Improving the weights near the horizon H + : The redshift . 121

r²|Ω∇/₄Ψ−Ω∇/₃Ψ|²+1 r³|Ψ|² +(r−3M)²

r² 1

r|∇ψ|/ ²+1

r²|Ω∇/₄Ψ+Ω∇/₃Ψ|²

6C[F_u^T

0[Ψ](v₀, v)+F_v^T

0[Ψ](u₀, u)].

(268)

The degeneration near r=3M is the familiar trapping phenomenon and cannot be removed (although it can be improved to logarithmic loss; cf. [52]). The degeneration at the horizon however can be removed by exploiting the redshift. The weights near infinity can also be improved. We turn to these two refinements in§11.3and§11.4below.

11.3. Improving the weights near the horizonH⁺: The redshift

Given Proposition11.1.1 and estimate (268), the argument exploiting the redshift iden-tity, as described in§2.3.1and§2.3.2for the scalar wave equation (48) (cf. [25]), can be immediately adapted to Ψ.

In particular, one upgrades Proposition 11.1.1 to the non-degenerate boundedness statement

Fu[Ψ](v0, v)+Fv[Ψ](u0, u).Fv₀[Ψ](u0, u)+Fu₀[Ψ](v0, v), (269) where these are nownon-degeneratenull-fluxes defined in (226) and (227), and the esti-mate (268) itself to the improved(²⁷)integrated decay estimate

Z u

u0

Z v

v0

Z

S²_u,¯¯v

d¯u d¯v sinθ dθ dφΩ² 1

r²|Ω∇/₄Ψ−Ω∇/₃Ψ|²+1 r³|Ψ|² +(r−3M)²

r² 1

r|∇Ψ|/ ²+1

r²|Ω∇/₄Ψ|²+ 1

r²|Ω⁻¹∇/₃Ψ|²

.Fu0[Ψ](v0, v)+Fv0[Ψ](u0, u).

(270)

Note that, taking the limitu, v!∞, the left-hand side is preciselyIdeg[Ψ]. Hence, (269) and (270) prove the following result.

(²⁷) In the sense that the regular transversal derivative (1/Ω)∇/₃Ψ is also controlled near the horizon–the degeneracy nearr=3M remains.

Proposition 11.3.1. The Ψin Theorem 1satisfies the boundedness estimate sup

u

Fu[Ψ](v0,∞)+sup

v

Fv[Ψ](u0,∞).Fv₀[Ψ](u0,∞)+Fu₀[Ψ](v0,∞), (271) and the integrated decay estimate

Ideg[Ψ].F_v0[Ψ](u₀,∞)+Fu0[Ψ](v₀,∞), (272) provided the initial energies on the right-hand side are finite.

Note that higher-order versions of the above proposition are immediate from Lie differentation with the Killing fields of§4.2.2, i.e.LT as well as LΩ_i. We note also the following result.

Corollary 11.1. The Ψin Theorem 1 satisfies Indeg[Ψ].

1

X

i=0

Fv₀[TⁱΨ](u0,∞)+

1

X

i=0

Fu₀[TⁱΨ](v0,∞), (273) provided the initial energies on the right-hand side are finite. Here, the left-hand side denotes the non-degenerate (near 3M) integrated decay energy

Indeg[Ψ] :=

Z ∞

u₀

Z ∞

v₀

Z

S²_u,¯¯v

d¯u d¯v dvol_S2Ω² 1

r³|Ψ|²+1

r|∇Ψ|/ ²+1

r²|Ω∇/₄Ψ|²+1

r²|Ω⁻¹∇/₃Ψ|²

.

Proof. By the remark following Proposition11.3.1, the right-hand side of (273) con-trolsIdeg[Ψ]+Ideg[TΨ]. In particular, both|TΨ|²and|R^?Ψ|²(and of course|Ψ|²) are now controlled without degeneration at 3M. To control also the term|∇Ψ|/ ²non-degenerately near 3M integrate the multiplier identity (259) with f=1/r and use Proposition 11.3.1 to estimate the boundary terms that appear.

11.4. Improving the weights near null infinityI⁺: The r^p hierarchy

Ther^p hierarchy of [22] recalled in§2.3.3in the context of the scalar wave equation (48) can also now be adapted to Ψ.

