

eM(x0−s)|d(·,s, ·)|2L2 ds,(9.32)

whereM depends on|b|L∞ .

Proof. –Note thataj |χ=0 = 0 andaj |x0<0 = 0. In the usual way take theL2(χ,x′′,θj) innerproduct of both sides of (9.31) withaj and integrate by parts, using the anti-self-adjointness ofiD2

χD−1θ . To conclude use|Ej (ba)|L2 6C|b|L∞ |a|L2 and Gronwall’s inequality. 2

Remark9.1. – Similar estimates for|∂kx0a(·,x0, ·)|2L2(χ, Hm(x′′,θ)) follow for sufficiently

regulara by differentiation of (9.31) with respect to (x0,x′′,θ) (not χ). These estimates, whileuseful for the linear higher order profile equations, don’t help with the semilinear problem fora0

because they give noL∞ control inχ.

PROPOSITION 9.8. – WithT0 as in Proposition 9.6 there existT ′0, 0< T ′0 6 T0 and a solutionc0 ∈ Γ∞χ,0(T

′0,ξ′) to the semilinear system(8.36). c0 vanishes to infinite order atχ= 0.

Proof. –1. In this step we prove Proposition 8.1, which implies the key estimate (8.46).Let W1 (respectivelyW2) equal the left (respectively right) side of (8.45). Straightforward

computation shows that bothWi satisfy


418 M. WILLIAMS(∂t − i

D2χD−1θ

cj

)Wi = 2e

itD2χD−1θ

cjDχD

−1θ

cjk,

Wi|χ=0 = 0,(9.33)

Wi = 0 in x0 < 0.

W1−W2 satisfies the hypotheses of Proposition 9.7, and since (9.32) continues to hold ifXj isreplaced by∂t in the definition ofLj , we haveW1 =W2.

2. The functionk(χ,x,θ) in (8.38) lies inΓ∞χ,0(T ,ξ′) and vanishes to infinite order atχ= 0.From (8.46) and the fact that

eix0

D2χD−1θ

cj :D∗→D∗

we deduce the same properties forc0.3. With c0 = (c0,j(χ,x,θj), j ∈ G(ξ′)) in (8.36) andm> (N + 2)/2, use the estimate (8.46),

the Moser-type estimates (7.18), (7.20), and iteration to find a solutionc0 ∈ C0(T ′0,Γm) (7.22)for some 0< T ′0 < T0. A continuation principle based on (7.21) yieldsc0 ∈ C0(T ′0,Γ∞), andthen the equation givesc0 ∈ C∞0 (T ′0,Γ∞) ⊂ Γ∞χ,0(T

′0,ξ′). c0 vanishes to infinite order atχ = 0

because all the iterates do.2This completes the determination of theMχσ0,j ∈ Γ∞χ,0(T

′0,ξ′) and hence the construction of

the leading profilea0(χ,x,θ)∈ Pχ(T ′0,ξ′).

9.4. Construction of the higher order profiles

The construction of the higher profiles parallels the construction ofa0, but is much simplersince the equations are all linear. For example, there is no need for the cutoffρ(χ〈Dθ〉3/2) or forthe use of Wiener algebrasAθ. Thus, we’ll focus here mainly on the final step in the constructionof a1, the solution of (8.30) forMχσ1,j , j ∈ G(ξ′) with boundary data atχ= 0 determined from(8.27). At this stage the other pieces ofa1 have already been determined and lie inPχ,0(T0,ξ′),whereT0 is the time of existence ofa0.

PROPOSITION 9.9. – The system(8.30) for Mχσ1,j , j ∈ G(ξ′), with Mχσ1,j |χ=0 = gj(x′,θj) ∈ C∞0 (T0) given has a unique solution inΓ∞χ,0(T0,ξ′), whereT0 is the time of existenceof a0.

Proof. –The first term on the right in (8.30) may be written

MχEjf ′(a0)a1 = Ejf ′(a0)

( ∑j∈G(ξ′)

Mχσ1,jrj

)+Fj(χ,x,θj),(9.34)

whereFj ∈ Γ∞χ,0(T0,ξ′) is known. Thus, the problem has the same form as (9.31), but withboundary conditionsaj |χ=0 = gj . Write aj = tj + uj , where

Ljtj = 0,(9.35)

tj |χ=0 = gj ,

Ljuj = Ej [b(t+ u)] + dj ,(9.36)

uj |χ=0 = 0.

4e SÉRIE– TOME 33 – 2000 –N 3


The solutiont to (9.35) can be written down using the Fourier transform (see e.g. (8.40)) andlies in Γ∞χ,0(T0,ξ′). Standard linear arguments using (9.32) and the higher derivative estimatesdescribed in Remark 9.1 yield a solutionu to (9.36) in

C∞0((−∞,T0] :L2

(χ,H∞(x′′,θ)

))(new but obvious notation).

Eq. (9.36) then gives

u ∈C∞0((−∞,T0] : H∞(χ,x′′,θ)

).

Finally, to seeu ∈ C∞0 ((−∞,T0] : Γ∞)⊂ Γ∞χ,0(T0,ξ′) we commute powers ofχ through (9.36)using [

χ,D2χD−1θ

]u= 2∂χD

−1θ u,(9.37)

and apply the above argument to deduce successivelyχu ∈ C∞0 ((−∞,T0 : H∞(χ,x′′,θ)),χ2u ∈C∞0 ((−∞,T0) :H∞), etc. 2

Remark9.2. – (a) The profilesa2, . . . ,aM are constructed by the same analysis as fora1.Note that in the interior equations forM∞σk,j , j ∈ P(ξ′)∪N (ξ′) the termx[(M−1)/2]

N R0,j(x,θj)

in (9.12) may be replaced byx[(M−1)/2]−(k/2)N Rk,j (x,θj). As observed in Remark 8.30(a) we

need only determine

(I −E)aM−1, MχaM−1, MχσM−1,j , j ∈O(ξ′)∪ I(ξ′)∪P(ξ′)∪N (ξ′),

and (I −E)aM in order to solveI0, . . . , IM−22

.

