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arXiv:1107.4597v1 [math.AP] 22 Jul 2011

A DECAY ESTIMATE FOR A WAVE EQUATION WITH

TRAPPING AND A COMPLEX POTENTIAL

LARS ANDERSSON, PIETER BLUE, AND JEAN-PHILIPPE NICOLAS

Abstract. In this brief note, we consider a wave equation that has both

trapping and a complex potential. For this problem, we prove a uniform bound

on the energy and a Morawetz (or integrated local energy decay) estimate.

The equation is a model problem for certain scalar equations appearing in the

Maxwell and linearised Einstein systems on the exterior of a rotating black

hole.

1. Introduction

We consider the Cauchy problem

?−∂2

u(0,x,ω) = ψ0(x,ω),

t+ ∂2

x+ V (∆ω− N) + iǫW?u = 0,

∂tu(0,x,ω) = ψ1(x,ω),

(1)

(2)

on (t,x,ω) ∈ M = R×R×S2with smooth, compactly supported initial data. Here

u is a complex function u = v + iw,

V =

1

x2+ 1,

W is a smooth, real-valued, compactly supported function which is nonvanishing at

x = 0 and uniformly bounded by 1, and ǫ > 0 is a small parameter. Finally, ∆ωis

the Laplacian in the angular variables and N is a number chosen to be sufficiently

large to allow us to avoid certain technical problems.

The equation (1) has both trapping, which occurs at x = 0, and a complex

potential, which does not vanish at the trapped set. The interaction of these creates

problems, which appear to frustrate the use of energy and Morawetz estimates at

the classical level. By adapting known, pseudodifferential methods, we show how

to overcome these problems. We now state our main result in terms of the energy

E(t) =1

2

?

{t}×R×S2|∂tu|2+ |∂xu|2+ V (|∇ωu|2+ N|u|2) dxd2ω,

Theorem 1. There is a constant C such that, if ψ0and ψ1are such that E(0) is

finite then

∀t ∈ R :

?|∇ωu|2

?

M

E(t) ≤ CE(0),

?

(3a)

?

M

|∂xu|2

x2+ 1+

x2

1 + x2

1 + |x|3+|∂tu|2

|u||∂tu|

1 + |x|3dtdxd2ω ≤ CE(0).

x2+ 1

+

|u|2

1 + |x|3dtdxd2ω ≤ CE(0),(3b)

(3c)

Date: July 25, 2011 File:ModelProblem.tex.

1

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2 L. ANDERSSON, P. BLUE, AND J.-P. NICOLAS

Since the equation (1) has t independent coefficients, one might naively think

that Noether’s theorem provides a positive conserved energy. However, for the

Lagrangian L1[u,∂u] = −(∂tu)2+ (∂xu)2+ V (∇ωu · ∇ωu + Nu2) − iǫWu2, which

has the wave equation (1) as its Euler-Lagrange equation, the conserved quantity

associated to the time translation symmetry is indefinite, being approximately the

energy of the real component of u minus the energy of the imaginary component

(plus ǫ times a term involving Wvw). On the other hand, a Lagrangian of the form

L2[u,∂u] = −|∂tu|2+|∂xu|2+V (|∇ωu|2+N|u|2), which corresponds to the energy

expression considered above, does not yield equation (1) as its Euler-Lagrange

equation.

The wave equation (1) is a model for equations arising in the study of the Maxwell

and linearised Einstein equations outside a Kerr black hole. The Kerr black holes

are a family of Lorentzian manifolds arising in general relativity, and they are

characterised by a mass parameter M and an angular momentum parameter a.

Black holes are believed to be the enormously massive objects at the center of

most galaxies. The case |a| ≤ M is the physically relevant. The case a = 0 is the

Schwarzschild class of black holes.

It is expected that every uncharged black hole will asymptotically approach a

Kerr solution under the dynamics generated by the Einstein equations of general

relativity. The wave, Maxwell, and linearised Einstein equations on a fixed Kerr

geometry are a sequence of increasingly accurate models for these dynamics. By

projecting on a null tetrad, the Maxwell and linearised Einstein fields can be decom-

posed into sets of complex scalars, the Newman-Penrose (NP) scalars [22, 23, 14].

