Amplification of pulses in nonlinear geometric optics

42 

Loading.... (view fulltext now)

Loading....

Loading....

Loading....

Loading....

Texte intégral

(1)

HAL Id: hal-00855803

https://hal.archives-ouvertes.fr/hal-00855803

Submitted on 30 Aug 2013

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires

Amplification of pulses in nonlinear geometric optics

Jean-François Coulombel, Mark Williams

To cite this version:

Jean-François Coulombel, Mark Williams. Amplification of pulses in nonlinear geometric optics. Journal of Hyperbolic Differential Equations, World Scientific Publishing, 2014, 11 (4), pp.749-793. �hal-00855803�

(2)

Amplification of pulses in nonlinear geometric optics

Jean-Fran¸cois Coulombel∗, Mark Williams† August 30, 2013

Abstract

In this companion paper to our study of amplification of wavetrains [CGW13], we study weakly stable semilinear hyperbolic boundary value problems with pulse data. Here weak sta-bility means that exponentially growing modes are absent, but the so-called uniform Lopatinskii condition fails at some boundary frequency in the hyperbolic region. As a consequence of this degeneracy there is again an amplification phenomenon: outgoing pulses of amplitude O(ε2) and wavelength ε give rise to reflected pulses of amplitude O(ε), so the overall solution has amplitude O(ε). Moreover, the reflecting pulses emanate from a radiating pulse that propagates in the boundary along a characteristic of the Lopatinskii determinant.

In the case of N×N systems considered here, a single outgoing pulse produces on reflection a family of incoming pulses traveling at different group velocities. Unlike wavetrains, pulses do not interact to produce resonances that affect the leading order profiles. However, pulse interactions do affect lower order profiles and so these interactions have to be estimated carefully in the error analysis. Whereas the error analysis in the wavetrain case dealt with small divisor problems by approximating periodic profiles by trigonometric polynomials (which amounts to using a high frequency cutoff), in the pulse case we approximate decaying profiles with nonzero moments by profiles with zero moments (a low frequency cutoff). Unlike the wavetrain case, we are now able to obtain a rate of convergence in the limit describing convergence of approximate to exact solutions.

Contents

1 Introduction 2

1.1 Assumptions . . . 4

1.2 Main results . . . 9

1.3 Exact solutions and singular systems . . . 11

1.4 Derivation of the profile equations . . . 12

1.5 Error analysis . . . 15

2 Analysis of profile equations 17 2.1 The key subsystem forU0 . . . 17

2.2 The first corrector U1 . . . 21

CNRS and Universit´e de Nantes, Laboratoire de math´ematiques Jean Leray (UMR CNRS 6629), 2 rue de la

Houssini`ere, BP 92208, 44322 Nantes Cedex 3, France. Email: jean-francois.coulombel@univ-nantes.fr.

University of North Carolina, Mathematics Department, CB 3250, Phillips Hall, Chapel Hill, NC 27599. USA.

Email: williams@email.unc.edu. Research of M.W. was partially supported by NSF grants number DMS-0701201 and DMS-1001616.

(3)

3 The exact solution of the singular system 23

4 Error analysis 26

4.1 Moment-zero approximations . . . 26

4.2 Proof of Theorem 1.14 . . . 28

5 Extension to the general N × N system 33 A Estimates of interaction integrals 35 B Singular pseudodifferential calculus for pulses 36 B.1 Symbols . . . 37

B.2 Definition of operators and action on Sobolev spaces . . . 37

B.3 Adjoints and products . . . 38

B.4 Extended calculus . . . 39

1

Introduction

In this paper we study the reflection of pulses in weakly stable semilinear hyperbolic boundary value problems. The problems are weakly stable in the sense that exponentially growing modes are absent, but the uniform Lopatinskii condition fails at a boundary frequency β in the hyperbolic region1H. As a consequence of this degeneracy in the boundary conditions, there is an amplification phenomenon: boundary data of amplitude ε in the problem (1.2) below gives rise to a response of amplitude O(1).

On Rd+1+ = {x = (x′, x

d) = (t, y, xd) = (t, x′′) : xd ≥ 0}, consider the N × N semilinear

hyperbolic boundary problem for v = vε(x), where ε > 02:

(a) L(∂)v + f0(v) = 0 (b) φ(v) = ε2G(x′,x ′· β ε ) on xd= 0 (c) v = 0 and G = 0 in t < 0, (1.1)

where L(∂) := ∂t+Pdj=1Bj∂j, the matrix Bd is invertible, and both f0(v) and φ(v) vanish at

v = 0. The function G(x′, θ0) is assumed to have polynomial decay in θ0, and the frequency

β ∈ Rd\ {0} is taken to be a boundary frequency at which the uniform Lopatinskii condition fails. A consequence of this failure is that the choice of the factor ε2in (1.1)(b) corresponds to the weakly

nonlinear regime for this problem. The leading order profile is coupled to the next order profile in the nonlinear system (1.36) derived below.

Before proceeding we write the problem in an equivalent form that is better adapted to the boundary. After multiplying the first equation by (Bd)−1 we obtain

˜ L(∂)v + f (v) = 0 φ(v) = ε2G(x′,x ′· β ε ) on xd= 0 v = 0 and G = 0 in t < 0,

1See Definition 1.3 and Assumption 1.5 for precise statements. 2We usually suppress the subscript ε.

(4)

where we have set ˜ L(∂) := ∂d+ d−1 X j=0 Aj∂j with Aj := Bd−1Bj for j = 0, . . . , d− 1 , ∂0:= ∂t.

Setting v = εu and writing f (v) = D(v)v, φ(v) = ψ(v)v, we get the problem for u = uε(x)

(a) ˜L(∂)u + D(εu)u = 0 (b) ψ(εu)u = ε G(x′,x

· β

ε ) on xd= 0 (c) u = 0 in t < 0.

(1.2)

For this problem we pose the two basic questions of rigorous nonlinear geometric optics: (1) Does an exact solution uε of (1.2) exist for ε∈ (0, 1] on a fixed time interval [0, T0]

indepen-dent of ε?

(2) Suppose the answer to the first question is yes. If we let uappε denote an approximate

solution on [0, T0] constructed by the methods of nonlinear geometric optics (that is, solving eikonal

equations for phases and suitable transport equations for profiles), how well does uappε approximate

uε for ε small? For example, is it true that

lim

ε→0|uε− u app

ε |L∞ → 0 ?

(1.3)

The amplification phenomenon was studied in a formal way for several different quasilinear problems in [AM87, MA88, MR83]. In [MR83] amplification was studied in connection with Mach stem formation in reacting shock fronts, while [AM87] explored a connection to the formation of instabilities in compressible vortex sheets. Both papers derived equations for profiles using an ansatz that exhibited amplification; however, neither of the two questions posed above were addressed. The first rigorous amplification results were proved in [CG10] for linear wavetrains. That paper provided positive answers to the above questions by making use of approximate solutions of high order, and showed in particular that the limit (1.3) holds.

An amplification result for wavetrains in weakly stable semilinear problems was proved in [CGW13], and in this paper we prove the analogous result for pulses. Our approximate solution has the form

(1.4) uappε (x) =U0(x, θ0, ξd)|θ0=φ0 ε ,ξd=xdε , where3 U0(x, θ0, ξd) = X m∈I νkm X k=1 σm,k(x, θ0+ ωmξd) rm,k. (1.5)

Here each pulse profile σm,k(x, θ) is polynomially decaying in θ ∈ R, rather than periodic in θ

as in the wavetrain case. From the explicit and rather simple equations satisfied by the profiles appearing in U0, we read off the main amplification phenomenon for solutions of (1.2): in t > 0

pulses of amplitude O(1), which emanate from a pulse in the boundary that propagates along a

(5)

characteristic of the Lopatinskii determinant, radiate into the interior at distinct group velocities determined by β.

In the case of N × N systems considered here, a single outgoing pulse produces on reflection a family of incoming pulses traveling at different group velocities. Unlike wavetrains, pulses do not interact to produce resonances that affect the leading order profiles. One effect of this is that we do not need to use Nash-Moser iteration to construct the leading pulse profiles as we did in the wavetrain case. The construction of exact solutions is again based on the study of an associated singular system (1.23). This system is solved by a Nash-Moser iteration argument based on a tame estimate for the linearized singular system (3.6). The proof of that estimate and the Nash-Moser iteration can be carried over verbatim from the corresponding argument in [CGW13] for wavetrains4. The estimate for the linearized singular system also plays a key role in the error analysis here.

The main differences between the analysis of this paper and that of [CGW13] occur in the study of the profile equations and in the error analysis. Although pulses do not produce resonances, pulse interactions do affect lower order profiles and so these interactions have to be estimated carefully in the error analysis. Unlike the wavetrain case, we are now able to obtain a rate of convergence in the limit (1.3). Whereas the error analysis in the wavetrain case dealt with small divisor problems by approximating periodic profiles by trigonometric polynomials (which amounts to using a high frequency cutoff), in the pulse case we approximate decaying profiles with nonzero moments by profiles with zero moments (using a low frequency cutoff)5. We provide a more detailed summary

of the argument after stating our main assumptions.

1.1 Assumptions

We make the following hyperbolicity assumption on the system (1.1):

Assumption 1.1. There exist an integer q ≥ 1, some real functions λ1, . . . , λq that are analytic on

Rd\ {0} and homogeneous of degree 1, and there exist some positive integers ν1, . . . , νq such that:

∀ ξ = (ξ1, . . . , ξd)∈ Rd\ {0} , det h τ I + d X j=1 ξjBj i = q Y k=1 τ + λk(ξ) νk .