(b) By finite propagation speed every profile has compact support inx′ and is compactlysupported inxN if not independent ofxN .

9.5. Analysis of the remainders

The proof of Theorem 6.1 is completed in the next Proposition.

PROPOSITION 9.10. – The remaindersRε(x), rε(x′) in (6.1) lie in Dm1ρ,0(T0) ∩ Bm1

ρ,0(T0),Dm1ρ,0(T0)∩ Bm1

ρ,0(T0) respectively for someρ > 0.

Proof. –The errorLuε− f (uε, u) may be written as a sum of two pieces

x(M−1)/2N R

(x, (θj)j∈P∪N

)|θj=

φjε

+ ε(M−1)/2Sε(χ,x,θ).(9.38)

The first piece is the error in solving the complex transport equations for elliptic modes (e.g.(9.12)), and the second is the usual error coming from Taylor remainders and reflecting thefailure of IM−1

2, IM

2to hold. Note thatR ∈ C∞0 (T ,ξ′), Sε ∈ Pχ(T ,ξ′) with smooth, bounded

dependence on√ε, and both have the support property described in Remark 9.2(b). Thus,

Sε|χ=xN√ε

, θ=φε∈Dm1

ρ,0(T0)∩ Bm1ρ,0(T0)

for someρ.Since

x(M−1)/2N R= ε(M−1)/2

(xNε

)(M−1)/2

R and Imφjε

= bj

(xNε

)ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE

420 M. WILLIAMS

with bj > 0 for j ∈ P ∪N , we have as well

(xNε

)(M−1)/2

R(x,

(φjε

)j∈P∪N

)∈Dm1

ρ,0(T0)∩Bm1ρ,0(T0)(9.39)

for someρ.The boundary remainderrε is handled similarly. 2Remark9.3 (Weakly regular boundary frequencies). – Supposeξ′ = (ξ0,η) ∈RN \0 satisfies

(3.13), but all the associated elliptic modesβj(ξ′) = (ξ′,ξjN (ξ′)), j ∈ P(ξ′) ∪ N (ξ′) are onlyrequired to beweakly regular(Definition 3.5). Letξ = (ξ0,η,ξN )=(ξ′,ξjN (ξ′)) for somej ∈P(ξ′) ∪ N (ξ′). Part (c) of Definition 3.5 allows us to defineπj as before (Notation 4.1). Thefunctionξ0(η,ξN ) in part (a) of Definition 3.5 is real analytic inη and analytic inξN near (η,ξN ),so the complex vector fieldXξ may be defined as in (3.9) and (3.7) is still valid. Part (b) ofDefinition 3.5 implies thatE+(ξ′) is spanned by eigenvectors (not just generalized eigenvectors)of A(ξ′), a point that is important in our solution of the profile equations.

The operatorsE, Ej , andQ are defined just as before. Instead of (8.10) we now have

Eak(χ,x,θ) = ak(χ,x) +∑

j∈G∪O∪Iσk,j (χ,x,θj)rj +

∑j∈P∪N

Ak,j (χ,x,θj)(9.40)

whereAk,j takes values in the eigenspace ofA(ξ′) associated toξN (which equals the eigenspaceof A(η,ξN ) associated to−ξ0). In (9.11)(a), for example,σ0,jrj should be replaced byA0,j .Since the elliptic transport equations are again governed by complex vector fields likeXξ, theiranalysis goes through with no substantial changes.

10. Theorem 6.2: Exact solutions near approximate ones

The proof is similar to an argument in [4] but simpler. We give a brief version here for the sakeof completeness.

Notation 10.1. – (a) Forα ∈ 0, 1, 2,. . . ∂αx is any operator of the form∂α0x0· · ·∂αNxN such that

α0 + · · ·+ αN = α. ∂αx′ is an analogous tangential operator.(b) Forα ∈ 1, 2, 3,. . . denote by∂〈α〉φ any product of the form (∂α1φi1) · · · (∂αrφir ) where

16 r 6 α, α1 + · · ·+ αr = α, αi > 1. If α= 0 set∂〈0〉φ= 1.(c) ‖w‖′m,µ,λ is the analogue of (8.54) defined by replacing∂αx with ∂αx′ .(d) Let |u|∗ = |u|L∞ .

LEMMA 10.1 ([4]). – Let α1 + · · ·+ αr 6 k 6m, αi ∈ 0, 1, 2,. . . , andui ∈ Hm0 (ΩT ) ∩

L∞(ΩT ). For 16 λ6 µ

µm−1∣∣(∂α1u1) · · · (∂αrur)

∣∣0,µ,λ6C

r∑i=1

|ui|m,µ,λ

(∏j 6=1

|ui|∗).

Proof of Proposition(8.52)(a). – 1. For 06 k 6m1 we will apply theL2 estimate (8.50) tothe problem satisfied byµm−k∂kx′wn+1, and then use the equation to estimate∂xN derivatives.

Below we’ll write m = m2, wn+1 = w, wn = b, C will always be a constant depending on(K,ρ), φ is always someC∞ function of its arguments, and 0< ε6 16 εµ.