It is well-known that the NP scalars with extreme spin weights satisfy decoupled

wave equations, known as the Teukolsky equations, and that the solutions to these

reduced equations can be used to reconstruct the full system [29].

For the Maxwell field on the Kerr background, the spin weight 0 NP scalar can be

treated in the same way, and the resulting equation is known as the Fackerell-Ipser

equation [11]. For linearized gravity on the Schwarzschild background, it is also

well known that the imaginary part of the spin weight 0 NP scalar is governed by a

wave equation, the Regge-Wheeler equation [26, 24]. The corresponding equation

for the real part is more complicated, cf. [33, 21], see also [1].

It was recently shown [1] that in the general (|a| < M) Kerr case, by imposing a

gauge condition related to the wave coordinates gauge, the equation for both the real

and imaginary parts of the spin weight 0 NP scalar of the linearized gravitational

field may be put in a form analogous to the Regge-Wheeler and Fackerell-Ipser

equations. Explicitely (in the Kerr spacetime with signature −+++, working in

Boyer-Lindquist coordinates) these take the form

?

∇α∇α+ 2s2

M

(r − iacosθ)3

?

u = 0,(4)

where s = 0 corresponds to the free scalar wave equation, s = 1 corresponds to the

Maxwell (Fackerell-Ipser) case, while s = 2 corresponds to the linearized gravity

(generalized Regge-Wheeler) case. In particular, for the a ?= 0 cases, the analogues

of the Regge-Wheeler equations have complex potentials, with the imaginary part

depending continuously on a.

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COMPLEX POTENTIALS AND TRAPPING3

For the wave equation in the Schwarzschild case, the use of the energy estimate

[31], like (3a) with C = 1, and Morawetz estimates1are well established [17, 3,

4, 6, 8]. In the Morawetz estimate (3b), there is a loss of control of time and

angular derivatives near x = 0, in the sense that the integrand cannot control

|x|p(|u|qt|∂tu|2−qt+ |u|qω|∇ωu|2−qω) with both p = 0 and either qt= 0 or qω= 0.

The presence of trapping makes some loss unavoidable [25]. By applying “angular

modulation” and “phase space analysis”, the range for the angular parameter qω

can be refined to p = 0 and q > 0 [5]. This type of refinement is crucial in the

current paper. Alternatively, certain pseudodifferential operators have been used

to obtain refinements near x = 0, to p > 0 and qt= qω= 0 [20].

For the wave equation in the general (|a| < M) Kerr case, it is possible to apply

Fourier transforms first in the φ and t variables2and then the remaining variables.

The individual φ modes decay pointwise [12]. Although the problem has a time-

translation symmetry, because the generator of time translations fails to be a time-

like vector with respect to the Lorentzian inner product of the Kerr geometry, there

is no positive, conserved energy. A major advance was the proof that, in the slowly

rotating case |a| ≪ M, there is a uniform energy bound, like estimate (3a). The

first proof used an estimate similar to (3b), but with additional restrictions on the

support of the Fourier transform [7]. Independent work [28] established estimates

similar to (3a) and (3b), but with no restriction on the Fourier support, and there

were subsequent pseudodifferential refinements [30]. Also, the first two authors

have proved similar results using methods which require two additional levels of

regularity but which completely avoid the use of Fourier transforms. Morawetz

estimates and refinements are a crucial step in proving pointwise decay estimates

[5, 6, 8, 9, 18, 19] and Strichartz estimates [20, 30], including the long-conjectured,

inverse-cubic, Price law [16, 27].

The study of the Maxwell and linearised Einstein systems in the Kerr geometry

is still in its infancy. For the general Kerr case, a certain transformed, separated

version of the Teukolsky system has no exponentially growing modes [32]. In the

Schwarzschild case, the φ modes of the Teukolsky equation decay pointwise [13].

Recently, improved decay estimates for the Regge-Wheeler type equation (4) on the

Schwarzschild background, giving decay rates of t−3, t−4and t−6, respectively, for

s = 0,1,2, have been proved [10]. The Regge-Wheeler equation has been used with

the full system to prove energy and Morawetz (and pointwise decay) estimates for

the Maxwell system [2] and, more recently, under assumptions on the asymptotic

behaviour, for the full (not merely linearised) Einstein equation [15].