Moreover the eigenvalues λ1(ξ), . . . , λq(ξ) are semi-simple (their algebraic multiplicity equals their

geometric multiplicity) and satisfy λ1(ξ) <· · · < λq(ξ) for all ξ ∈ Rd\ {0}.

For simplicity, we restrict our analysis to noncharacteristic boundaries and therefore make the following:

Assumption 1.2. The matrix Bd is invertible and the matrix B := ψ(0) has maximal rank, its

rank p being equal to the number of positive eigenvalues of Bd (counted with their multiplicity).

Moreover, the integer p satisfies 1≤ p ≤ N − 1.

In the normal modes analysis for (1.2), one first performs a Laplace transform in the time variable t and a Fourier transform in the tangential space variables y. We let τ − i γ ∈ C and

4Thus, we do not repeat those (lengthy) arguments here.

5We borrow the idea of introducing low frequency cutoffs from [AR03] where single phase pulses in free space are

(6)

η∈ Rd−1denote the dual variables of t and y. We introduce the symbol A(ζ) := −i B−1d  (τ − iγ) I + d−1 X j=1 ηjBj   , ζ := (τ − iγ, η) ∈ C × Rd−1.

For future use, we also define the following sets of frequencies:

Ξ :=n(τ − iγ, η) ∈ C × Rd−1\ (0, 0) : γ ≥ 0o, Σ :=nζ∈ Ξ : τ2+ γ2+|η|2= 1o, Ξ0:=

n

(τ, η)∈ R × Rd−1\ (0, 0)o= Ξ∩ {γ = 0} , Σ0 := Σ∩ Ξ0.

Two key objects in our analysis are the hyperbolic region and the glancing set that are defined as follows:

Definition 1.3. • The hyperbolic region H is the set of all (τ, η) ∈ Ξ0 such that the matrix

A(τ, η) is diagonalizable with purely imaginary eigenvalues.

• Let G denote the set of all (τ, ξ) ∈ R × Rd such that ξ 6= 0 and there exists an integer

k∈ {1, . . . , q} satisfying:

τ + λk(ξ) =

∂λk

∂ξd

(ξ) = 0 .

If π(G) denotes the projection of G on the d first coordinates (in other words π(τ, ξ) = (τ, ξ1, . . . , ξd−1) for all (τ, ξ)), the glancing set G is G := π(G) ⊂ Ξ0.

We recall the following result that is due to Kreiss [Kre70] in the strictly hyperbolic case (when all integers νj in Assumption 1.1 equal 1) and to M´etivier [M´et00] in our more general framework:

Proposition 1.4 ([Kre70, M´et00]). Let Assumptions 1.1 and 1.2 be satisfied. Then for all ζ ∈ Ξ\ Ξ0, the matrix A(ζ) has no purely imaginary eigenvalue and its stable subspace Es(ζ) has

dimension p. Furthermore, Es defines an analytic vector bundle over Ξ\ Ξ0 that can be extended

as a continuous vector bundle over Ξ.

For all (τ, η) ∈ Ξ0, we let Es(τ, η) denote the continuous extension of Es to the point (τ, η). The

analysis in [M´et00] shows that away from the glancing set G ⊂ Ξ0, Es(ζ) depends analytically on

ζ, and the hyperbolic region H does not contain any glancing point.

To treat the case when the boundary operator in (1.2)(b) is independent of u, meaning ψ(εu) ψ(0) =: B, we make the following weak stability assumption on the problem (L(∂), B).

Assumption 1.5. • For all ζ ∈ Ξ \ Ξ0, KerB∩ Es(ζ) ={0}.

• The set Υ0 := {ζ ∈ Σ0 : KerB∩ Es(ζ) 6= {0}} is nonempty and included in the hyperbolic

region H.

• For all ζ ∈ Υ0, there exists a neighborhood V of ζ in Σ, a real valued C∞ function σ defined

on V, a basis E1(ζ), . . . , Ep(ζ) of Es(ζ) that is of classC∞with respect to ζ ∈ V, and a matrix

P (ζ)∈ GLp(C) that is of class C∞ with respect to ζ∈ V, such that

∀ ζ ∈ V , B E1(ζ) . . . Ep(ζ)= P (ζ) diag γ + i σ(ζ), 1, . . . , 1.

(7)

Definition 1.6 ([Kre70]). As before let p be the number of positive eigenvalues of Bd. The problem

( ˜L(∂), B) is said to be uniformly stable or to satisfy the uniform Lopatinskii condition if B : Es(ζ)−→ Cp

is an isomorphism for all ζ ∈ Σ.

Remark 1.7. Observe that if ( ˜L(∂), B) satisfies the uniform Lopatinskii condition, continuity implies that this condition still holds for ( ˜L(∂), B+ ˙ψ), where ˙ψ is any sufficiently small perturbation of B. Hence the uniform Lopatinskii condition is a convenient framework for nonlinear perturbation. The analogous statement may not be true when ( ˜L(∂), B) is only weakly stable. Remarkably, weak stability persists under perturbation in the so-called WR class exhibited in [BGRSZ02], and Assumption 1.5 above is a convenient equivalent definition of the WR class (see [CG10, Appendix B] or [BGS07, Chapter 8]). In order to handle general nonlinear boundary conditions as in (1.2) we shall strengthen Assumption 1.5 in Assumption 1.12 below.

Boundary and interior phases. We consider a planar real phase φ0 defined on the boundary:

(1.6) φ0(t, y) := τ t + η· y , (τ , η)∈ Ξ0.

As follows from earlier works, see e.g. [MA88], oscillations or pulses on the boundary associated with the phase φ0 give rise to oscillations or pulses in the interior associated with some planar

phases φm. These phases are characteristic for the hyperbolic operator L(∂) and their trace on the

boundary{xd= 0} equals φ0. For now we make the following:

Assumption 1.8. The phase φ0 defined by (1.6) satisfies (τ , η)∈ Υ0. In particular (τ , η)∈ H.

Thanks to Assumption 1.8, we know that the matrixA(τ, η) is diagonalizable with purely imaginary eigenvalues. These eigenvalues are denoted i ω1, . . . , i ωM, where the ωm’s are real and pairwise

distinct. The ωm’s are the roots (and all the roots are real) of the dispersion relation:

dethτ I + d−1 X j=1 η jBj+ ω Bd i = 0 .

To each root ωm there corresponds a unique integer km∈ {1, . . . , q} such that τ + λkm(η, ωm) = 0.

We can then define the following real6 phases and their associated group velocities: (1.7) ∀ m = 1, . . . , M , φm(x) := φ0(t, y) + ωmxd, vm :=∇λkm(η, ωm) .

Let us observe that each group velocity vm is either incoming or outgoing with respect to the space

domain Rd

+: the last coordinate of vm is nonzero. This property holds because (τ , η) does not

belong to the glancing set G. We can therefore adopt the following classification:

Definition 1.9. The phase φm is incoming if the group velocity vm is incoming (that is, when

∂ξdλkm(η, ωm) > 0), and it is outgoing if the group velocity vm is outgoing (∂ξdλkm(η, ωm) < 0). 6If (τ , η) does not belong to the hyperbolic regionH, some of the phases ϕ

m may be complex, see e.g. [Her12,

(8)

In all that follows, we let I denote the set of indices m ∈ {1, . . . , M} such that φm is an incoming

phase, andO denote the set of indices m ∈ {1, . . . , M} such that φm is an outgoing phase. If p≥ 1,

thenI is nonempty, while if p ≤ N − 1, O is nonempty (see Lemma 1.11 below). We will use the notation: L(τ, ξ) := τ I + d X j=1 ξjBj, ˜L(β, ωm) := ωmI + d−1 X k=0 βkAk, β := (τ , η), x′ = (t, y), φ0(x′) = β· x′.

For each phase φm, dφm denotes the differential of the function φm with respect to its argument

x = (t, y, xd). It follows from Assumption 1.1 that the eigenspace of A(β) associated with the

eigenvalue i ωm coincides with the kernel of L(dφm) and has dimension νkm. The following result

is a direct consequence of the diagonalizability of A(β), see [CG10] for the proof: Lemma 1.10. The space CN admits the decomposition:

(1.8) CN =Mm=1 ker L(dφm)

and each vector space in (1.8) admits a basis of real vectors. If we let P1, . . . , PM denote the

projectors associated with the decomposition (1.8), then there holds Im ˜L(dφm) = ker Pm for all

m = 1, . . . , M .

The following well-known lemma, whose proof is also recalled in [CG10], gives a useful decom-position of Es in the hyperbolic region.

Lemma 1.11. The stable subspace Es(β) admits the decomposition:

(1.9) Es(β) =m∈I ker L(dφm) ,

and each vector space in the decomposition (1.9) admits a basis of real vectors.

By Assumption 1.5 we know that the vector space ker B ∩ Es(β) is one-dimensional; moreover,

it admits a real basis because B has real coefficients and Es(β) has a real basis. This vector space is therefore spanned by a vector e ∈ RN \ {0} that we can decompose in a unique way by using

Lemma 1.11:

(1.10) ker B ∩ Es(β) = Span {e} , e = X

m∈I

em, Pmem = em.

Each vector em in (1.10) has real components. We also know that the vector space B Es(β) is

(p− 1)-dimensional. We can therefore write it as the kernel of a real linear form:

(1.11) B Es(β) =   X∈ C p, b· X = p X j=1 bjXj = 0   , for a suitable vector b∈ Rp\ {0}.