4e SÉRIE– TOME 33 – 2000 –N 3


We will show

‖w‖m,µ,λ 6C

λ‖b‖m,µ,λ +

m∑k=0

εM1−m(εµ)m−kρ(10.1)

for λ> λ0(K,ρ), and then setµ= λ/ε to get (8.55). The first step is to show

‖w‖′m,µ,λ 6C

λ‖b‖mµ,λ +

m∑k=0

εM1−m(εµ)m−kρ(10.2)

for λ> λ′0(K,ρ).2. Set∂kx′ = ∂k. For 06 k 6m the problem satisfied by∂kw is

L∂kw = ∂k[f (u+ b)− f (u)

]− εM1Rε

,

Bw|xN=0 =−εM1∂krε,(10.3)

w= 0 in x0 < 0.

Write

f (u+ b)− f (u) = g(u,b)b.

Then|µm−k∂k[f (u+ b)− f (u)]|0,µ,λ yields terms of the form

µm−k∣∣φ(u,b)∂〈α〉u∂〈β〉b∂γb|0,µ,λ,(10.4)

whereα+ β + γ = k. u ∈Dmρ,0(T ), so

(10.4)6 µm−k+α(εµ)−αC|∂〈β〉b∂γb|0,µ,λ,(10.5)

with β + γ = k−α. Lemma 10.2 implies

(10.5)6C|b|m,µ,λ.(10.6)

We have

µm−k∣∣∂k(εM1Rε)

∣∣0,µ,λ6 εM1−m(εµ)m−kρ

sinceRε ∈ Bmρ,0(T ), and a similar estimate holds forεM1∂krε. This gives (10.2). Using theequation to estimate normal derivatives gives (10.1).2

PROPOSITION 10.1 (Boundedness of the iterates‖wn‖m1, λ/ε, λ). – There existλ1(K,ρ)>λ0(K,ρ) andε1(λ) such that forλ> λ1, ε ∈ (0,ε1] thewn defined by(8.52)satisfy

(a) ‖w‖m1, λ/ε, λ 6 2εM1−m1φ(λ) for all n,(10.7)

(b) |wn|∗ 6K for all n.

Proof. –(a) follows immediately from (8.55) forλ > λ1 large enough. (b) follows from (a)for ε1 small enough, since foru ∈Hm1

0 (ΩT ), 0< δ <m1− ((N + 1)/2), we have the Sobolevinequality [4]

|u|∗ 6 µ−δC(λ)|u|m1,µ,λ.(10.8)

2ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE

422 M. WILLIAMS

PROPOSITION 10.2 (Convergence of (wn)). – There existλ2(K,ρ)> λ1(K,ρ) andε2(λ) 6ε1(λ) such that forλ> λ2 the sequence(wn) converges in the‖ ‖m1, λ/ε, λ norm, uniformly withrespect toε ∈ (0,ε2]. The limitw satisfies

‖w‖m1, λ/ε, λ 6 2εM1−m2φ(λ).(10.9)

Hence(w,w|xN=0) ∈ εM1Bm1σ,0(T )× εM1Bm1

σ,0(T ) for someσ > 0.

Proof. –∆n+1 =wn+1−wn satisfies

L∆n+1 = f (u+wn)− f (u+wn+1) = ∆ng(u,wn,wn−1),

B∆n+1|xN=0 = 0,(10.10)

∆n+1 = 0 in x0 < 0.

By the argument that produced (10.1) we use (10.7)(a) to show

‖∆n+1‖m1, λ/ε, λ 6C

λ|∆n|m1, λ/ε, λ +Cφ1(λ)|∆n|∗(10.11)

for λ large andε ∈ (0,ε1(λ)]. (10.11) and (10.8) allow us to chooseλ2(K,ρ), ε2(λ) so that

‖∆n+1‖m1, λ/ε, λ 612‖∆n‖m1, λ/ε, λ

for λ> λ2, ε ∈ (0,ε2]. 2

11. Generic validity of the small divisor properties

We first recall a result from [20] which gives generic validity of the small divisor properties inthe hyperbolic region

H=ξ′ ∈RN \ 0:M(ξ′) =O(ξ′)∪ I(ξ′)

.

For ξ′ ∈H we have

φ(x) =(x′ξ′ + xN ξ

iN (ξ′)

)i∈O(ξ′)∪I(ξ′)

.

Setki(ξ′) = ξiN (ξ′).

PROPOSITION 11.1. – LetO ⊂H be open. Suppose for eachj ∈ 1, . . . ,m there existsl 6= jsuch that them− 2 vectors

dξ′

(ki − kjkl − kj

), i ∈ 1, . . . ,m \ l,j,

are linearly independent inRN for eachξ′ ∈O. Then for almost everyξ′ ∈O, ξ′ (or φ) satisfiesthe small divisor condition(4.12), (4.13).

In this section, as in this whole paper, we are mainly interested inξ′ that correspond to glancingand elliptic modes as well as hyperbolic modes. To simplify the exposition we’ll focus here ona 6× 6 example where all three types of modes are present simultaneously. It will be clear that

4e SÉRIE– TOME 33 – 2000 –N 3


similar propositions can be formulated and proved in the same way for systems of any dimension.Proposition 11.1 represents the simplest case in this collection of results.

The proof depends on the following classical result in simultaneous diophantine approxima-tion.

THEOREM 11.1 ([1]). – (a)Fix ε > 0. For almost allθ = (θ1, . . . ,θn) ∈ Rn there exists aconstantCθ > 0 such that

|α1θ1 + · · ·+ αnθn −αn+1|>Cθ|α|−n−ε(11.1)

for all α ∈ Zn+1 \ 0.(b) In particular (11.1)holds when1,θ1, . . . ,θn are algebraic numbers linearly independent

over the rationals.

