14. Proof of Theorem 4
14.2. Integrated local energy decay
(Ω trχ);
• Proposition13.5.12for the metric coefficients. Note that the initial terms appear-ing in (400) and (401) can be estimated byE0 using the round sphere condition (191) and (192).
Remark 14.1. Proposition 14.1.3and Corollary14.4can be paraphrased by saying that the solution in the future gaugeS∧=
S∨ +G∧satisfies the same boundedness estimates as the solution
S∨ in the initial data gauge. In particular, there is no loss of derivatives at the level of flux bounds. It is important to note, however, that the estimates we obtain for five angular derivatives of(1)η and Ω−1
(1)
Ω are slightly weaker in terms of their r-decay towards null infinity than those for (1)η[S∨ ] and Ω−1
(1)
Ω[S∨ ]. This is because the estimate establishing decay of fv (413) is not optimal in terms of regularity. Compare the estimates (412) and (414).
14.2. Integrated local energy decay
We now turn to show integrated decay statements for the quantities associated withS∧. Recall from our comments above that, in addition to the estimates for the gauge-invariant quantities, the results of §13.3, in particular the integrated decay statement Proposition13.3.1, remain valid for
(1)
χ[bS∧].
The first quantity for which one could not obtain integrated decay in the initial data normalised gauge was the quantity
(1)
χ[b
S∨ ]. In§14.2.1below, we will succeed to obtain an integrated local energy decay statement for
(1)
χ[bS∧], obtaining also a similar estimate for
∇/4
(1)
χb in §14.2.2. We will then finally unravel the decay hierarchy, proving successively integrated local energy decay for(1)%,(1)σand
(1)
β in§14.2.3,(1)η and(1)η in§14.2.4and
(1)
(Ω trχ) in
§14.2.5. Finally, we will obtain various refined statements for higher angular derivatives in§14.2.6, which allow to infer integrated local energy decay for
(1)
β.
14.2.1. Integrated decay for angular derivatives of
(1)
χb
Using the new horizon fluxes obtained in the previous section, we will now obtain global control on angular derivatives of
(1)
χ.b
Proposition 14.2.1. We have the following integrated decay estimate for the ingo-ing shear bχof the solution
S∧in Theorem 4. For any v>v0 and i=4,5,we have In view of Proposition 12.3.3, the first term can be dropped for i=4.
Proof. To show the statement for i=5, we contract the r /D?2-commuted (358) by r−2−εA[3]
Absorbing the first term on the right by the left-hand side, and using the estimate (243) of Theorem2(for the underlined quantities), we obtain an integrated decay estimate for A[3]
(1)
Y after integrating the estimate over spacetime, observing that the flux term on the horizon has the wrong sign, but is controlled from Propositions12.2.3and 14.1.1using the definition of
(1)
Y, (218). On the left-hand side, we use again (218) and that we control A[3]
(1)
ψfrom the integrated decay statement of Theorem2applied withn=2 to descend to the desired estimate for angular derivatives of
(1)
χ. This establishes the estimate claimedb with an additional termkA[5]Ω−1
(1)
χ[bS∨ ]k2S2
∞,v0
on the right-hand side, which entered when we applied Proposition14.1.1. Corollary 13.3allows to replace it by
A[3] The statement for i=4 is proven entirely analogously directly from (358), without an additionalr /D?2-commutation.
14.2.2. Integrated decay for angular derivatives of∇/4
(1)
χb
Proposition 14.2.2. We have the following integrated decay estimate for the quan-tity ∇/4χb of the solution S∧in Theorem 4. For any v>v0,
Proof. We commute (365),
We absorb the first term on the right by the left-hand side. Upon integration over a spacetime region, for the last term on the right we use the integrated decay estimate on
(1)
Ψ of Theorem 1, and the integrated decay estimates on
(1)
ψ and (1)α of Theorem 2. The flux term on the horizon (which has a bad sign) arising from the first term on the left is controlled by inserting the commuted (370), which reads
A[2]Ω∇/4 Note that, when considering the horizon flux, the first term on the right vanishes, the second is controlled by Corollary14.1withi=4, and the last two by (242) of Theorem2 forn=1. This produces an integrated decay for∇/4
(1)
Y. To descend to the desired quantity A[4]rΩ∇/4(r
(1)
χ/Ω), we use once more the identity (416), now expressing the second term onb the right in terms of everything else, in conjunction with the integrated decay estimate of Proposition14.2.1, the definition of
(1)
Y (see (218)), and the estimate (243) of Theorem2 (againn=1 is actually sufficient). This establishes the estimate with the additional term kA[4]
(1)
χ[bS∨ ]Ω−1k2S2
∞,v0
on the right (having entered through Corollary14.1). Using (218) to re-express
ψ, and applying Propositions12.3.3and13.4.1 (note Remark13.6) we deducekA[4]
(1) With the integrated decay estimates on
(1)
χ, we can easily show decay of all curvatureb components. Recall that, for (1)α and(1)α, we already have these statements from Propo-sition 12.3.2. The estimate for
(1)
β will be proven in Proposition 14.2.8 after we have estimated four angular derivatives of
(1)
χ. (Theb i=2 non-degenerate version for
(1)
β can be proven at this point already; cf. the proof of Proposition14.2.8.)