(9)

To formulate our last assumption we observe first that for every point ζ ∈ H there is a neigh-borhoodV of ζ in Σ and a C∞ conjugator Q

0(ζ) defined on V such that

Q0(ζ)A(ζ)Q−10 (ζ) =    iω1(ζ)In1 0 . .. 0 iωJ(ζ)InJ    =: D1(ζ), (1.12)

where the ωj are real when γ = 0 and there is a constant c > 0 such that either

Re (iωj)≤ −c γ or Re (iωj)≥ c γ for all ζ ∈ V.

In view of Lemma 1.11 we can choose the first p columns of Q−10 (ζ) to be a basis of Es(ζ), and we write

Q−10 (ζ) = [Qin(ζ) Qout(ζ)].

Choose J′ so that the first Jblocks of D

1 lie in the first p columns, and the remaining blocks in

the remaining N− p columns. Thus, Re (iωj)≤ −c γ for 1 ≤ j ≤ J′.

Observing that the linearization of the boundary condition in (1.2) is ˙u7−→ ψ(εu) ˙u + [dψ(εu) ˙u] εu ,

we define the operator

B(v1, v2) ˙u := ψ(v1) ˙u + [dψ(v1) ˙u] v2,

(1.13)

which appears in Assumption 1.12 below. For later use we also define D(v1, v2) ˙u := D(v1) ˙u + [dD(v1) ˙u] v2,

(1.14) as well as

B(v1) :=B(v1, v1) , D(v1) :=D(v1, v1) .

(1.15)

We now state the weak stability assumption that we make when considering the general case of nonlinear boundary conditions in (1.2).

Assumption 1.12. • There exists a neighborhood O of (0, 0) ∈ R2N such that for all (v

1, v2)∈

O and all ζ ∈ Ξ \ Ξ0, ker B(v1, v2)∩ Es(ζ) ={0}. For each (v1, v2)∈ O the set Υ(v1, v2) :=

{ζ ∈ Σ0 : ker B(v1, v2)∩ Es(ζ)6= {0}} is nonempty and included in the hyperbolic region H.

Moreover, if we set Υ :=(v1,v2)∈OΥ(v1, v2), then Υ⊂ H (closure in Σ0).

• For every ζ ∈ Υ there exists a neighborhood V of ζ in Σ and a C∞ function σ(v

1, v2, ζ) on

O × V such that for all (v1, v2, ζ)∈ O × V we have Ker B(v1, v2)∩ Es(ζ)6= {0} if and only if

ζ ∈ Σ0 and σ(v1, v2, ζ) = 0.

Moreover, there exist matrices Pi(v1, v2, ζ) ∈ GLp(C), i = 1, 2, of class C∞ on O × V

such that ∀ (v1, v2, ζ)∈ O × V

(10)

For nonlinear boundary conditions, the phase φ0 in (1.6) is assumed to satisfy (τ , η)∈ Υ(0, 0),

or in other words the intersection ker B∩ Es(τ , η) is not reduced to{0}. The set Υ

0 in Assumption

1.5 is a short notation for Υ(0, 0). The phases φm are still defined by (1.7) and thus only depend

on L(∂) and B and not on the nonlinear perturbations f0 and ψ(ε u)− ψ(0) added in (1.2).

Remark 1.13. 1) The properties stated in Assumption 1.12 are just a convenient description of the requirements for belonging to the WR class of [BGRSZ02]. Like the uniform Lopatinskii condition, Assumption 1.12 can in practice be verified by hand via a “constant-coefficient” computation. More precisely, for (v1, v2) near (0, 0) ∈ R2N and ζ ∈ Σ, one can define (see, e.g., [BGS07, chapter 4])

a Lopatinskii determinant ∆(v1, v2, ζ) that is C∞ in (v1, v2), analytic in ζ = (τ − iγ, η) on Σ \ G,

continuous on Σ, and satisfies

∆(v1, v2, ζ) = 0 if and only if KerB(v1, v2)∩ Es(ζ)6= {0}.

In particular, ∆(v1, v2,·) is real-analytic on H.

It is shown in Remark 1.13 of [CGW13] that Assumption 1.12 holds provided (1.17) ∅ 6= {ζ ∈ Σ : ∆(0, 0, ζ) = 0} ⊂ H and ∆(0, 0, ζ) = 0 ⇒ ∂τ∆(0, 0, ζ)6= 0,

and it thus only involves a weak stability property for the linearized problem at (v1, v2) = (0, 0).

In-stead of assuming (1.17), we have stated Assumption 1.12 in a form that is more directly applicable to the error analysis of Theorem 4.10.

2) To prove the basic estimate for the linearized singular system, Proposition 3.3, and to construct the exact solution Uεto the singular system (1.23) below, it is enough to require that the

analogue of Assumption 1.12 holds when B(v1, v2) is replaced by B(v1) given in (1.15). However,

for the error analysis of section 4 in the case of nonlinear boundary conditions, we need Assumption 1.12 as stated.

1.2 Main results

For each m∈ {1, . . . , M} we let

rm,k, k = 1, . . . , νkm

denote a basis of ker L(dφm) consisting of real vectors. In section 4 we shall construct a “corrected”

approximate solution uc ε of (1.2) of the form (1.18) ucε(x) =hU0(x, θ0, ξd) + εU1(x, θ0, ξd) + ε2Up2(x, θ0, ξd) i |θ0=φ0 ε ,ξd=xdε , where U0(x, θ0, ξd) = X m∈I νkm X k=1 σm,k(x, θ0+ ωmξd) rm,k, U1(x, θ0, ξd) = X m∈I νkm X k=1 τm,k(x, θ0+ ωmξd) rm,k− R∞  ˜ L(∂)U0+ D(0)U0, (1.19)

and R∞ is the integral operator defined in (1.32). The σm,k’s and τm,k’s are scalar functions

decaying in θm := θ0+ ωmξd. The σm,k’s satisfy

Z

R

(11)

and describe the propagation of oscillations with phase φm and group velocity vm. The second

corrector ε2U2

p(x, θ0, ξd) in (1.18) is a more complex object constructed in the error analysis of

section 4.

The following theorem, our main result, is an immediate corollary of the more precise Theorem 4.10. Here we let ΩT := {(x, θ0) = (t, y, xd, θ0) ∈ Rd+1× R1 : xd ≥ 0, t < T }, and bΩT :=

{(t, y, θ0)∈ Rd× R1: t < T}. The spaces ETs appearing in the theorem are

(1.20) EsT := C(Rxd, H

s(bΩ

T))∩ L2(Rxd, H

s+1(bΩ T)),

while the Sobolev spaces with θ-weights ΓsT and bΓsT are defined in Definition 2.6. With [r] denoting the smallest integer greater than r, we define

(1.21) a0 :=  d + 1 2  , a1 :=  d + 1 2  + M0, a := max(2a0+ 3, a1+ 1), ˜a := 2a− a0,

where M0 := 3 d + 5 is fixed according to the regularity requirements in the singular

pseudodiffer-ential calculus (see Appendix B).

Theorem 1.14. We make Assumptions 1.1, 1.2, 1.5, and 1.8 when the boundary condition in (1.2) is linear (ψ(εu)≡ ψ(0)); in the general case we substitute Assumption 1.12 for Assumption 1.5. Fix T > 0, set M0:= 3 d + 5, and let a and ˜a be as in (1.21).

Consider the semilinear boundary problem (1.2), where G∈ bΓa˜

T. There exists ε0> 0 such that

if hGia+2

T is small enough, there exists a unique function Uε(x, θ0)∈ E

a−1

T satisfying the singular

system (1.23) on ΩT such that

uε(x) := Uε

 x,φ0

ε 

is an exact solution of (1.2) on (−∞, T ] × Rd+ for 0 < ε ≤ ε0. In addition there exists a profile

U0(x, θ0, ξd) as in (1.19), whose components σm,k lie in Γa+1T , such that the approximate solution

defined by uappε :=U0  x,φ0 ε , xd ε  satisfies lim ε→0|uε− u app ε |L∞ = 0 on (−∞, T ] × Rd+.

In fact we have the rate of convergence |uε− uappε |L∞ ≤ C ε 1 2M1+3, where M1 :=  d 2 + 3  .

Observe that although the boundary data in the problem (1.2) is of size O(ε), the approximate solution uappε is of size O(1), exhibiting an amplification due to the weak stability at frequency

β. The main information provided by Theorem 1.14 is that this amplification does not rule out existence of a smooth solution on a fixed time interval, that is it does not trigger a violent instability, at least in this weakly nonlinear regime. Although the first and second correctorsU1 andU2

p do not

appear in the statement of Theorem 1.14, they are essential in the error analysis. As an application of Theorem 1.14, an easy analysis of the equations for the leading profile U0 (see (2.11) or (5.2))

(12)

Remark 1.15. a) In order to avoid some technicalities we have stated our main result for the problem (1.2) where all data is 0 in t < 0. This result easily implies a similar result in which outgoing pulses defined in t < 0 of amplitude O(ε) and wavelength ε give rise to reflected pulses of amplitude O(1). In either formulation, inspection of the profile equations shows that the pulses of amplitude O(1) emanate from a radiating pulse that propagates in the boundary along a characteristic of the Lopatinskii determinant.

b) We have decided to fix T > 0 at the start and choose data small enough so that a solution to the nonlinear problem exists up to time T . With only minor changes one can fix the data in the problem (G in (1.2)) at the start, and then choose T small enough so that a solution to the nonlinear problem exists up to time T .