Notation 11.1. – (a) LetL as in (1.1) be a 6× 6 system,p(ξ) = p(ξ0,η,ξN ) = detL(ξ),andB ⊂ RN−1 \ 0 an open set satisfyingB ∩ −B = ∅. Supposeξ0 :B → R is aC∞ functionhomogeneous of degree 1 such that forξ′(η) = (ξ0(η),η), η ∈ B,

p(ξ′(η),ξN ) =(ξN − k1(ξ′)

)2(ξN − k2(ξ′)

)(ξN − k3(ξ′)

)×(ξN − k+

4 (ξ′))(ξN − k

+

4 (ξ′)).(11.2)

Here theki areC∞ functions on the submanifold ofRN parametrized byξ′(η), k1,k2,k3 arereal and distinct for eachη ∈ B, and Imk4 > 0 onB. For η ∈ B defineki(−ξ′(η)) =−ki(ξ′(η)),i = 1, 2, 3, andk4(−ξ′(η)) = −k4(ξ′(η)). Note that (11.2) holds now withξ′(η) replaced by−ξ′(η).

(b) Fori= 1,. . . , 4 andξ′(η) = (ξ0(η),η),

φi(x) = x′ξ′ + xNki(ξ′(η)

); φ5(x) =−x′ξ′ − xNk4

(ξ′(η)

).

Write k4 = σ4 + iτ4.

PROPOSITION 11.2. – (a)In the notation above suppose there existsl ∈ 1, 2, 3 such thatthe2 vectors

dη

(ki − σ4

kl − σ4

), i∈ 1, 2, 3, 4\ l, 4,(11.3)

are linearly independent inRN−1 for eachη ∈ B. Then for almost everyη ∈ B, ξ′ = (ξ0(η),η)satisfies the first small divisor condition(4.12).

(b) Assume(11.3). Suppose also that for eachj ∈ 1, 2, 3 there existsl ∈ 1, 2, 3 \ j suchthat the vector

dη

(ki − kjkl − kj

), i∈ 1, 2, 3 \ j, l,(11.4)

is nonzero for eachη ∈ B. Then for almost everyη ∈ B, ξ′ = (ξ0(η),η) satisfies the second smalldivisor condition(4.13).

Proof. –(a) Fix ε > 0. Forα= (α1, . . . ,α5) ∈ Z5 \ 0

dx(α · φ) =((α1 + α2 + α3 + α4− α5)ξ′,α1k1 +α2k2 + α3k3

+ (α4− α5)σ4 + i(α4 + α5)τ4)

≡ β(α,η).(11.5)


424 M. WILLIAMS

If α1 + α2 + α3 + α4 − α5 6= 0 or α4 + α5 6= 0 then |β(α,η)| > C|η|. If both equal 0, then(α1,α2,α3) 6= 0 and withl= 1 say in (11.3)

|β(α,η)|= |α1(k1− σ4) + α2(k2− σ4) + α3(k3− σ4)∣∣

= |k1− σ4|∣∣∣∣α1 + α2

k2− σ4

k1− σ4+α3

k3− σ4

k1− σ4

∣∣∣∣.(11.6)

Define theC∞ mapf :B→R2 by

f (η) =

(k2− σ4

k1− σ4,k3− σ4

k1− σ4

).

LetE be the set of measure zero inR2 where (11.1) fails to hold. By hypothesisf is a submersionat everyη ∈ B, so the preimagef−1(E) has measure zero. Thus, for almost everyη ∈ B thereexistsCη > 0 such that the right side of (11.6)>Cη|α1,α2,α3|−2−ε.

(b) Fix ε > 0. We now considerα ∈ Z(ξ′) \ 0, soα4> 0,α5> 0. It suffices to prove the resultwith B replaced by an arbitrary open setO ⊂⊂B.

DefineS = ξ = (ξ′,ξN ) ∈RN ×C: Im ξN > 0, |ξ|= 1 and letP : (RN ×C) \0→ S be themapP (ξ) = ξ/|ξ|. Forδ > 0 define

Eδ =ξ ∈ S: d(ξ,p−1(0)∩ S)> δ

,(11.7)

whered indicates distance measured onS.There existsCδ > 0 such that

|p(ξ)|>Cδ for ξ ∈ Eδ.(11.8)

For η ∈ O let ξ′ = (ξ0(η),η) or (−ξ0(η),−η). Providedδ1 > δ2 > 0 are chosen small enough, ifP (ξ′,ξN ) ∈ (Eδ2)

c there exists an uniquel ∈ 1, 2, 3, 4 andC > 0 such that

(a) P(ξ′,ξN + s(kl(ξ

′)− ξN ))∈ (Eδ1)

c for 06 s6 1,(11.9)

(b)

∣∣∣∣ ∂p∂ξN∣∣∣∣>C atP (ξ′,ξN ) if l 6= 1,

(c)

∣∣∣∣ ∂2p

∂ξ2N

∣∣∣∣>C atP (ξ′,ξN ) if l= 1,

(d) ImξN = 0 if l ∈ 1, 2, 3 and (ξ′,ξN ) = β(α,η).

Suppose first thatP (β(α,η))∈ Eδ2. Then by (11.8)∣∣p(β(α,η))∣∣>Cδ2|β|6.(11.10)

Note that if bothα1 + α2 + α3 + α4 − α5 = 0 andα4 + α5 = 0, thenα1 + α2 + α3 = 0 now.Thus, the argument of part (a) shows that except forη in a set of measure zero

|β|6>Cδ2

(Cη|α|−1−ε)6

(11.11)

in this case.