From the Regge–Wheeler equation for Ψ we derive the identity (for 16p62 and k>1)

∂u

r^p

(1−µ)^k|Ω∇/₄Ψ|²

+∂v

r^p

(1−µ)^k−1|∇Ψ|/ ²

+∂v

r^p V

(1−µ)^k|Ψ|²

−∂u

r^p (1−µ)^k

|Ω∇/₄Ψ|²−∂v

V r^p (1−µ)^k

|Ψ|² +

(2−p)r^p−1(1−µ)^1−k+r^p(k−1)(1−µ)^−k2M r² rv

|∇Ψ|/ ²≡0,

(274)

where we have used

−2∇/₄Ψ·∆Ψ/ ≡∇/₄|∇ψ|/ ²−2[∇/₄, /∇]Ψ·∇Ψ =/ ∇/₄|∇Ψ|/ ²+trχ|∇Ψ|/ ²,

and≡indicates that (274) becomes an identity after integration against sinθ dθ dφ.

For our current purposes, it will be sufficient to integrate (274) for 16p62 with respect to the measuredu dvsinθ dθ dφin a region

R={(u, v)∈ M:r(u, v)>R, u06u6ufinal andv06v6vfinal},

for sufficiently large R, and ufinal and vfinal arbitrarily large. Precisely, we choose R sufficiently large (depending only onM) such that

−∂u

r^p (1−µ)^k

>1

2r^p−1 for all 16p62 andk65.

Note also that

−∂v

V r^p (1−µ)^k

=−∂_v

4r^p−2−6M r^p−3 (1−µ)^k−1

= rv

(1−µ)^k

(4(2−p)r^p−3−6M(3−p)r^p−4)(1−µ)+(k−1)2M

r² (4r^p−2−6M r^p−3)

= rv

(1−µ)^k[4(2−p)r^p−3+M r^p−4(8k+14p−42)+12M²r^p−5(4−p−k)]

holds, which means that, given any 16p62, the choicek=4 ensures that also the estimate

−∂v

V r^p (1−µ)^k

>2M r^p−4

holds inR, for sufficiently largeR(depending only onM). Therefore, integrating (274) forp=2 with respect to the measuredu dv sinθ dθ dφ, we first obtain the estimate

Z

R

du dvsinθ dθ dφ(r|Ω∇/₄Ψ|²+r^1−ε|∇Ψ|/ ²+r^−1−ε|Ψ|²) 6C

Z ∞

v0

dv Z

S²

sinθ dθ dφ(r²|Ω∇/₄Ψ|²)(u0, v)+C(Fu₀[Ψ](v0, v)+Fv₀[Ψ](u0, u)), (275) forε=1, where the last two terms on the right-hand side account for the terms arising on the timelike hypersurface atr=R, which can be controlled by the Morawetz estimate (270), after averaging inR. To show that the estimate (275) holds for our fixed 0<ε<¹₈,

we integrate (274) forp=2−εwith respect to the measuredu dvsinθ dθ dφand add it to thep=2 estimate. Note that the constant in (275) is independent of bothufinalandvfinal, and that the estimate hence holds forRreplaced byM∩{u>u0}∩{v>v0}∩{r>R}.

At the same time, the integration of (274) overRproduces good boundary terms (fluxes) onu=ufinalandv=vfinalfrom the terms in the first line of (274). Taking suprema, we deduce both (238) and then=0 part of (240), after recalling the shorthand notation (230), (231) for the energies.

With these bounds established, we deduce the following result.

Corollary 11.2. Under the assumptions of Theorem 1,we also have the estimate sup

u>u₀ v>v₀

kr⁻¹·Ψk²_S2 u,v.F_u^I

0[Ψ](v₀,∞)+F_v^I₀[Ψ](u₀,∞). (276)

Proof. The fundamental theorem of calculus in the∇/₄-direction and the Cauchy–

Schwarz inequality using the flux (228) gives this bound with an additional (initial) term sup_ukr⁻¹·Ψk²_S2

u,v0

on the right-hand side. Applying 1-dimensional Sobolev embedding onv=v0, shows that this initial term is controlled byF_v^I₀[Ψ](u0,∞).

We finally note that integrating (274) withp=1 andk=4 (instead ofp=2 andk=4, as done to derive (275)) leads to additional estimates which together with the choice p=0 and k=0 (for which the identity (274) also holds) constitute the Regge–Wheeler analogue of ther^p-hierarchy for the wave equation in [22].

11.5. Higher-order estimates and polynomial decay

In this section we will extend the above weighted estimates to higher order and then infer polynomial decay.

We note the trivial fact that the Regge–Wheeler equation (255) commutes with Lie differentation with the Killing fields of §4.2.2, i.e. LT as well asLΩ_i.(²⁸) Recalling the commuted energies (233), we hence immediately conclude the following corollary, which provides the estimate (239) and then>0 part of the estimate (240) in Theorem 1.