In the remainder of this introduction, we discuss the construction of exact solutions, the con-struction of the approximate solutionU0, and the error analysis. The proofs are given in Sections 2,

3, and 4. We hope that the analysis developed in this article will be useful in justifying quasilinear amplification phenomena such as the Mach stems or kink modes formation of [MR83, AM87].

1.3 Exact solutions and singular systems

The theory of weakly stable hyperbolic initial boundary value problems fails to provide a solution of the system (1.2) that exists on a fixed time interval independent of ε7. In order to obtain such an exact solution to the system (1.2) we adopt the strategy of studying an associated singular problem first used by [JMR95] for an initial value problem in free space. We look for a solution of the form

uε(x) = Uε(x, θ0)|θ

0=φ0(x′)ε

, (1.22)

where Uε(x, θ0) satisfies the singular system derived by substituting (1.22) into the problem (1.2).

Recalling that ˜L(∂) = ∂d+Pd−1j=0Aj∂j, we obtain:

∂dUε+ d−1 X j=0 Aj  ∂j+ βj∂θ0 ε  Uε+ D(εUε) Uε=: ∂dUε+ A  ∂x′ +β∂θ0 ε  Uε+ D(εUε) Uε= 0 ψ(εUε) Uε|xd=0= ε G(x ′, θ 0), Uε= 0 in t < 0. (1.23)

In the wavetrain case the function Uε(x, θ0) is taken to be periodic in θ0, but in the case of pulses

the variable θ0∈ R is treated as just another tangential variable on an equal footing with x′. The

special difficulties presented by such singular problems when there is a boundary are described in detail in the introductions to [Wil02] and [CGW11]; these difficulties are present for both wavetrains and pulses. In particular we mention: (a) symmetry assumptions on the matrices Bj appearing in

the problem (1.1) equivalent to (1.2) are generally of no help in obtaining an L2 estimate for (1.23) (boundary conditions satisfying Assumption 1.5 can not be maximally dissipative, see [CG10]); (b) one cannot control L∞ norms just by estimating tangential derivatives ∂α

(x′

0)Uε because (1.23) is

not a hyperbolic problem in the xd direction8; moreover, even if one has estimates of tangential

7This would be true even for problems ( ˜L(∂), B) that are uniformly stable in the sense of Definition 1.6.

8For initial value problems in free space, one can control Lnorms just by estimating enough derivatives tangent

(13)

derivatives uniform with respect to ε, because of the factors 1/ε in (1.23) one cannot just use the equation to control ∂dUε and thereby control L∞ norms.

In [CGW12] a class of singular pseudodifferential operators, acting on functions U (x′, θ 0) and

having the form (1.24) pDU (x′, θ0) = 1 (2 π)d+1 Z Rd×R eix′·ξ′+iθ0kp  εV (x′, θ0), ξ′+ kβ ε , γ  b U (ξ′, k) dξ′dk, γ ≥ 1,

was introduced to deal with these difficulties. Observe that the differential operator A appearing in (1.23) can be expressed in this form. Operators like (1.24), but with the integral over k replaced by a sum over k∈ Z, were first used in [Wil02] to prove uniform estimates for quasilinear singular systems arising in the uniformly stable wavetrain case. The calculi of [CGW12] are inspired by the calculus of [Wil02], but incorporate improvements needed for the study of amplification in both the wavetrain and pulse cases.

Under the weak stability assumptions of the present paper, Assumption 1.5 or Assumption 1.12, the basic L2estimate for the problem ( ˜L(∂), B) and for its singular analogue both exhibit loss of one tangential derivative. To derive energy estimates with a loss of derivative, one needs the remainder in the composition of two zero-order singular operators to be a smoothing operator, and not just bounded on L2 with small norm as in [Wil02]. The calculi constructed in [CGW12] have this smoothing property. The improved wavetrain calculus was used in [CGW13] to study amplification of wavetrains. As noted in that paper, the arguments used to prove the main L2 estimate for the linearized singular problem and the higher order tame estimates carry over verbatim to the pulse case. The Nash-Moser iteration used in [CGW13] to construct an exact solution Uε(x, θ0) of (1.23)

on ΩT also carries over verbatim to the pulse case. In section 3 we state the main estimates for

the linearized singular problem in the pulse case along with the existence theorem for the nonlinear singular system (1.23), and refer to [CGW13] for the proofs. The tame estimate for the linearized singular problem is given in terms of ETs spaces (recall (1.20)), and these spaces are used as well in the Nash-Moser iteration. The singular pulse calculus summarized in Appendix B must, of course, be substituted for the wavetrain calculus used in [CGW13].

1.4 Derivation of the profile equations

At first we work formally, looking for a corrected approximate solution of the form ucε(x) =hU0(x, θ0, ξd) + εU1(x, θ0, ξd) + ε2U2(x, θ0, ξd) i |θ0=φ0 ε ,ξd= xd ε .

Plugging into the system (1.2) and setting coefficients of successive powers of ε equal to zero, we obtain interior profile equations

ε−1: L(∂˜ θ0, ∂ξd)U 0= 0 ε0 : L(∂˜ θ0, ∂ξd)U 1+ ˜L(∂)U0+ D(0)U0 = 0 ε1 : L(∂˜ θ0, ∂ξd)U 2+ ˜L(∂)U1+ D(0)U1+ (dD(0)U0)U0 = 0, (1.25) where ˜ L(∂θ0, ∂ξd) := ∂ξd+ d−1 X j=0 βjAj∂θ0 = ∂ξd+ iA(β)∂θ0.

(14)

Similarly, we obtain the boundary equations on {xd= 0, ξd= 0},

ε0 : BU0 = 0

ε1 : BU1+ (dψ(0)U0)U0 = G(x′, θ0).

(1.26)

For ease of exposition we will begin by studying these equations in the 3× 3 strictly hyperbolic case, which contains all the main difficulties. In section 5 we describe the (minor) changes needed to treat the general case. We define the boundary phase φ0 := β· x′, eigenvalues ωj, and interior

phases φj := φ0+ ωjxd as in Section 1.1, where β ∈ H. We suppose ω1 and ω3 are incoming (or

causal) and ω2 is outgoing. The corresponding right and left eigenvectors of iA(β) are rj and lj,

j = 1, 2, 3, normalized so that lj· rk = δjk.

A general function W (x, θ0, ξd) valued in R3 can be written

W = W1(x, θ0, ξd)r1+ W2(x, θ0, ξd)r2+ W2(x, θ0, ξd)r3

where we refer to Wi, i = 1, 3 as the incoming components and W2 as the outgoing component of

W . With F (x, θ0, ξd) =P3i=1Fi(x, θ0, ξd)ri valued in R3 consider now an equation of the form

˜

L(∂θ0, ∂ξd)W = F.

Using the matrixr1 r2 r3to diagonalize iA(β), we find that the Wi must satisfy

(∂ξd− ωi∂θ0)Wi(x, θ0, ξd) = li· F := Fi(x, θ0, ξd), i = 1, 2, 3.

(1.27)

The general solution to (1.27) is

Wi(x, θ0, ξd) = τi∗(x, θ0+ ωiξd) +

Z ξd

0

Fi(x, θ0+ ωi(ξd− s), s) ds,

(1.28)

where τi∗ is arbitrary. This can be rewritten

Wi(x, θ0, ξd) = τi∗(x, θ0+ ωiξd) + Z R+ Fi(x, θ0+ ωi(ξd− s), s) ds − Z +∞ ξd Fi(x, θ0+ ωj(ξd− s), s) ds := τi(x, θ0+ ωiξd)− Z +∞ ξd Fi(x, θ0+ ωi(ξd− s), s) ds , (1.29)

provided the integrals in (1.29) exist.

Definition 1.16 (Functions of typeF). Suppose F (x, θ0, ξd) = 3 X i=1 Fi(x, θ0, ξd)ri, (1.30)

where each Fi has the form

Fi(x, θ0, ξd) = 3 X k=1 fki(x, θ0+ ωkξd) + 3 X l≤m=1 gl,mi (x, θ0+ ωlξd) hil,m(x, θ0+ ωmξd), (1.31)

(15)

where the functions fki(x, θ), gl,mi (x, θ), hil,m(x, θ) are real-valued, C1, and decay along with their first order partials at the rate O(hθi−2) uniformly with respect to x. We then say that F is of type F. For such functions F , define

EF (x, θ0, ξd) := 3 X j=1  lim T →∞ 1 T Z T 0 lj· F (x, θ0+ ωj(ξd− s), s) ds  rj.

Remark 1.17. For F as in (1.31), we have EF =

3

X

i=1

˜

Fi(x, θ0, ξd) ri, where ˜Fi := fii(x, θ0+ ωiξd) + gi,ii (x, θ0+ ωiξd)hii,i(x, θ0+ ωiξd).

Remark 1.18. The definition of E can be extended to more general functions. For example, if F =

3

X

i=1

Fi(x, θ0+ ωiξd) ri,

where the Fi(x, θ) are arbitrary continuous functions (not necessarily decaying with respect to θ),

the limits that define EF exist and we have EF = F .

For another example, suppose F is of type F and satisfies EF = 0. Define R∞F (x, θ0, ξd) :=− 3 X i=1 Z +∞ ξd Fi(x, θ0+ ωi(ξd− s), s) ds  ri. (1.32)

Then the limits defining R∞F and ER∞F exist and we have ER∞F = 0.

The following two results are proved in [CW13].

Proposition 1.19 ([CW13]). Suppose F is of type F and EF = 0. Then R∞F is bounded and

˜

L(∂θ0, ∂ξd)R∞F = R∞L(∂˜ θ0, ∂ξd)F = F = (I− E)F.