4e SÉRIE– TOME 33 – 2000 –N 3


Suppose next thatP (β(α,η)) ∈ (Eδ2)c, but β /∈ p−1(0), and apply (11.9)(a) withβ(α,η)

in place of (ξ′,ξN ). If l = 1 in (11.9)(a) Taylor’s formula gives, withβ(α,η) = (β1,β2) andM = α1 + α2 + α3 + α4− α5,

p(β(α,η)

)= p(β(α,η)

)− p(Mξ′,k1(Mξ′)

)=

12∂2p

∂ξ2N

(β1,k1(β1) + s

[β2− k1(β1)

])·(β2− k1(β1)

)2(11.12)

for some 0< s < 1. By (11.9)(c) the first factor on the right in (11.12) had modulus>C|η|m−2.The second factor, in the case whereM > 0 equals[

α2(k2− k1) +α3(k3− k1)]2

(11.13)

since (11.9)(d) impliesα4 = α5 = 0. Arguing just as in (a) we see that except forη in a set ofmeasure zero, this factor has modulus> Cη|α|−2−ε. The caseM < 0 is treated similarly (seebelow).

Whenl= 2 or 3 in (11.9)(a), argue as above using (11.9)(b) instead of (11.9)(c).Finally, if l = 4 in (11.9)(a), one obtains a product similar to (11.12) in which the first factor

has modulus>C|η|m−1 by (11.9)(b). The second factor (β2− k4(β1)) in the caseM > 0 equals[α1(k1− σ4) + α2(k2− σ4) + α3(k3− σ4) + iτ4(2α5− α1− α2− α3)

].(11.14)

If 2α5− α1− α2− α3 6= 0, the modulus of (11.14) is> C|η|. If 2α5− α1− α2− α3 = 0, then(α1,α2,α3) 6= 0 and|β2− k4(β1)| reduces again to (11.6).

WhenM < 0 write

k4(β1) = k4(Mξ′) = k4(−|M |ξ′

)=−|M |k4(ξ′).

The second factor now equals[α1(k1− σ4) + α2(k2− σ4) +α3(k3− σ4) + iτ4(2α4 +α1 +α2 + α3)

](11.15)

and is handled like (11.14).2Remark11.1. – (a) In the analogue of Proposition 11.2 for the case when preciselyk glancing

modes are present instead of just one,B should be replaced by an open subset ofRN−k \ 0.(b) Part (b) of Theorem 11.1 may be used to identify explicit choices ofξ′ for which the small

divisor condition holds.

Part IIIExamples of blow-up

12. Examples of blow-up caused by high-order glancing modes

In Theorems 6.1 and 6.3 we assumed thatp(ξ′,ξN ) = 0 had real rootsξN of multiplicity atmost two. Here we present examples showing that when real roots of order> 3 are present, thetime of existenceTε can shrink to 0 asε→ 0.

Example1. – We work onR2+1+ = (x, t,y): x > 0 and denote dual variables by (ξ,τ ,η).

In this example part of the boundary data oscillates at a boundary frequency (τ0,η0) such thatp(ξ,τ0,η0) = 0 has a triple real rootξ0.


426 M. WILLIAMS

Let P (D) be a third-order scalar operator with total symbol

p(ξ,τ ,η) = (ξ − η)3 + 3τ (ξ − η)2 +(3τη2− τ3

)(12.1)

and consider the semilinear system foruε,vε

(a) P (D)uε = 0,

uε|x=0 = 0,(12.2)

Dxuε|x=0 = εG(t,y)ei(τ0,η0)·(t,y)/ε,

uε = 0 in t <−T (T > 0),

(b) (∂t + ∂x)vε = |∂2xuε|βv2

ε,

vε|x=0 =H(t,y)ei(τ1,η1)·(t,y)/ε, H(0)> 0,

vε = 0 in t <−T.Here (τ0,η0) = (0, 1), (τ1,η1) 6= 0 is arbitrary,H andG are supported in−T 6 t 6 T , and forγ < 1 to be chosen we take

G(t,y) = eγtg1(t)g2(y),(12.3)

whereg1(t), g2(y) are smooth functions, and

(a) g1(τ )> 0, g1 > 1 on [−b,b](b) g2(η)> 0, suppg2(η)⊂

|η|6 γ1+δ

for some(12.4)

0< δ < 1/3, g2(η)> 1 on|η|6 γ1+δ/2.

LEMMA 12.1 (Cardano). –Consider the cubicp(ζ) = ζ3 + 3pζ + q where p,q ∈ R. Ifq2 + 4p3 < 0 (respectively> 0, = 0), thenp(ζ) has3 distinct real (respectively,1 real and2complex conjugate, repeated real) roots.

The lemma applied to (12.1) implies thatP (D) is strictly hyperbolic with respect tot.Thus, (12.2) is a system with 2 blocks, with each block satisfying the Kreiss condition andnoncharacteristic with respect tox= 0.

Remark12.1. – To see that two is the correct number of boundary conditions foruε in (12.2),observe that forγ > 0 p(ξ,−iγ, 0)= 0 has 2 roots with Imξ > 0, one with Imξ < 0. The sametherefore holds for the roots ofp(ξ,τ − iγ,η) = 0 for any (τ ,η) 6= 0.

LEMMA 12.2. – Regarded as a cubic inξ, p(ξ,τ ,η) as in (12.1) has 3 distinct real(respectively,1 real and2 complex conjugate, repeated real) roots, when

−τ2(τ2− 3η2

)(τ2 + η2

)< 0 (respectively> 0,= 0).