Corollary 11.3. If the Ψin Theorem 1 satisfies F^{n,T , /}0 ^∇[Ψ]<∞ for some integer n>0,then we have the estimate

I^{n,T , /}_I,ε ^∇[Ψ]+I^{n,T , /}_deg^∇[Ψ]+F^{n,T , /∇}[Ψ].F^{n,T , /}0 ^∇[Ψ]. (277)

(²⁸) As in [18], we could alternatively commute “tensorially” withr /∇_Aand estimate the lower-order terms inductively.

As an immediate consequence of Corollary 11.2, we also have sup

u>u₀ v>v₀

kr⁻¹·A^[n]Ψk²_S2

u,v.F^{n,T , /}0 ^∇[Ψ].

We can in fact show an analogue of the above for annth-ordernon-degenerateenergy, where higher derivatives have moreover additional weights inv. This is a straightforward adaptation of the procedure appearing in [68] and [57] to Ψ, and follows by commuting the equation with the redshift operator Ω⁻¹∇/₃ near the horizon and with the weighted operatorrΩ∇/₄near null infinity, and observing that the terms non-controllable by (277) occur with favourable signs. We will simply state the estimate arising. Define the energy

Fⁿ[Ψ] : = X

i+j+k6n

sup

u

F_u^I[(Ω⁻¹∇/₃)ⁱ(rΩ∇/₄)^j(r /∇_A)^kΨ](v0,∞)

+ X

i+j+k6n

sup

v

F_v^I[(Ω⁻¹∇/₃)ⁱ(rΩ∇/₄)^j(r /∇_A)^kΨ](u0,∞),

(278)

with initial energy

Fⁿ0[Ψ] : = X

i+j+k6n

F_u^I

0[(Ω⁻¹∇/₃)ⁱ(rΩ∇/₄)^j(r /∇_A)^kΨ](v₀,∞)

+ X

i+j+k6n

F_v^I

0[(Ω⁻¹∇/₃)ⁱ(rΩ∇/₄)^j(r /∇_A)^kΨ](u₀,∞).

(279)

We have the following result.

Corollary 11.4. If theΨin Theorem 1satisfies Fⁿ0[Ψ]<∞,then we have,for any n>0 and non-negative integers i, j and k with i+j+k6n,the estimate

Fⁿ[Ψ].Fⁿ0[Ψ], Ideg[(Ω⁻¹∇/₃)ⁱ(rΩ∇/₄)^j(r /∇_A)^kΨ]+I^Iε[(Ω⁻¹∇/₃)ⁱ(rΩ∇/₄)^j(r /∇_A)^kΨ].Fⁿ0[Ψ].

We will in fact only use Corollary 11.4later to optimise decay statements already obtained.

Exploiting ther^p-hierarchy for the Regge–Wheeler equation discussed in§11.4, poly-nomial decay estimates can be obtained for Ψ exactly as in [22] for the case of the scalar wave equation (cf. the discussion in§2.3.3). We only give the most elementary statement here.

Let us fixr₀:=r(u₀, v₀)>2M and denote byu(v, r₀) theu-value corresponding to the sphere of intersection between ther=r₀ hypersurface and the constant v hypersurface.

Note thatv∼u(v, r₀), for largev. Applying step by step the method of [22] (for details on the method of [22] in more general settings see [68], [57]), we obtain the following result.

Proposition11.5.1. Fix r0=r(u0, v0)and v>v0 and suppose the Ψin Theorem1 satisfies F^2,T0 [Ψ]<∞ initially. Then, for any V>v and any U>u(v, r0), we have

FU[Ψ](v,∞)+FV[Ψ](u(v, r0),∞). 1

v²·F^2,T0 [Ψ].

The constant implicit in .depends on r₀, and we recall the non-degenerate energy fluxes (226)and (227).

Proof. Since the proof is entirely analogous to that in [22], we only provide a sketch from which the reader can easily fill in the details. Adding to (275) the estimate of Corollary11.1(and theirT-commuted analogues), we find

Z ∞

u₀

Z ∞

v₀

Z d

S²_u,¯¯v

¯

u d¯v dvol_S2Ω² 1

r²|Ψ|²+|∇Ψ|/ ²+r|Ω∇/₄Ψ|²+1

r²|Ω⁻¹∇/₃Ψ|²

+ Z ∞

u₀

Z ∞

v₀

Z

S_u,¯²_¯v

d¯u d¯v dvol_S2Ω² 1

r²|TΨ|²+|∇T/ Ψ|²+r|Ω∇/₄TΨ|²+ 1

r²|Ω⁻¹∇/₃TΨ|²

.F^2,T0 [Ψ].