For the spin-weight 0 equations arising from the Maxwell and linearised Einstein

equations with a ?= 0, the presence of complex potentials in the reduced equations

(4) prevents the existence of a positive conserved energy and means that an unre-

fined Morawetz estimate, such as (3b), is insufficient to control the growth of the

energy, which is why we also prove estimate (3c). In the Kerr geometry, there is

no positive, conserved energy because the generator of the time-translation sym-

metry fails to be timelike everywhere. In contrast, for the equation (1), there is no

positive, conserved energy because the complex potential prevents any variational

approach from providing an energy-momentum tensor that satisfies the dominant

1These are also called integrated local energy decay estimates.

2Here φ is the azimuthal angle, which would be one component of ω in the notation of this

paper.

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4L. ANDERSSON, P. BLUE, AND J.-P. NICOLAS

energy condition. The method of this paper combines a Fourier-transform-in-time

technique (as in [7, 28]) with a “modulation” (or Fourier-rescaling) technique (from

[5]).

As is common, C will be used to denote a constant which may vary from line to

line, but which is independent of the choice of u or T. The notation A ? B is used

to denote that there is some C such that A < CB, with C independent of u and

T, and similarly for ?.

2. A preliminary energy estimate

We derive an estimate for an energy for the wave equation (1) by integrating

by parts against ∂t¯ u and following the standard procedure for getting an energy

estimate:

0 = Re?(∂t¯ u)(−∂2

= −1

2∂t|∂tu|2

+ ∂xRe((∂t¯ u)∂xu) −1

2∂t|∂xu|2

+ ∇ω· Re((∂t¯ u)∇ωu) −1

t+ ∂2

x+ V (∆ω− N) + iǫW)u?

2∂t

?V (|∇ωu|2+ N|u|2)?

− ǫWIm((∂t¯ u)u).

Introducing an energy which we denote by

?

E(t) =1

2

{t}×R×S2|∂tu|2+ |∂xu|2+ V (|∇ωu|2+ N|u|2) dxd2ω,

assuming that u decays sufficiently rapidly as |x| → ∞, and integrating the previous

formula over a region [t1,t2] × R × S2, we find

?

E(t2) − E(t1) =

[t1,t2]×R×S2−ǫWIm((∂t¯ u)u) dtdxd2ω.(5)

In particular, note that the energy fails to be conserved and that an estimate of the

form (3b) would be insufficient to control the right-hand side. There is, however, a

trivial exponential bound:

E(t2) ≤ eǫ(t2−t1)E(t1).

3. The Morawetz estimate

Following the standard procedure for investigating the wave equation, we derive

a Morawetz estimate by multiplying the wave equation by (f(x)∂x¯ u+q(x)¯ u), where

f and q are real-valued functions.

In performing this calculation, it is useful to observe that

q′(x)Re(¯ u∂xu) = ∂x

?q′

2¯ uu

?

−1

2q′′¯ uu.

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COMPLEX POTENTIALS AND TRAPPING5

Using this and applying the product rule term-by-term, one finds

Re?(f∂x¯ u + q¯ u)?−∂2

= ∂tpt+ ∂xpx+ ∇ωpω

?

??1

?

− ǫfWIm((∂x¯ u)u),

tu + ∂2

xu + V (∆ω− N)u + iǫWu??

?

?

??1

+−1

2f′+ q|∂tu|2−

?1

2f′+ q

?

?

|∂xu|2

+

2f′− qV +1

?

2f(∂xV )

V +1

|∇ωu|2

?

+N

2f′− q

2f(∂xV )+1

2q′′

?

|u|2

(6)

where

pt= pt(f,q;u) = −Re((f(∂x¯ u) + q¯ u)(∂tu)),

px= px(f,q,u) =1

2f|∂tu|2+1

2f|∂xu|2−1

2fV |∇ωu|2

+ qRe(¯ u∂xu) −1

2(NfV + q′)|u|2,

pω= pω(f,q;u) = fV Re((∂x¯ u)(∇ωu)) + qV Re(¯ u∇ωu).

We take f = −arctan(x), for which f′= −(x2+1)−1= −V , f′′= 2x(x2+1)−2,

and f′′′= −2(3x2− 1)(x2+ 1)−3. We take q = f′/2 + δ(1 + x2)−1arctan(x)2for

some sufficiently small δ.

We use the notation

?