(1.33)

Proposition 1.20 ([CW13]). Let F (x, θ0, ξd) be a function of type F. (a) Then the equation

˜

L(∂θ0, ∂ξd)U = F has a solution bounded in (θ0, ξd) if and only if EF = 0.

(b) When EF = 0, every C1 solution bounded in (θ

0, ξd) has the form

U =

3

X

i=1

τi(x, θ0+ ωiξd) ri+ R∞F with τi(x, θ)∈ C1 and bounded.

Here EU =P3i=1τi(x, θ0+ ωiξd) ri and (I− E)U = R∞F .

(c)If U is of type F then

E ˜L(∂θ0, ∂ξd)U = ˜L(∂θ0, ∂ξd)EU = 0.

(1.34)

By applying the operators E and R∞ to the equations (1.25) formally (for now) using the

properties (1.33),(1.34), we obtain: (a) EU0 =U0, (b) E( ˜L(∂)U0+ D(0)U0) = 0, (c) BU0= 0 on xd= 0, ξd= 0, (d)U0= 0 in t < 0 (1.35)

(16)

and (a) (I − E)U1+ R∞( ˜L(∂)U0+ D(0)U0) = 0, (b) EL(∂)˜ U1+ D(0)U1+ (dD(0)U0)U0= 0, (c) BEU1 = G− (dψ(0) U0)U0− B(I − E)U1 on xd= 0, ξd= 0, (d)U1 = 0 in t < 0. (1.36)

To constructU1 we writeU1= EU1+ (I− E)U1 where

(I− E)U1 =−R∞( ˜L(∂)U0+ D(0)U0) = R∞F0,

(1.37)

and EU1 satisfies

(a) E( ˜L(∂) + D(0))EU1 =−E((dD(0) U0)U0)− E( ˜L(∂) + D(0))R∞F0,

(b) BEU1= G− (dψ(0) U0)U0− R∞F0 on xd= 0, ξd= 0

(c) EU1= 0 in t < 0. (1.38)

1.5 Error analysis

The error analysis is done by studying the singular problem (4.17) satisfied by the difference Wε(x, θ0) = Uε(x, θ0)− Uε(x, θ0),

where Uε is the exact solution of the semilinear singular problem (1.23) and Uε is a corrected

approximate solution of the form Uε(x, θ0) =U0(x, θ0, xd ε ) + εU 1(x, θ 0, xd ε ) + ε 2U2 p(x, θ0, xd ε).

Here U0 andU1 have the forms given in (1.19) and satisfy the profile equations (1.35), (1.36), and the second corrector U2

p remains to be chosen.

Let us denote U0

ε(x, θ0) :=U0(x, θ0,xεd) and similarly define Uε1,Up,ε2 . The idea is to apply the

tame estimate of Proposition 3.6 to the linear singular problem (4.17) satisfied by Wε. To apply

this estimate we need the functions Uε0,Uε1, and Up,ε2 to lie in ETs spaces for appropriate choices of s. Thus, roughly speaking, these functions must decay in (x′, θ

0) “like functions in Hs(x′, θ0)”.

The pulse profiles σm,k(x, θm) definingU0(x, θ0, ξd) have good decay in θm, so it is not hard to

see thatUε0 lies in an ETs space (Proposition 4.3). Moreover, we show that the σm,k(x, θm) actually

have moment zero, and this can be used to show thatU1

ε, whose definition involves taking integrals

as in (1.32) of an expression that is linear inU0, also lies in an Es

T space (see, for example, Lemma

2.9 and Proposition 2.10). If the σm,k(x, θm) did not have moment zero,Uε1 would fail to lie in any

Es

T space.

However, the second corrector presents greater difficulties. Considering the ε1 profile equation in (1.25), we see that a natural first choice for the second corrector would be

U2 =−R∞(I − E)

 ˜

L(∂)U1+ D(0)U1+ (dD(0)U0)U0. (1.39)

Since the primitive Rθ+∞f (s)ds of a function f that decays, say, like |s|−2 as|s| → +∞, itself

(17)

U1 nor the term (dD(0)U0)U0 in (1.39) has moment zero9, we see that this choice of U2

ε generally

cannot lie in any ETs space. A natural idea then is to replace U0 and U1 by approximations that

involve profiles having moment zero. More precisely, we take for exampleU0

p to be as in (1.19), but

where the profile σm,k(x, θm) is replaced by the moment-zero approximation σm,k,p(x, θm) whose

Fourier transform in θm is defined by

ˆ

σm,k,p(x, η) = χp(η)ˆσm,k(x, η).

Here χp(η) is a smooth cutoff function vanishing on a neighborhood of 0 of size O(|p|) and equal

to one outside a slightly larger neighborhood10. In place of (1.39) one could then try

Up2 =−R∞(I − E)  ˜ L(∂)Up1+ D(0)Up1+ (dD(0)Up0)Up0  . (1.40) The contributions toU2

p,ε involving Up1 now do lie in suitable ETs spaces, but some of the

contribu-tions from the quadratic term do not lie in any Es

T space. The difficulty is caused by products of

the form σm,k,pσm,k,p, and the resolution is to redefineUp2as in (1.40) after replacing these terms by

(σm,k,pσm,k′,p)p, see (4.10). We note that the contributions to Up,ε2 from “transversal interactions”

like σm,k,pσm′,k,p, where m6= m′ do already lie in appropriate Es

T spaces, see Proposition A.1.

These moment-zero approximations introduce errors that blow up as p → 0, of course, but taking p = εb for an appropriate b > 0, one can hope to control these errors using the factor ε2 in ε2Up2. This strategy works and is carried out in the proof of Theorem 4.10 in section 4.2. The proof relies on the machinery of moment-zero approximations developed in section 4.1 and the quadratic interaction estimates of Appendix A.

Remark 1.21. 1) We learned the idea of using moment-zero approximations to construct correctors in pulse problems from the paper [AR03].

2) Together with [CW13], which dealt with reflecting pulses in quasilinear uniformly stable boundary problems, this paper completes the first stage of our project to rigorously justify, when that is possible, the formal constructions of [HMR86, AM87, MA88, MR83] in boundary problems involving multiple interacting pulses. The operators E, R∞, and the machinery of moment-zero

approximations developed in these papers provide a set of tools for rigorously constructing leading profiles and correctors. The estimates for weakly stable singular systems (1.23) given in [CGW13]11 and for uniformly stable quasilinear singular systems in [CW13] provide the basis for showing that approximate solutions are close to exact solutions for ε small. The approach to error analysis based on singular systems is especially well-suited to situations involving several pulses traveling at distinct group velocities. In these situations “one-phase methods” that depend on constructing high order approximate solutions, or which use estimates of conormal or striated type, seem inapplicable. Finally, we stress that pulses do interact; the interactions do not produce resonances that affect leading order profiles as in the wavetrain case, but the interactions, whether of transversal or self-interaction type, do affect correctors to the leading order profiles. We have treated the analogous problems for wavetrains in [CGW11, CGW13].

9More precisely, we refer here to the profiles like τ

m,k(x, θm) or products of profiles that appear in these terms. 10The cutoff renders harmless the small divisor that appears when one writes the Fourier transform of the θ

m

-primitive of σm,k,p in terms of ˆσm,k,p(x, η).

11Recall that the wavetrain estimate in [CGW13] applies verbatim to the pulse case, as long as one uses the singular

(18)

2

Analysis of profile equations

For ease of exposition we continue as in Section 1.4 to consider the 3× 3 strictly hyperbolic case. The minor changes needed to treat the general case are described in section 5.

2.1 The key subsystem for U0

Let us writeU0, the solution of the ε−1 equation in (1.25), as

U0 = ˜σ1(x, θ0, ξd) r1+ ˜σ2(x, θ0, ξd) r2+ ˜σ3(x, θ0, ξd) r3.

As we saw in (1.28), the ˜σi’s have the form

˜

σi(x, θ0, ξd) = σi(x, θ0+ ωiξd) for some σi(x, θi).

Recall that we assume that the phase φ2 is outgoing while the other two phases φ1, φ3 are incoming

(the case where only one phase is incoming and all other phases are outgoing is actually simpler). Our first task is to show that σ2 ≡ 0. We will need the following version of a classical lemma by

Lax [Lax57], see [CGW11, Lemma 2.11] for the proof.

Proposition 2.1. Let W (x, θ0, ξd) =P3i=1wi(x, θ0, ξd)ri be any C1 function. Then

˜ L(∂)W = 3 X i=1 (Xφiwi) ri+ 3 X i=1  X k6=i Vkiwk  ri,

where Xφi is the characteristic vector field

Xφi := ∂xd+ d−1 X j=0 −∂ξjωi(β)∂xj = 1 vi· ed (∂t+ vi· ∇x) ,

and Vki, for k6= i, is the tangential vector field

Vki:=

d−1

X

l=0

(liAlrk) ∂xl.

The relation (1.35)(b) can be simplified by using Proposition 2.1. More precisely, we recall the notation F0 :=−( ˜L(∂)U0+D(0)U0), see (1.37). The next Proposition is an immediate consequence of Proposition 2.1.

Proposition 2.2. The function F0 reads F0=P3i=1Firi, with

Fi(x, θ0, ξd) =−Xφiσi− X k6=i Vkiσk+ 3 X k=1 eikσk,

for some real constants ei

(19)

In particular, the requirement EF0 = 0 in (1.35)(b) reads

(2.1) Xφσi− eiiσi= 0, i = 1, 2, 3.

Since the outgoing mode σ2 is zero in t < 0, this implies σ2 ≡ 0 by integrating along the

character-istics.