From this, we extract a dyadic sequence (v_i)_i with associated ingoing cone

Cev_i= [u(vi, r0),∞)×{vi}×S²

and outgoing coneCe_u(vi,r0)={u(vi, r0)}×[vi,∞)×S² such that for eachvi we have Z ∞

u(v_i,r₀)

d¯u dvolS²Ω²

|Ω⁻¹∇/₃Ψ|²+|∇Ψ|/ ²+ 1 r²|Ψ|² +|Ω⁻¹∇/₃TΨ|²+|∇T/ Ψ|²+ 1

r²|TΨ|²

(¯u, v_i) +

Z ∞

vi

d¯v dvol_S2

r|Ω∇/₄Ψ|²+|∇Ψ|/ ²+ 1 r²|Ψ|² +r|Ω∇/₄TΨ|²+|∇T/ Ψ|²+ 1

r²|TΨ|²

(u(v_i, r₀),v)¯ .F^2,T0 [Ψ]

vi

.

Note that, in particular, the ingoing non-degenerate energy (of both Ψ andTΨ) on Ce_vi and the outgoing energy (of both Ψ andTΨ) onCe_u(vi,r0) are decaying. Hence, applying Proposition 11.3.1, now from each dyadic pair of cones Ce_vi∪Ce_u(vi,r0) (instead of from Ce_v0∪Ce_u0) yields, using the previous estimate for the right-hand side in Proposition11.3.1

for anyv>v0, the estimate Z ∞

u(v,r0)

d¯u dvol_S2Ω²

|Ω⁻¹∇/₃Ψ|²+|∇Ψ|/ ² +|Ψ|²

r² +|Ω⁻¹∇/₃TΨ|²+|∇T/ Ψ|²+ 1 r²|TΨ|²

(¯u, v) +

Z ∞

v

d¯v dvol_S2

r |Ω∇/₄Ψ|²+|∇Ψ|/ ²+|Ψ|² r² +r |Ω∇/₄TΨ|²+|∇T/ Ψ|²+1

r²|TΨ|²

(u(v, r0),v)¯ .F^2,T0 [Ψ]

v (280) without the boxedr-weight. In addition, we obtain (cf. (270))

Z ∞

v

d¯v Z ∞

u(v,r₀)

d¯u Z

S²_u,¯¯v

dvol_S2Ω²[e[Ψ]+e[TΨ]].F^2,T0 [Ψ]

v , for allv>v0, where

e[Ψ] = 1

r²|Ω∇/₄Ψ−Ω∇/₃Ψ|²+1

r³|Ψ|²+(r−3M)² r²

1

r|∇Ψ|/ ²+1

r²|Ω∇/₄Ψ|²+1

r²|Ω⁻¹∇/₃Ψ|²

.

We now integrate (274) (and itsT-commuted version) withp=1 andk=4 inRfrom each C_u(vi,r0)to improve (280) toinclude the boxedr-weight. In addition, we obtain a good spacetime term in the regionRwhich can be combined with the estimate of Corollary11.1 (exchange the initial conesCe_v0∪Ce_u0 by the dyadic pair of conesCe_vi∪Ce_u(vi,r0), and use (280) without the boxedrfor the right-hand side in Corollary11.1) to obtain

Z ∞

v

d¯v Z ∞

u(v,r₀)

d¯u Z

S²_¯u,¯v

dvolS²Ω² 1

r²|Ψ|²+|∇Ψ|/ ²+|Ω∇/₄Ψ|²+ 1

r²|Ω⁻¹∇/₃Ψ|²

.F^2,T0 [Ψ]

v (281) for all v>v0. From this, we find a (potentially different) dyadic sequence (vi)i along which

Z ∞

u(v_i,r₀)

d¯u dvol_S2Ω²(|Ω⁻¹∇/₃Ψ|²+|∇Ψ|/ ²+|Ψ|² r² )(¯u, vi) +

Z ∞

v_i

d¯v dvol_S2(|Ω∇/₄Ψ|²+|∇Ψ|/ ²+|Ψ|²

r² )(u(v_i, r₀),¯v).F^2,T0 [Ψ]

(vi)² .

Finally, applying Proposition11.3.1from each of these dyadic pair of conesCe_vi∪Ce_u(vi,r0) towards the future yields the statement of the proposition.

Corollary 11.5. Under the assumptions of the previous proposition, we have the integrated decay estimate

Z ∞

v

d¯v Z ∞

u(v,r0)

d¯u Z

S_u,¯²_¯v

dvol_S2Ω²

Dans le document The linear stability of the Schwarzschild solution to gravitational perturbations (Page 121-128)