Proposition 14.2.3. For i=3 the following holds for the geometric quantities of S∧in Theorem 4:
Z ∞
v0
dv Z ∞
u0
du Ω2 r1+ε
1−3M
r 2
×(kr−1·A[i]D/?2D/?1(r3(1)% , r3(1)σ)k2S2
u,v+kr−1·A[i+1]D/?2(
(1)
β r2Ω−1)k2S2
u,v).E0. For i=2,this estimate holds without the degenerating factor of (1−3M/r)2.
Proof. The bound on (r3(1)% , r3(1)σ) is a direct consequence of the identity (374) applied withi=3 (ori=2), the estimate (240) of Theorem1applied withn=2 (note this provides a degenerate (near r=3M) integrated decay estimate for A[3]P and a non-degenerate estimate forA[2]P), Propositions 14.2.1(applied withi=4) and13.3.3.
For the bound on
(1)
β, we use the identity (375) applied withi=4 (i=3) and rewrite the term A[4]
(1)
ψ (A[3]
(1)
ψ) using the A[2] (A[1]) commuted (303). All terms can then be estimated by the integrated decay estimates of Theorems1and2and Proposition14.2.1 applied withi=4.
14.2.4. Integrated decay forD/?2(1)η and D/?2(1)η
Proposition 14.2.4. We have the following integrated decay estimate on the solu-tion S∧in Theorem 4:
Z ∞
v0
dv Z ∞
u0
duΩ2r1−ε(kr−1·A[3]r /D?2(1)ηk2S2
u,v+kr−1·A[3]r /D?2(1)ηk2S2
u,v).E0.
Proof. This follows directly from (379) and (380) applied with i=3 using the inte-grated decay estimates of Propositions14.2.1(withi=4),14.2.2and13.3.3.
14.2.5. Integrated decay for angular derivatives of
(1)
(Ω trχ)
Proposition 14.2.5. We have the following integrated decay estimate for the solu-tion
S∧in Theorem 4: for i=2,3, Z ∞
v0
dv Z ∞
u0
du Ω2 r1+ε
r−1·A[i]r3D/?2∇/
(1)
(Ω trχ) Ω2
2
S2u,v
.kA[i]
(1)
ψΩ−1k2S2
∞,v0
+E0. In view of Proposition 12.3.3, the first term can be dropped for i=3.
Proof. Follows directly from (383) using the integrated decay estimates of Proposi-tions14.2.1,14.2.4and (theA[2]commuted version of)12.2.2(or apply Theorem2).
14.2.6. Refined estimates for higher angular derivatives and integrated decay for
(1)
β
We easily derive the following analogue of Proposition 13.5.6 by integrating both inu andv (and not invonly as in the derivation of Proposition13.5.6).
Proposition 14.2.6. We have, for n>0, any i∈N and any u>u0, the estimate
We remind the reader that, following our notation, (1)η+(1)η on the right denote the geometric quantities of
S∧. In the above Proposition we have used that
(1)
Z[ S∧]=
(1)
Z[S∨ ] holds on v=v0. To deduce the latter, note that
(1)
Z[
G∧]=0 holds onv=v0, which follows from Proposition9.3.1and Lemma6.1.1. (Indeed, note thatη[
G∧]=0 onv=v0by (214) and that the expressionr /∇A
(1)
(Ω trχ)−2Ω2(1)η vanishes for any pure gauge solution in Lemma6.1.1.) Recall also that
(1)
Z∼Ω2holds onv=v0 near the horizon by Proposition9.4.3, so the first term in the second line is indeed finite for smooth data. We can reinsert the definition of
(1)
Z for the second term on the left to obtain an estimate for i angular derivatives of
(1)
Again, consistent with our notation, (1)η+(1)η above denotes the geometric quantities of
S∧. We conclude the following result.
Proposition 14.2.7. We have the integrated decay estimate for the geometric quantities of S∧in Theorem 4.
Proof. Apply Corollary 14.5with n=2+ε and i=3, and use Proposition 14.2.4 to obtain the first part of the estimate. Then, use identity (389) in conjunction with Propo-sition14.2.4, Theorem2and Proposition13.3.3to obtain the second part.
Remark 14.2. The weights near infinity are far from optimal. The weight near infinity forA[5]
(1)
χbcan be improved a posteriori from the transport equation (139) and the (degenerate nearr=3M) integrated decay estimate forA[5](1)α. See also Corollary 13.10.
The weights are sufficient to prove the integrated decay estimate of Proposition14.2.3 for the missing curvature component
(1)
β.
Proposition 14.2.8. For the geometric quantities of
S∧in Theorem 4,we have the following integrated decay estimate for i=3:
Z ∞
For i=2, this estimate holds without the degenerating factor of (1−3M/r)2 on the left, and without the last term on the right.
Proof. For i=3, this follows from the identity (395) and application of Proposi-tion14.2.7 (to estimate the 32%A[4](Ω
(1)
χ) term), as well as using the identity (300) com-b muted with A[2] to estimate the A[4](Ω
(1)
ψ) term (through application of Theorems 1 and2). Fori=2, this also follows from (395), but observing that now Proposition13.3.3 already estimates three angular derivatives of
(1)
χ, while (the twice angular commuted)b Proposition12.2.2estimates (non-degenerately) three angular derivatives of
(1)
ψ.
Finally, repeating the arguments(35) leading to Propositions 13.5.9, 13.5.10 and 13.5.11, we can prove integrated decay for five derivatives of(1)η and(1)η of the solutionS∧,
(35) All that is required is to insert an additionalu-integration everywhere.
i.e. the estimate