Using (1.26) (ε0) and σ2 = 0, we find the boundary condition

B (σ1(x′, 0, θ0) r1+ σ3(x′, 0, θ0) r3) = 0 ,

and we recall that the vectors r1, r3 span the stable subspace Es(β), see Lemma 1.11. Thus, by

(1.10) we have

σ1(x′, 0, θ0) r1+ σ3(x′, 0, θ0) r3= a(x′, θ0) e , for some scalar function a(x′, θ0).

Since e = e1+ e3, where ei ∈ span {ri}, we deduce

(2.2) σi(x′, 0, θ0) ri = a(x′, θ0) ei, i = 1, 3 .

Using (2.1) and (2.2), we have derived the main subsystem governing U0:

Xφiσi− e i iσi= 0 for i = 1, 3 σi(x′, 0, θ) ri = a(x′, θ) ei, σi = 0 in t < 0 , (2.3)

where the scalar function a(x′, θ) remains to be determined.

Boundary equation From the definition of F0, the fact that U0 is purely incoming (σ

2 ≡ 0),

and Remark 1.17, we see that the outgoing component of the right side of (1.38)(a) is zero. Writing EU1 =

3

X

i=1

τi(x, θ0+ ωiξd) ri

and using Proposition 2.1, we find Xφ2τ2− e

2

2τ2 = 0, τ2 = 0 in t < 0 ,

and thus τ2 ≡ 0.

Since EU1is purely incoming, it is valued in Es(β). Therefore, by (1.11), the equation (1.38)(b)

is solvable if and only if the condition

b·G− (dψ(0) U0)U0− B (I − E) U1= 0 on xd= ξd= 0 ,

(2.4)

holds. Since b· Bri = 0, i = 1, 3, using (1.37) we see that

b· B (I − E)U1(x′, 0, θ0, 0) = b· B R∞F0 =−b · B r2 Z R+F2(x ′, 0, θ 0− ω2s, s) ds  . Thus, the solvability condition (2.4) becomes

b·  G− (dψ(0) U0)U0+ Br2 Z R+ F2(x′, 0, θ0− ω2s, s) ds  = 0. (2.5)

We now rewrite this equation in terms of the function a(x′, θ

0) appearing in (2.3). We will use

(20)

Proposition 2.3 ([CG10]). There holds

(2.6) b· B X

m∈{1,3}

RmAjem= κ∂ηjσ(β), j = 0, . . . , d− 1, where κ ∈ R \ 0, ∂τσ(β) = 1,

the function σ is defined in Assumption 1.5, and Rm :=Pk6=mωmP−ωk k, so

12 R1= P2 ω1− ω2 + P3 ω1− ω3 , R3 = P1 ω3− ω1 + P2 ω3− ω2 . Thus, b· BX m∈I RmL(∂)e˜ m = κ  ∂τσ(β)∂t+ d−1 X j=1 ∂ηjσ(β)∂xj   := XLop. (2.7)

Our main result in this paragraph reads as follows.

Proposition 2.4. There are real constants c and d such that taking the θ0-derivative of equation

(2.5) is equivalent to the following equation for the function a(x′, θ

0) in (2.3):

XLopa + c a + d ∂θ0(a

2) =−b · ∂ θ0G .

(2.8)

Proof. 1.) Let us first observe that U0 = a(x, θ

0) e on xd = ξd = 0, so the bilinear term

(dψ(0)U0)U0 in (2.5) reads a2(dψ(0) e)e, which gives an a2 contribution (up to a real

multiplica-tive constant). This term contributes to the ”Burgers like” term d ∂θ0(a

2) in (2.8) after taking the

θ0-derivative.

2.) Let us recall the definition F0 = −( ˜L(∂)U0 + D(0)U0), and F

2 := l2· F0. Another easy

contribution to (2.5) is thus given when computing Z R+ l2· D(0)(σ1(x′, 0, θ0+ (ω1− ω2) s) r1+ σ3(x′, 0, θ0+ (ω3− ω2) s) r3) ds = Z R+ l2· D(0)(a(x′, θ0+ (ω1− ω2) s) e1+ a(x′, θ0+ (ω3− ω2) s) e3) ds .

Taking the θ0 derivative of this integral gives a contribution of the form c a(x′, θ0) for a suitable

real constant c.

3.) It remains to examine the last integral term in (2.5), that is I := Z R+ l2· ˜L(∂)U0(x′, 0, θ0− ω2s, s) ds = Z R+ l2· ˜L(∂)(σ1(x′, 0, θ0+ (ω1− ω2) s) r1+ σ3(x′, 0, θ0+ (ω3− ω2) s) r3) ds .

Since l2· ri = 0, i = 1, 3, the coefficient of the ∂xd term in the latter integral vanishes and we obtain

I r2=   d−1 X j=0 (l2Aje1) Z R+ ∂ja(x′, θ0+ (ω1− ω2)s) ds   r2 +   d−1 X j=0 (l2Aje3) Z R+ ∂ja(x′, θ0+ (ω3− ω2)s) ds   r2. (2.9) 12Recall P iX= (li· X)ri.

(21)

LetAm(x′, θ0), m = 1, 3 be the unique antiderivative(s) of a in θ0 such that

Am(x′, (ωm− ω2)∞) = 0 .

The integrals in (2.9) can be computed and the function I r2 can be rewritten

I r2 = d−1 X j=0 (l2Aje1) ∂jA1(x′, θ0) ω1− ω2 r2+ d−1 X j=0 (l2Aje3) ∂jA3(x′, θ0) ω3− ω2 r2. (2.10)

Using (2.6), (2.7), and the fact that b· Bei = 0, i = 1, 3, we obtain from (2.10) that the final

contribution in taking the θ0-derivative of (2.5) reads

b· B∂θ0(I r2) = XLopa.

Combining (2.3) with Proposition 2.4, we have the following key subsystem satisfied by (σ1, σ3):

(a) Xφ1σ1− e11σ1= 0 (b) Xφ3σ3− e 3 3σ3 = 0 (c) XLopa + c a + d ∂θ0(a 2) =−b · ∂ θ0G on {xd= 0, ξd= 0}. (2.11) where σi(x′, 0, θ0)ri = a(x′, θ0)ei (2.12)

and all unknowns are zero in t < 0. Recall that the outgoing component σ2 of U0 vanishes. In

Proposition 2.8 we solve for a by Picard iteration, and then obtain σ1 and σ3 in the interior by

simply integrating along the appropriate characteristics.

Remark 2.5. In the case of wavetrains the interior equation for τ2 is coupled to σ1 and σ3 by a

nonlinear interaction integral whenever there is a resonance of the form nφ1+ n3φ3 = n2φ2, with ni∈ Z.

Thus, τ2, the outgoing component of EU1 is not necessarily zero, and this leads to the presence of

an extra term of the form ∂θ0τ2 in the last equation of (2.11). That extra term leads to a loss of

derivatives in the estimate for the linearized system, and forced us to use Nash-Moser iteration to solve for (σ1, σ3, τ2) in [CGW13].

We shall work with profiles lying in a class of Sobolev spaces weighted in θ. Let ΩT ={(x, θ) ∈ R

d+1

+ × R : t < T },

and introduce the following spaces.

Definition 2.6. For s∈ N, define the spaces

(22)

with norms |a|s,T = X |β|=|β1,β2,β3|≤s |θβ1β2 x ∂θβ3a|L2(Ω T). (2.14)

The analogous norms of functions of b(x′, θ) defined on bΩT := {(x′, θ) ∈ Rd× R : t < T } are

denoted with brackets: hbis,T. We denote the corresponding spaces by bΓsT.

We will let Hs

T denote the usual Sobolev space on ΩT with norm defined just as in (2.14) but

without the θ weights. These spaces and those below have the obvious meanings when a function is vector-valued.

Remark 2.7. By Sobolev embedding if the functions fki, gl,mi , hil,m appearing in (1.31) lie in ΓsT for s > d+22 + 3, then F as in (1.30) is of typeF.

Proposition 2.8. Let T > 0, m > d+12 + 1 and suppose G ∈ bΓm+1T and vanishes in t < 0. Then provided hGibΓm+1T is small enough, the system (2.11), (2.12) has a unique solution satisfying

a∈ bΓm

T and σi∈ ΓmT, i = 1, 3.

Proof. First one solves (2.11)(c) using the iteration scheme XLopan+1+ 2 d an∂θ0a

n+1=−b · ∂

θ0G− c a

n.

The standard proof showing convergence towards a solution a ∈ Hm

T (x′, θ0) is easily adapted to

the weighted spaces bΓmT. Details of a very similar argument are given in the proof of [CW13, Proposition 3.6]. The conclusion for σ1 and σ3 follows easily.

2.2 The first corrector U1

To understand the properties of U1 we need:

Lemma 2.9. The functions σ1, σ3, and a all have moment zero.

Proof. Let a(x) =RRa(x′, θ

0) dθ0. Taking the moment of equation (2.11)(c), we obtain the problem

for a:

XLopa + c4a = 0, a = 0 in t < 0.

Thus a = 0 and so the equations for σi, i = 1, 3 imply their moments are zero as well.

Proposition 2.10. Let σ(x, θ) ∈ Γs

T, s > d2 + 3, have moment zero and let σ∗(x, θ) denote the

unique θ-primitive of σ that decays to zero as |θ| → ∞. Then σ∗ ∈ Γs−1 T .