Ef∂x+q(t) =

{t}×R×S2Re(f(∂x¯ u)∂tu) + Re(q¯ u(∂tu)) dxd2ω,

and observe that, by a simple Cauchy-Schwarz argument, there is the estimate

|Ef∂x+q| ≤ CE.

Observing that the left-hand side of (6) vanishes, we have

0 = ∂tpt+ ∂xpx+ ∇ωpω

+ δarctan(x)2

1 + x2

?xarctan(x) − δ arctan(x)2

(1 + x2)2

?

− ǫfWIm((∂x¯ u)u)

|∂tu|2+

1

1 + x2(1 − δ arctan(x)2)|∂xu|2

?

?

+

|∇ωu|2

+N

?xarctan(x) − δarctan(x)2

(1 + x2)2

+1

2q′′

?

|u|2

Taking ǫ sufficiently small, N sufficiently large, and δ sufficiently small, the factors

in front of |∂xu|2and |u|2are nonnegative and one can dominate the term involving

W using these two terms. (These estimates are uniform, in the sense that, if the

estimate holds for choices of ǫ0, N0, and δ0, then it remains valid for ǫ < ǫ0, N = N0,

and δ = δ0.) Thus, by integrating over a time-space slab M[t1,t2]= [t1,t2]×R×S2,

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6L. ANDERSSON, P. BLUE, AND J.-P. NICOLAS

one can conclude that there is a constant C such that

E(t2) + E(t1)

?

M[t1,t2]

?

|∂xu|2

x2+ 1+ |arctan(x)|2

?|∇ωu|2

1 + |x|3+|∂tu|2

x2+ 1

?

+

|u|2

1 + |x|3dtdxd2ω.(7)

4. Pseudodifferential refinements

4.1. The wave equation for an approximate solution. We define a smooth

characteristic function of an interval [a,b] to be a function which is identically 1 on

[a,b], which is supported on [a − 1,b + 1], and which is monotonic on each of the

intervals [a−1,a] and [b,b+1]. A smooth characteristic function of a collection of

intervals, each of which are separated by distance at least two, is defined to be the

sum of the smooth characteristic functions of each interval.

Let T > 0 be a large constant. (Here large means larger than −log|ǫ| and

2.) Let χ1 be a smooth characteristic function on [0,T], and let χ2be a smooth

characteristic function of [−1,0]∪ [T,T +1]. Let χ|x|≤2be a smooth characteristic

function of [−1,1]. We will use χ1, χ2, and χ|x|≤2to denote χ1(t), χ2(t), and

χ|x|≤2(x) respectively.

Since χ1is smooth, there is a uniform bound on its derivative and second deriv-

ative, each of which are supported on [0,1] ∪ [T,T + 1], so that there is a constant

C such that |∂tχ1| + |∂2

The functions

tχ1| ≤ Cχ2.

u1= χ1χ|x|≤2u,

u2= χ2χ|x|≤2u,

u3= χ1u,

satisfies the equation

?−∂2

where

t+ ∂2

x+ V (∆ω− N) + iǫW?u1= F(u2,∇u2,t,x) + G(u3,∇u3,t,x), (8)

F(u2,∇u2,t,x) = −2(∂tχ1)(∂tu2) − (∂2

G(u3,∇u3,t,x) = 2(∂xχ|x|≤2)(∂xu3) + (∂2

tχ1)u2

xχ|x|≤2)u3.

Since all functions of t in this equation are smooth and supported in t ∈ [−2,T +2],

they are Schwartz class in t, so we may take the Fourier transform in t and remain

in the Schwartz class. We will use ? to denote the Fourier transform in t, and τ

describe u, u1, u2, and u3 and the words “Fourier transforms” to describe their

Fourier transforms. We will use L2to denote L2(dωdxdt) for functions and to

denote L2(dωdxdτ) for Fourier transforms. We will use ? · ? for ? · ?L2 unless

otherwise specified.

We introduce the following space-time integrals

?T+2

−2−2

?

for the argument of such functions. We will typically use the word “functions” to

I(T) =

?2

?

S2x2|∂tu|2+ |∂xu|2+ |u|2dωdxdt,

J(T) =

R×R×S2|τ|6/5|?

u1|2dωdxdτ.