Indeed, ifτ = 0, ξ = η is a triple root, while ifτ 6= 0, τ2− 3η2 = 0, thenξ = η is a double rootandξ = η− 3τ a simple root.

Proof. –Set β = ξ − η in (12.1) and note thatβ3 + 3τβ2 + (3τη2 − η3) takes the formζ3 + 3pζ + q as in Lemma 12.1 ifζ = β + τ andp=−τ2, q = τ3 + 3τη2. Thus,

q2 + 4p3 =−3τ2(τ2− 3η2

)(τ2 + η2

).

The last sentence of the Lemma is obvious.24e SÉRIE– TOME 33 – 2000 –N 3


Remark12.2. – β0 = (ξ0,τ0,η0) = (1, 0, 1) is a glancing mode of order 3 forP (D). That is,ξ0 = 1 is a triple root ofp(ξ,τ0,η0) = 0.

The solution to (12.2)(a) is given by

uε(x, t,y) =

∫eit(τ−iγ+

τ0ε )+iy(η+

η0ε )εG(τ − iγ,η)

eixξ1(X) − eixξ2(X)

ξ1(X)− ξ2(X)dτ dη.(12.5)

HereX = ((τ0,η0)/ε) + (τ − iγ,η), ξ1(X), ξ2(X) are the roots ofp(ξ,X) = 0 with positiveimaginary parts, and we ignore factors of 1/2π. Fora > 0, b1 > 0, b2 > 0 small and|t|6 b1ε

a,x∼ ε(2+a)/3, |y|6 b2, we estimate

|∂2xuε|= etγ |

∫eitτ+iyηεg1(τ )g2(η)Q(X)dτ dη|(12.6)

from below. In (12.6)

Q(X) =ξ2

1(X)eixξ1(X) − ξ22(X)eixξ2(X)

ξ1(X)− ξ2(X).(12.7)

Remark12.3. – Blow-up will be caused by the fact that forτ ,γ,ε small enough,|ξ1(X)−ξ2(X)| ∼ ε−2/3 sinceβ0 is glancing of order 3. Ifβ0 were hyperbolic or glancing of order 2, wewould haveε−1 instead. Along with the factor ofε in (12.6), this would exactly cancel the 1/ε2

from ξ21(X) andξ2

2(X), and blow-up would not occur. Roughly speaking, the boundary data in(12.2)(a) is oscillating with a single linear phase that “resonates” with a structural defect in thesolution operator defined by (12.5). This mechanism is completely different from the focusingmechanisms in free space identified in [8]. The latter mostly involve phases with critical points.

1. With Y0 = (τ0,η0), Y = (τ0,η0) + ε(τ − iγ,η) we use a classical argument (e.g., Sakamoto[16]) to compare the zeros ofp(ξ,Y ) to those ofp(ξ,Y0). We have

p(ξ,Y ) = (ξ − ξ0)3 + h2(Y )(ξ − ξ0)2 + h1(Y )(ξ − ξ0) + h0(Y ),(12.8)

wherehj(Y0) = 0, j = 0, 1, 2. Hereh0(Y ) = p(ξ0,Y ) satisfies

∂h0

∂τ(Y0) = 3η2

0 ≡C0 > 0.(12.9)

Write

h0(Y ) =C0ε(τ − iγ) +∂h0

∂η(Y0)εη+ O

(|ε(τ − iγ,η)|2

)and setq(ξ,Y ) = (ξ − ξ0)3 +C0ε(τ − iγ). Forγ < 1, |η|6 γ1+δ (as is the case on suppg2(η)),C1 >C0, and|ξ − ξ0|6 (C1ε|τ ,γ|)1/3 we obtain∣∣p(ξ,Y )− q(ξ,Y )

∣∣6C(ε4/3|τ ,γ|4/3 + εγ1+δ + ε2|τ ,γ|2),(12.10)

whereC will always denote a constant independent of (ε,τ ,γ,η).Suppose now

|τ |< ε−a for some 0< a <13.(12.11)


428 M. WILLIAMS

(12.10) implies there exists anε0(γ) such that forε < ε0(γ)∣∣p(ξ,Y )− q(ξ,Y )∣∣6 3Cεγ1+δ.(12.12)

Write the roots ofq(ξ,Y ) as

ξj = ξ0 +[C0ε(iγ − τ )

]1/3

j, j = 1, 2, 3,

where Imξj > 0 for j = 1, 2. Letk ∈ 1, 2, 3, fix M > 0, and note that forγ(M ) < 1 smallenough, if|ξ− ξk|=Mε1/3γ(1/3)+δ , then|ξ− ξ0|6 (C1ε|τ ,γ|)1/3 and|ξ− ξj |> (C0ε|τ ,γ|)1/3,for j 6= k. Thus,

|Q(ξ,Y )|=3∏j=1

|ξ − ξj |>(C0ε|τ ,γ|

)2/3 ·Mε1/3γ1/3+δ >C2/30 Mεγ1+δ.(12.13)

ForM such thatC2/30 M > 3C (C as in (12.12)), (12.12), (12.13) and Rouché’s Theorem imply

ξj(Y ) = ξj + O(ε1/3γ1/3+δ

)= ξ0 +

[C0ε(iγ − τ )

]1/3

j+ O

(ε1/3γ1/3+δ

).(12.14)

Thus,

ξj(X) =ξ0

ε+ ε−2/3

[C0(iγ − τ )

]1/3

j+ ε−2/3O

(γ1/3+δ

)(12.15)

for

γ = γ(M ), ε < ε0(γ), |η|6 γ1+δ, |τ |6 ε−a, 0< a< 1/3.(12.16)