Proof. We have σ ∈ Γs

T if and only if ∂mk∂xα(mpσ(x, m))ˆ ∈ L2(x, m) for all k, α, p such that k +

|α| + p ≤ s. Since ˆσ(x, 0) = 0 we have ˆ σ(x, m) = m Z 1 0 (∂mσ)(x, sm) dsˆ so c σ∗(x, m) = 1 imˆσ(x, m) =−i Z 1 0 (∂mˆσ)(x, sm) ds,

(23)

Proposition 2.11. Suppose G∈ bΓs+1T , s > d2+ 4, and vanishes in t < 0. Then the first corrector U1 in (1.36) satisfies (2.15) (I− E)U1(x, θ0, ξd) = 3 X i=1 Wi(x, θ0, ξd) ri, where Wi(x, θ0, ξd) = X k6=i bk(x, θ0+ ωkξd) , with bk∈ Γs−2T .

Proof. We have (I− E)U1= R∞F0 = R∞(I − E)F0, so using Proposition 1.37 we obtain

((I− E)U1)i=− X k6=i eik Z +∞ ξd σk(x, θ0+ ωiξd+ s(ωk− ωi)) ds +X k6=i Z +∞ ξd Vkiσk(x, θ0+ ωiξd+ s(ωk− ωi)) ds .

Now σk∈ ΓsT has moment zero, so

Z +∞

ξd

σk(x, θ0+ ωiξd+ s(ωk− ωi)) ds = σk∗(x, θ0+ ωkξd)/(ωk− ωi),

where σ∗k(x, θk) ∈ Γs−1T by Proposition 2.10. The other terms are treated similarly using Vkiσk ∈

Γs−1T .

Proposition 2.12. Suppose G ∈ bΓs+1T , s > d2 + 4, and vanishes in t < 0. Then there exists a

solution U1 to (1.36) that satisfies

EU1(x, θ0, ξd) = τ1(x, θ0+ ω1ξd)r1+ τ3(x, θ0+ ω3ξd)r3 with τk∈ Γs−3T .

Proof. We have already shown τ2 = 0. Since the σi ∈ ΓsT, i = 1, 3, have moment zero, and the

functions bkin (2.15) lie in Γs−2T , it follows that the right side of (1.38)(a) has components given by

functions in Γs−3T . Similarly, the right side of the boundary equation (1.38)(b) has components given by functions in bΓs−3T . The solvability condition (2.4) holds, so we may make a choice of boundary data EU1(x′, 0, θ0, 0) satisfying (1.38)(b) whose components are elements of bΓs−3T . Thus, solving

the system (1.38), we obtain a solution EU1 with components given by τ

k∈ Γs−3T , k = 1, 3.

Definition 2.13. Suppose W (x, θ0, ξd) = EW + (I− E)W and

EW (x, θ0, ξd) = 3 X i=1 ai(x, θ0+ ωiξd)ri, with ai(x, θi)∈ ΓsT; (I − E)W (x, θ0, ξd) = 3 X i=1 wiri, where wi(x, θ0, ξd) = X k6=i bk(x, θ0+ ωkξd) with bk(x, θk)∈ ΓsT. Then we write W ∈ ˜Γs T.

We may summarize our construction ofU0 and U1 as follows.

Theorem 2.14. Suppose G ∈ bΓs+1T , s > d2 + 4, and vanishes in t < 0. Then provided hGis+1 T

is small enough, there exist solutions U0 ∈ ˜Γs

T, U1 ∈ ˜Γs−3T , satisfying the profile equations (1.35),

(24)

3

The exact solution of the singular system

In this section we state the main estimates for the linearized singular system and also the existence theorem for the nonlinear singular system (1.23). We begin by gathering the notation for spaces and norms that is needed below.

Notations 3.1. Here we take s∈ N = {0, 1, 2, . . . }.

(a) Let Ω := Rd+1+ × R, ΩT := Ω∩ {−∞ < t < T }, bΩ := Rd× R, bΩT := bΩ∩ {−∞ < t < T },

and set ωT := Rd+1+ ∩ {−∞ < t < T }.

(b) Let Hs ≡ Hs(bΩ), the standard Sobolev space with norm hV (x, θ

0)is. For γ ≥ 1 we set

Hγs:= eγtHs and hV is,γ :=he−γtVis.

(c) L2Hs≡ L2(R

+, Hs(bΩ)) with norm |U(x, θ0)|L2Hs ≡ |U|0,s given by

|U|20,s=

Z ∞

0 |U(x ′, x

d, θ0)|2Hs(bΩ)dxd.

The corresponding norm on L2Hγs is denoted |V |0,s,γ.

(d) CHs≡ C(R

+, Hs(bΩ)) denotes the space of continuous bounded functions of xdwith values

in Hs(bΩ), with norm |U(x, θ0)|CHs = |U|∞,s := supx

d≥0|U(., xd, .)|Hs(bΩT).The corresponding

norm on CHs

γ is denoted |V |∞,s,γ.

(e) Let M0 := 3d + 5 and define C0,M0 := C(R+, CM0(bΩ)) as the space of continuous bounded

functions of xdwith values in CM0(bΩ), with norm|U(x, θ0)|C0,M0 :=|U|L∞WM0,∞. Here L∞WM0,∞

denotes the space L∞(R

+; WM0,∞(bΩ))13.

(f ) The corresponding spaces on ΩT are denoted L2HTs, L2Hγ,Ts , CHTs, CHγ,Ts and CT0,M0 with

norms |U|0,s,T, |U|0,s,γ,T,|U|∞,s,T,|U|∞,s,γ,T, and |U|C0,M0 T

respectively. On bΩT we use the spaces

HTs and Hγ,Ts with norms hUis,T and hUis,γ,T.

(g) All constants appearing in the estimates below are independent of ε, γ, and T unless such dependence is explicitly noted.

The following spaces appear in the statement of the main existence theorem and are used in its proof.

Definition 3.2. For s∈ {0, 1, 2, . . . }, let

ETs := CHTs ∩ L2HTs+1, with the norm|U(x, θ0)|Es

T :=|U|∞,s,T +|U|0,s+1,T,

Eγ,Ts := CHγ,Ts ∩ L2Hγ,Ts+1, with the norm|U(x, θ0)|Es

γ,T :=|U|∞,s,γ,T +|U|0,s+1,γ,T.

The linearization of the singular problem (1.23) at U (x, θ0) has the form

(a) ∂dU˙ε+ A  ∂x′+β∂θ0 ε  ˙ Uε+D(εU) ˙Uε = f (x, θ0) on Ω , (b)B(εU) ˙Uε|xd=0 = g(x ′, θ 0) , (c) ˙Uε= 0 in t < 0, (3.1) 13The size of M

(25)

where the matrices B(εU), D(εU) are defined in (1.15)14. Instead of (3.1), consider the equivalent problem satisfied by ˙Uγ := e−γtU :˙ ∂dU˙γ+ A  (∂t+ γ, ∂x′′) +β ∂θ0 ε  ˙ Uγ+D(εU) ˙Uγ= fγ(x, θ0) , B(εU) ˙Uγ|xd=0= g γ(x, θ 0) , ˙ Uγ = 0 in t < 0 . (3.2)

Below we let ΛD denote the singular Fourier multiplier (see (B.2)) associated with the symbol

Λ(X, γ) := γ2+ ξ′+ k βε 2!1/2 , X := ξ′+k β ε .

The basic estimate for the linearized singular problem (3.2) is given in the next Proposition. Observe that the estimate (3.3) exhibits a loss of one “singular derivative” ΛD. This is quite a high price to

pay, which counts as a factor 1/ε. In view of [CG10, Theorem 4.1], there is strong evidence that the loss below is optimal.

Proposition 3.3 (Main L2linear estimate). We make the structural assumptions of Theorem 1.14, let s0 := d+12 + 1, and recall M0 = 3d + 5. Fix K > 0 and suppose |ε ∂dU|C0,M0−1+|U|C0,M0 +

|U|CHs0 ≤ K for ε ∈ (0, 1]. There exist positive constants ε0(K) > 0, C(K) > 0 and γ0(K)≥ 1

such that sufficiently smooth solutions ˙U of the linearized singular problem (3.1) satisfy15:

| ˙Uγ|0,0+h ˙U γi 0 √γ ≤ C(K)  |ΛDfγ|0,0+|fγ/ε|0,0 γ2 + hΛDgγi0+hgγ/εi0 γ3/2  (3.3) for γ≥ γ0(K), 0 < ε≤ ε0(K).

The same estimate holds if B(εU) in (3.1) is replaced by B(εU, εU) and D(εU) is replaced by D(εU, εU) as long as |ε∂d(U,U)|C0,M0−1+|U, U|C0,M0 +|U, U|CHs0 ≤ K for ε ∈ (0, 1].

The proof is identical to the proof of Proposition 2.2 in [CGW13]. Here our hypothesis on |U|CHs0 is needed to allow the pulse calculus summarized in Appendix B to be used in exactly

the same way that the wavetrain calculus was used in [CGW13]. The corollary below follows from Proposition 3.3 by the same argument used to derive Corollary 2.3 from Proposition 2.2 in [CGW13].

Corollary 3.4 (Main H1

tan linear estimate). Under the same assumptions as in Proposition 3.3,

smooth enough solutions ˙U of the linearized singular problem (3.1) satisfy: | ˙Uγ|∞,0+| ˙Uγ|0,1+h ˙U

γi 1 √γ ≤ C(K)  |ΛDfγ|0,1+|fγ/ε|0,1 γ2 + hΛDgγi1+hgγ/εi1 γ3/2  (3.4) for γ≥ γ0(K), 0 < ε≤ ε0(K).