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COMPLEX POTENTIALS AND TRAPPING7

The dependence of J upon T is through the smooth cut-off χ1in u1. Typically, the

argument T will be clear from context and will be omitted. From the Morawetz

estimate (7) and the exponential bound on the energy, it follows that I ? E(T) +

E(0).

We now aim to prove a Morawetz estimate using the Fourier transform. We take

f = −arctan(|τ|αx),

q =f′

2

=1

2

|τ|α

1 + |τ|2αx2,

with α ∈ [0,1/2]. We multiply the Fourier transform of equation (8) by (f∂x+q)¯?

all the functions are compactly supported in time, so the Fourier transforms are

Schwartz class.

u1,

and integrate the real part over R × R × S2. This integral is convergent because

4.2. Controlling the terms arising from the cut-off. We consider first the

integral arising from the right-hand side of (8). This is

?

R×R×S2Re

??(f∂x+ q)¯?

u1

???F +?G

??

dωdxdτ ≤ ?(f∂x+ q)¯?

u1???F +?G?.

The terms on the right can be estimated by

?(f∂x+ q)¯?

?f∂x?

u1? ≤ ?f∂x?

u1? ? ?∂x?

?q?

u1? + ?q?

u1? ? ??

u1?,

u1? ? I1/2,

u1? ? ?|τ|α?

u1? + ?|τ|1/2?

u1? ? I1/2+ J1/2,

and

??F +?G? ≤ ??F? + ??G?.

Because G is supported only for t ∈ [−1,T + 1] and x ∈ [−2,2], we have

??G? ? I1/2.

Similarly, because F is supported only for t ∈ [−1,0] ∪ [T,T + 1] and x ∈ [−2,2],

we have that at each instant in t, the function F is bounded in L2(dxdω) by either

CE(0)1/2or CE(T)1/2. Since we are considering two intervals in t of length 1, we

have

??F? = ?F? ? C(E(0)1/2+ E(T)1/2).

Thus, the terms on the right-hand side of the Fourier transform of (8) are bounded

by

?

≤ C(E(0)1/2+ E(T)1/2+ J1/2)(E(0)1/2+ E(T)1/2).

R×R×S2Re?(f∂x+ q)¯?

u1

???F +?G

?

dωdxdτ

(9)

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8L. ANDERSSON, P. BLUE, AND J.-P. NICOLAS

4.3. The Morawetz estimate for the approximate solution. If we multiply

the left-hand side of the Fourier transform of the wave equation (8) by (f∂x+q)¯?

Re??f∂x¯?

?

??1

?

− ǫfWIm((∂x¯?

where

u1

and take the real part, then we have the analogue of (6)

??τ2?

u1+ q¯?

u1

u1+ ∂x?

−1

u1+ V (∆ω− N)?

2f′+ q

?

??1

u1+ iǫW?

?1

?

V +1

2f(∂xV )

u1

??

= ∂xpx+ ∇ωpω

+

?

|τ?

V +1

?

u1|2−

2f′+ q

?

|∂x?

u1|2

u1|2

+

2f′− q

2f(∂xV )|∇ω?

+N

2f′− q

?

+1

2q′′

?

|?

u1|2

u1)?

u1),(10)

px= px(f,q,?

+ qRe(¯?

pω= pω(f,q;?

u1) =1

2f|∂t?

u1) −1

u1|2+1

2f|∂x?

u1|2−1

2fV |∇ω?

u1|2

u1∂x?

u1) = fV Re((∂x¯?

u1?

2(NfV + q′)|?

u1)(∇ω?

u1=¯?

u1|2,

u1)) + qV Re(¯?

u1∇ω?

u1).

Note that there is no ptterm because, for Fourier transforms, the analogue of the

product rule is simply −iτ?

integrate to zero, and the remaining terms are all non-negative except for those

arising from q′′and from W. The integral of the term involving W is bounded by

I.

We now consider the term involving q′′:

u1iτ?

u1.

When this equality is integrated over a space-time slab, the px and pω terms

1

2q′′|?

u1|2= |τ|3α1 − 3|τ|2αx2

(1 + |τ|2αx2)3|?

u1|2.

From the positivity of the remaining terms and the bound (9) of the terms coming

from the right-hand side of the wave equation (8) for u1, we have that

?

≤ C(E(0)1/2+ E(T)1/2+ J1/2)(E(0)1/2+ E(T)1/2).