2. We now determine the magnitude and direction ofεQ(X) (12.7) whenX = ((τ0,η0)/ε) +(τ − iγ,η) is in the range defined by (12.16) and

x= λε(2+a)/3 for 0< δ1 < λ< δ2, δ1,δ2 small.(12.17)

(12.15) implies the main contribution toεQ(X) is given by

ε · ξ20

ε2eiλξ0ε

(a−1)/3

eiλC

1/30 εa/3(iγ−τ )1/3

1 − eiλC1/30 εa/3(iγ−τ )1/3

2

ε−2/3C1/30 [(iγ − τ )1/3

1 − (iγ − τ )1/32 ]

.(12.18)

The remainder is smaller by a factor ofc(ε,γ) 1.Since|τ |6 ε−a, for δ2 small enough the numerator inside the braces in (12.18) equals

iλC1/30 εa/3

[(iγ − τ )1/3

1 − (iγ − τ )1/32

]·F (ε,γ,τ ),(12.19)

where|F − 1|< 1/10 for (ε,γ,τ ) as in (12.16). Thus,

(12.18)=(iλξ2

0eiλξ0ε

(a−1)/3) · (ε(a−1)/3)·F.(12.20)

The important property of (12.20) is that it blows up likeε(a−1)/3 and maintains a nearlyfixeddirection (for fixed ε,λ) for all |τ | 6 ε−a. For ε1 < ε0(γ) small enough,εQ(X) has the sameproperty forε < ε1.

4e SÉRIE– TOME 33 – 2000 –N 3


3. Consider the integral (12.6) over the region|τ | 6 ε−a, |η| 6 γ1+δ for ε < ε1, |y| < b,t∼ ε(2+a)/3, andx as in (12.17). (12.4) and the fact thateitτ+iyη ∼ 1 imply there is essentiallyno cancellation of the contributions from (12.20) for|τ | 6 b. Thus, the modulus of the integralover|τ |6 ε−a, |η|6 γ1+δ is bounded below by

C(λ,γ,b)ε(a−1)/3, 0< a < 1/3.(12.21)

4. Sinceg1(τ ) is rapidly decreasing inτ and|eixξj (X)|6 1, any polynomial (in|τ ,ε−1|) upperbound on|ξ1(X)− ξ2(X)|−1 for |τ | > ε−a will imply that the contribution to (12.6) from theregion|τ |> ε−a, |η|6 γ1+δ is negligible.

Let

S =Z = (τ ,γ,η): |τ ,γ,η|= 1, γ > 0

.

For δ3 small let

S1 =Z ∈ S:

∣∣(γ,η)− (0,η0)∣∣6 |τ |4/36 δ4/3

3

S2 =

Z ∈ S:

∣∣(γ,η)− (0,η0)∣∣6 δ4/3

3 , |τ |> δ3.

Sincep(ξ,τ ,η) has triple zeros inξ only whenτ = 0, we have∣∣ξ1(Z)− ξ2(Z)∣∣>C > 0 forZ ∈ S2.(12.22)

ForZ ∈ S1 an argument using Rouché’s Theorem as in part1 of this proof shows∣∣ξ1(Z)− ξ2(Z)∣∣>C|τ |1/3 for Z ∈ S1.(12.23)

Using the homogeneity ofξj(X), we see that ifX = ((τ0,η0)/ε) + (τ − iγ,η) projects toS2

(respectivelyS1) underX→X/|X |, (12.22) and (12.23) imply∣∣ξ1(X)− ξ2(X)∣∣>C|X |

(12.24) (respectively

∣∣ξ1(X)− ξ2(X)∣∣>C|X |2/3|τ |1/3

).

Provided 0< a < 1/3 and (4/3)(1− a) < 1, the restrictions on (τ ,γ,η,ε) imply these are theonlyX we need to consider. From (12.21) and (12.24) we conclude for sucha∣∣∂2

xuε∣∣>C(λ,γ,b)ε(a−1)/3,(12.25)

for |t|6 b1εa, x= λε(2+a)/3, 0< δ1 < λ< δ2, |y|6 b2.

5. As in [8] we demonstrate blow-up invε using a simple fact about ODEs.

LEMMA 12.3. – Supposea(t) is a smooth nonnegative function on[0,∞) and∫∞

0 a(t)dt >1/y0. The maximal interval of existence[0,T0) for the initial value problem

y′ = a(t)y2, y(0) = y0 > 0

is given by∫ T0

0 a(t)dt= 1/y0.

Considervε on the characteristic of∂t + ∂x starting at (x,y, t) = 0. (12.25) and Lemma 12.3imply that if β is large enough (β > (2+ a)/(1− a)), the maximal time of existence on thischaracteristic is less thanCε(2+a)/3.


430 M. WILLIAMS

6. The example is not yet completely satisfactory since the data in (12.2) is supported int>−T , and we have not ruled out [−T ,−T/2], for example, as an interval of existence forvεindependent ofε. To rectify this defect replaceG(t,y) andH(t,y) in (12.2) by

Gk(t,y) =1kM

G(kt,y), Hk(t,y) =1kM

H(kt,y).

