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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,