R×R×S2|τ|3α1 − 3|τ|2αx2

(1 + |τ|2αx2)3|?

u1|2dωdxdτ

This can be combined with an additional factor of MI, where M is a large constant.

(700 is sufficient.) The integral I dominates the integral of (|τ|2+ 1)x2|?

?

R×R×S2

≤C(E(0)1/2+ E(T)1/2+ J1/2)(E(0)1/2+ E(T)1/2).

u1|2and is

bounded by C(E(T) + E(0)). Thus, we have

?

|τ|3α1 − 3|τ|2αx2

(1 + |τ|2αx2)3+ M(|τ|2+ 1)x2

?

|?

u1|2dωdxdτ

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COMPLEX POTENTIALS AND TRAPPING9

By considering the two cases |τ|α|x| < M−1/2and |τ|α|x| ≥ M−1/2, one can see

that if 2 − 2α = 3α (i.e. α = 2/5), then

?

|τ|3α1 − 3|τ|2αx2

(1 + |τ|2αx2)3+ M(|τ|2+ 1)x2

?

χ|x|≤2≥ C|τ|3αχ|x|≤2

and therefore we find

C(E(0)1/2+ E(T)1/2+ J1/2)(E(0)1/2+ E(T)1/2)

?

R×R×S2

≥

?

|τ|3α1 − 3|τ|2αx2

(1 + |τ|2αx2)3+ M(|τ|2+ 1)x2

?

|?

u1|2dωdxdτ ≥ J,

which implies

J ≤C(E(T) + E(0)). (11)

4.4. Closing the energy estimate. It is now possible to estimate the integral

on the right-hand side of the energy estimate (5). For |x| ≥ 1, the right-hand

side (using the compact support of W and the Cauchy-Schwarz estimate) can be

dominated by I ≤ C(E(T) + E(0)). For |x| ≤ 1, we would like to dominate the

integral over R × R × S2of |WIm( ¯ u1∂tu1)| by the integral J. However, this is not

entirely correct, because in J there is a contribution arising from the support of u

in the region t ∈ [−1,0] ∪ [T,T + 1]. The error in this approximation is bounded

by C(E(T) + E(0)). Thus, we have

?

E(T) − E(0) ≤ CǫE(T) + E(0) +

????

?

R×R×S2WIm( ¯ u1∂tu1)dωdxdt

????

?

.

We can also take the Fourier transform to obtain an estimate by

?

?

CǫE(T) + E(0) +

????

?

R×R×S2W Im(¯?

R×R×S2|τ||?

u1?

u1|2dωdxdt

∂tu1)dωdxdt

????

.

?

≤ CǫE(T) + E(0) +

?

?

The integrand is now controlled by I + J, which, from estimates (7) and (11), we

know can be estimated also by the sum of the initial and final energies. This leaves

the estimate

E(T) − E(0) ≤ Cǫ(E(T) + E(0)).

By taking ǫ sufficiently small relative to the constant, we obtain a uniform bound

on the energy

E(T) ≤ CE(0).

We note that, since all the constants were independent of T, the estimate holds

uniformly in T. This proves the first statement (3a), in theorem 1. Combining this

with estimate (7) (and estimating x2(1 + x2) ? arctan(x)2) gives the second, (3b).

Finally, the arguments of this section and the bound on I + J, from estimates (7)

and (11), give the third result (3c).

Remark 2. Using the method given in [5], the stronger Morawetz estimate (11)

can be improved to control the integral of |τ|2−ε|? u|2for any ε > 0. Because of the

presence of trapping, it is not possible to improve this to |τ|2|? u|2.

Page 10

10L. ANDERSSON, P. BLUE, AND J.-P. NICOLAS

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E-mail address: laan@aei.mpg.de

Albert Einstein Institute, Am M¨ uhlenberg 1, D-14476 Potsdam, Germany

E-mail address: P.Blue@ed.ac.uk

The School of Mathematics and the Maxwell Institute, University of Edinburgh,James

Clerk Maxwell Building, The King’s Buildings, Mayfield Road, Edinburgh, Scotland

EH9 3JZ, UK

E-mail address: Jean-Philippe.Nicolas@univ-brest.fr

Laboratoire de Math´ ematiques, Universit´ e de Brest, 6 avenue Victor Le Gorgeu,

29200 Brest, France