This data is supported int>−T/k and for anys, theHs norm of (Gk,Hk) is uniformly boundedwith respect tok ∈ 1, 2, 3,. . ., providedM =M (s) is big enough. Letuε,k, vε,k be the solutionto (12.2)k defined by this new data. The preceding argument shows

|∂2xuε,k|>C(k,λ,γ,b)ε(a−1)/3,(12.26)

and the maximal time of existence ofvε,k on the characteristic of∂t + ∂x through 0 is less thanC(k)ε(2+a)/3. Theorem 4.2 of [17] shows that ifp(ξ,τ ,η) = 0 had realξ roots of multiplicity atmost two, a uniform (ink) bound|Gk,Hk|Hs 6Cs for s > 3/2 would imply a time of existenceT (Cs) for solutions to (12.2)k independent of (k,ε).

Example2. – Consider (12.2) again where the boundary data forDxuε is replaced by

ε

∞∑k=M

1k1+µ

G(t,y)ei(τk,ηk)·(t,y)/ε,(12.27)

where (τk,ηk) = (1/k,η0 + (1/(k1+α)))→ (τ0,η0) = (0, 1) ask→∞ andM ,µ,α are positiveconstants to be chosen. In the notation of (3.11) we have fork large

E+(τk,ηk) =E+(β1(τk,ηk)

)⊕E+

(β2(τk,ηk)

),(12.28)

whereβ1(τk,ηk) is an outgoing hyperbolic mode andβ2(τk,ηk) an elliptic mode with Imβ2 > 0.Observe that the projectionsP (βi(τk,ηk)) of E+(τk,ηk) onto its components in (12.28) blow upask→∞ (recall Remark 3.2).

1. The solution to (12.2) with this new data isuε =∑∞k=M uε,k, where

∂2xuε,k =

1k1+µ

eitτkε +iy

ηkε +γt

∫eitτ+iyηεg1(τ )g2(η)Q(Xk)dτ dη,(12.29)

Xk = ((τk,ηk)/ε) + (τ − iγ,η), andQ(X) is defined by (12.7). For 0< a < 1/3 as in Example1, b1 > 0, b2 > 0 small, and|t|6 b1ε

a, x∼ ε(2+a)/3, |y|6 b2, we proceed to estimate

∞∑k=M

∂2xuε,k(12.30)

from below. Rewrite (12.30) as∑M<k6(4ε1−a)−1

∂2xuε,k +

∑k>(4ε1−a)−1

∂2xuε,k = S1 + S2.(12.31)

2. Chooseα such that

(1− a)(1+ α) = 1,(12.32)

4e SÉRIE– TOME 33 – 2000 –N 3


and note that fort,y as above andk > (4ε1−a)−1, eitτkε +iy

ηkε +γt stays close toeiy

η0ε . An

analysis similar to the one in parts 1–3 of Example 1 shows that each term inS2 has magnitude> (C/k1+µ)ε(a−1)/3, and these contributions have a nearly fixed direction independent ofk (butnotε). In this argument (1/k) + ε(τ − iγ) plays the role ofε(τ − iγ) in Example 1. Thus,

|S2|> ε(a−1)/3∑

k>(4ε1−a)−1

1k1+µ

=C

µε(a−1)(1/3−µ).(12.33)

3. We claim that each term inS1 has magnitude6

CBk1+µ

(k1/3 + εB

), for anyB > 0.(12.34)

TheεB piece corresponds to the integral over|τ |> ε−a (recall g1(τ ) is rapidly decreasing). Thek1/3 piece appears since

∣∣ξ1(Xk)− ξ2(Xk)∣∣∼ 1

k1/3for

1M

>1k> 4ε1−a, |τ |6 ε−a,

if M is large enough. Thus,

|S1|6C∑

M<k<(4ε1−a )−1

k(1/3)−1−µ =Cε(a−1)(1/3−µ)

13 − µ

.(12.35)

4. |S2| blows up asε→ 0 for 0< µ < 1/3 and is much larger than the right side of (12.35)for µ near 0. The argument of Example 1, Part5, shows thatvε blows up fort∼ ε(2+a)/3 in thisexample.

As in Part6 of Example 1, we may replaceG(t,y) in (12.27) andH(t,y) in (12.2)(b) byGj(t,y) = (1/jL)G(jt,y), Hj(t,y) = (1/jL)H(jt,y) to obtain data supported int > −T/j,j ∈ 1, 2, . . .. Set

gj(t,y,θ′) =

( ∞∑k=M

1k1+µ

Gj(t,y)ei(τk,ηk)·θ′ ,Hj(t,y)ei(τ1,η1)·θ′).(12.36)

For any s, |gj|Aθ′ (Hs) 6 Cs for all j, providedL = L(s) is large enough. If there were noglancing modes of order> 2, Theorem 4.2 of [17] would yield a time of existenceT (s), s > 3/2,independent of (j,ε) for solutions to (12.2)j defined by the data (12.36). In our example themaximal time of existence ofvε,j on the characteristic of∂t + ∂x through 0 is6C(j)ε(2+a)/3.

Remark12.4. – The blow-up in this example depends on a delicate balance between the sizeof the convergence factor 1/k1+µ in (12.27), the rate of blow-up of individual termsε(a−1)/3,and the rate of convergence (τk,ηk)→ (τ0,η0) (recall (1− a)(1 + α) = 1). For example, if(1− a)µ> 1/3, blow-up does not occur.

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432 M. WILLIAMS

[2] CHAZARAIN J., PIRIOU A., Introduction to the Theory of Linear Partial Differential Equations, NorthHolland, Amsterdam, 1982.

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192.

(Manuscript received January 4, 1999,accepted, after revision, June 4, 1999.)

Mark WILLIAMS

Department of Mathematics,University of North Carolina,Chapel Hill, NC 27599, USA

E-mail: [email protected]

4e SÉRIE– TOME 33 – 2000 –N 3

Boundary layers and glancing blow-up in nonlinear geometric optics

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