The next proposition localizes the estimate (3.4) to ΩT. By considering data εf , εg instead of

f, g we can recast (3.4) in a form where the loss of a singular derivative is replaced by loss of an ordinary derivative. The proof is identical to that of [CGW13, Proposition 2.8]. Let us write the linearized operators on the left sides of (3.1)(a) and (b) as L′(εU ) ˙U and B(εU ) ˙U respectively.

14Here and below we often suppress the subscript ε on ˙U. 15Note that the norms |u|

(26)

Proposition 3.5. Let s0=

d+1

2



+ 1, fix K > 0, and suppose|ε∂dU|C0,M0−1 T

+|U|C0,M0 T

+|U|CHTs0 ≤

K for ε ∈ (0, 1]. There exist positive constants ε0(K), γ0(K) such that solutions of the singular

problem L′ ε(U ) ˙U = εf in ΩT , B′ ε(U ) ˙U = εg on bΩT , ˙ U = 0 in t < 0, satisfy

| ˙U|∞,0,γ,T +| ˙U |0,1,γ,T +h ˙U|xd√=0i1,γ,T

γ ≤ C(K)  |f|0,2,γ,T γ2 + hgi2,γ,T γ3/2 

for 0 < ε≤ ε0(K), γ ≥ γ0(K), and the constant C(K) only depends on K.

The same estimate holds if B(εU) in (3.1) is replaced by B(εU, εU) given in (1.13), and D(εU) is replaced by D(εU, εU) given in (1.14), as long as there holds |ε∂d(U,U)|C0,M0−1

T

+|U, U|C0,M0 T

+ |U, U|CHs0T ≤ K for ε ∈ (0, 1].

We need the following higher derivative estimate in the error analysis. This estimate is also needed in the Nash-Moser iteration used to prove the nonlinear existence Theorem (3.7). We recall the definitions (1.21) for the indices a0, a1, a, ˜a.

Proposition 3.6 (Tame estimate for the linearized system). Fix K > 0 and suppose |ε ∂dU|C0,M0−1 T +|U|C0,M0 T +|U|CHs0T ≤ K for ε ∈ (0, 1]. (3.5)

Let µ0 := a0+2 and s∈ [0, ˜a], where ˜a is defined in (1.21). There exist positive constants γ = γ(K),

κ0(γ, T ), ε0, and C such that if

|U|0,µ0,γ,T +hU|xd=0iµ0,γ,T ≤ κ0,

then solutions ˙U of the linearized system (3.1) satisfy for 0 < ε≤ ε0:

| ˙U|Es

γ,T +h ˙U|xd=0is+1,γ,T

≤ Ch|f|0,s+2,γ,T+hgis+2,γ,T + (|f|0,µ0γ,T +hgiµ0,γ,T) (|U|0,s+2,γ,T +hU|xd=0is+2,γ,T)

i . The same estimate holds for 0 < ε ≤ ε0 if B(εU) in (3.1) is replaced by B(εU, εU) and D(εU) is

replaced by D(εU, εU) as long as |ε∂d(U,U)|C0,M0−1 T +|U, U| CT0,M0 +|U, U|CHs0 T ≤ K for ε ∈ (0, 1] and |U, U|0,µ0,γ,T +hU|xd=0,U|xd=0iµ0,γ,T ≤ κ0.

The proof is exactly the same as the proof of Proposition 2.15 in [CGW13]. Finally, we state the existence theorem for solutions of the singular system (1.23).

Theorem 3.7. Fix T > 0, define a, a0, and ˜a as in (1.21), and suppose G ∈ H˜a(bΩT). There

exists ε0 > 0 such that ifhGia+2 is small enough, there exists a solution Uε of the system (1.23) on

ΩT for 0 < ε ≤ ε0 with Uε ∈ Ea−1, Uε|xd=0 ∈ H

a. This statement remains true if a is increased

(27)

Proof. The proof is an exact repetition of the Nash-Moser argument used to prove Theorem 5.13 in [CGW13]. One fixes T > 0, K > 0 and γ = γ(K) as in Proposition 3.6, and uses the scale of spaces Es

γ,T (Definition 3.2) with γ T = 1. The smoothing operators Sθ are defined just as in Lemma 5.12

of [CGW13], taking account in the obvious way for the fact that now functions u(x, θ0) ∈ Eγ,Ts

are defined for θ0 ∈ R instead of θ0 ∈ T. The verification of the condition (3.5) needed to apply

the tame estimate of Proposition 3.6 in the induction argument is done just as in Lemma 5.20 of [CGW13].

4

Error analysis

4.1 Moment-zero approximations

When constructing a corrector to the leading part of the approximate solution we must take primitives in θ of functions f (x, θ) that decay to zero as |θ| → ∞. A difficulty is that such primitives do not necessarily decay to zero as |θ| → ∞, and this prevents us from using those primitives directly in the error analysis. The failure of the primitive to decay manifests itself on the Fourier transform side as a small divisor problem. To get around this difficulty we work with the primitive of a moment-zero approximation, because such a primitive does have the desired decay.

We will use the following spaces:

Definition 4.1. 1.) For s≥ 0, let ETs be given in Definition 3.2. 2.) Let Es

T :={U(x, θ0, ξd) :|U|ETs := supξd≥0|U(·, ·, ξd)|EsT <∞}.

Proposition 4.2. For s > (d + 1)/2, the spaces ETs andETs are Banach algebras.

Proof. This is a consequence of Sobolev embedding and the fact that L∞(bΩT)∩ Hs(bΩT) is a

Banach algebra for s≥ 0.

The proofs of the following two propositions follow directly from the definitions.

Proposition 4.3. (a) For s≥ 0, let σ(x, θ) ∈ HTs+1 and set ˜σ(x, θ0, ξd) := σ(x, θ0+ ωξd), ω∈ R.

Then there holds

|˜σ|Es T ≤ C|σ|Hs+1T . (b) Set ˜σε(x, θ0) := ˜σ(x, θ0,xεd). Then |˜σε|Es T ≤ |˜σ|E s T.

Definition 4.4 (Moment-zero approximations). Let 0 < p≤ 1, and let φ ∈ C∞(R) have supp φ

{m : |m| ≤ 2}, with φ = 1 on |m| ≤ 1. Set φp(m) = φ(mp) and χp = 1− φp. For σ(x, θ) ∈ L2,

define the moment zero approximation to σ, σp(x, θ) by

ˆ

σp(x, m) := χp(m) ˆσ(x, m),

(4.1)

where the hat denotes the Fourier transform in θ.

Proposition 4.5. For s≥ 1 suppose σ(x, θ) ∈ Γs+2 and ˜σ(x, θ

0, ξd) := σ(x, θ0+ ωξd). Then (a) |˜σ − ˜σp|Es T ≤ C |σ|Γs+2Tp, (b)|∂xdσ˜− ∂xdσ˜p|ETs−1 ≤ C |σ|Γs+2Tp.

(28)

Proof. 1.) Recall that σ ∈ Γs

T ⇔ θβ1∂ β2

x ∂θβ3σ(x, θ) ∈ L2(x, θ) for |β| ≤ s ⇔ ∂mβ1∂xβ2mβ3σ(x, m)ˆ ∈

L2(x, m) for |β| ≤ s. It follows that

σ ∈ Γs+2T ⇒ ˆσ(x, m) ∈ HTs+2(x, m)⊂ H1(m, Hs+1(x))⊂ L∞(m, Hs+1(x)). (4.2) 2.) We have |σ − σp|2Hs+1 T ∼ X |α|+k≤s+1 ∂xαmkσ(x, m)(1ˆ − χp(m)) 2 L2(x,m) = X |α|+k≤s+1 Z |m|≤2p Z |∂xαmkσ(x, m)(1ˆ − χp(m))|2dx dm ≤ C Z |m|≤2p|ˆσ(x, m)| 2 Hs+1(x)dm≤ C |σ|2 Γs+2T (2p),

where the last inequality uses (4.2). The conclusion now follows from Propostion 4.3. 3.) The proof of part (b) is essentially the same.

Proposition 4.6. Let σ(x, θ)∈ Hs

T, s≥ 0, and let σp be a moment-zero approximation to σ. We

have (a)p|Hs T ≤ C |σ|HTs , (b) If σ∈ ΓsT, then p|Γs T ≤ C ps|σ|ΓsT.

Proof. Part (b) follows from (4.1). Indeed, for|β| ≤ s, there holds |∂β1 m∂xβ2mβ3ˆσp(x, m)|L2 T ≤ C pβ1 |σ|Γ s T, since|∂β1

mχp| ≤ C/pβ1. Taking β1= 0 we similarly obtain part (a).

Next we consider primitives of moment-zero approximations.

Proposition 4.7. Let σ(x, θ)∈ ΓsT, s > d2+ 3. Let σp∗(x, θ) be the unique primitive of σp in θ that

decays to zero as |θ| → ∞. Then σ

p ∈ ΓsT with moment zero, and

(a) p|Hs T ≤ C |σp|Hs T p (b)p|Γs T ≤ C |σp|Γs T ps+1 . (4.3)

Proof. 1.) Since σp(x, θ)∈ ΓsT, s > d2 + 3, we have |σp(x, θ)| ≤ C hθi−2 for all (x, θ). The unique

θ-primitive of σp decaying to zero as |θ| → ∞ is thus

σ∗p(x, θ) = Z +∞ θ σp(x, s) ds = Z θ −∞ σp(x, s) ds. Moreover, we have ∂θσ∗p = σp⇔ im cσp∗ =cσp = χpσ, so cˆ σp∗ = χpσˆ im . (4.4)

Figure

Updating...

Références

Updating...

Sujets connexes :