Proof of Proposition 9.3

9.3 Preparations for the Proofs of Main Estimates for the Eikonal Function Quatities

9.3.1 Proof of Proposition 9.3

kξkL^Q_g/(St,u)= _Z

S²

|ξ|^Q_g/dϖ_g/

_Q¹

≈ _Z

S²

er²|ξ|^Q_g/dϖ_{e/ Q}¹

= er^Q2ξ

L^Qω(St,u). (9.47)

Proof of (9.30).

r⁻¹²ξ

2

= _Z

r⁻¹|ξ|²dϖ

≤ r⁻¹

_Z

|ξ|⁴dϖ ¹₂

≈ kξk² . (9.48)

We now provekξk_L4

g/(St,u).kξk_H1(Σt). For fixedtandu, letϕbe a cut-off function ofuverifying 0≤ϕ≤1, ϕ(u) =1 and supp(ϕ)⊂[u−^t−u₂ ,u+^t−u₂ ]. Integrating on the region ^S

u≤u⁰≤^t+u₂

S_t,u⁰, by the Fundamental Theorem of Calcules, we have:

kξk⁴_L4 g

/(St,u)=−

Z u u−^t−u₂

Z

S_t,u0

D

/_N|ϕξ|⁴+tr_g/θ|ϕξ|⁴

bdϖ_g/du⁰ (9.49)

=− Z u

u−^t−u₂ Z

S_t,u0

ϕ⁴|ξ|³·/D_Nξ+ϕ³|ξ|⁴·/D_Nϕ+tr_g/θ|ϕξ|⁴

bdϖ_g/du⁰.

For the first term on the RHS of (9.49), by bootstrap assumption (9.5b), the H¨older’s inequality and Sobolev embeddingL⁶,→H¹, we have:

Z _u u−^t−u₂

Z

S_t,u0

ϕ⁴|ξ|³·/D_Nξ

bdϖ_g/du⁰

.kD/_Nξk_L2(Σt)

|ξ|³

L²(Σ_t).kξk⁴_H1(Σt), (9.50) For the second term on the RHS of (9.49), by bootstrap assumption (9.5b), and noticing that|D/_Nϕ|.t−u¹ , we have

Z u u−^t−u₂

Z

S_t,u0

ϕ³|ξ|⁴·/D_Nϕ

bdϖ_g/du⁰

.

1 t−u

Z u u−^t−u₂

Z

S_t,u0

|ξ|⁴dϖ_g/du⁰

. (9.51)

By Lemma 9.4, Z

S_t,u0

|ξ|⁴dϖ_g/. er⁻¹²ξ

L²_g/(S_t,u0)

|ξ|³

L²_g/(S_t,u0)er¹² .kξk_H1(Σt)

|ξ|³

L²_g/(S_t,u0)er¹². (9.52) Then by (9.29), (9.51), (9.52), the H¨older’s inequality, the Sobolev embeddingL⁶,→H¹, we have:

1 t−u

Z u u−^t−u₂

Z

S_t,u0

|ξ|⁴dϖ_g/du⁰

. 1

t−ukξk_H1(Σ_t)

|ξ|³

L²(Σ_t)

_{Z u}

u−^t−u₂ er(t,u⁰)du⁰ 1/2

(9.53) .kξk⁴_H1(Σ_t).

Now we consider the third term on the RHS of (9.49), that is, ^R_u−^u t−u 2

R S_t,u0

tr_g/θ|ϕξ|⁴

bdϖ_g/du⁰. By bootstrap assumptions (6.13), (9.4) and (9.5b) and the H¨older’s inequality, we have:

Z u u−^t−u₂

Z

S_t,u0

tr_g/θ|ϕξ|⁴

bdϖ_g/du⁰. Z u

u−^t−u₂ Z

S_t,u0

er⁻¹|ϕξ|⁴

dϖ_g/du⁰ (9.54)

+ Z _u

u−^t−u₂ Z

S_t,u0

tr_g/θ−2 er

|ϕξ|⁴

dϖ_g/du⁰ .

Z _u u−^t−u₂

Z

S_t,u0

er⁻¹|ξ|⁴

dϖ_g/du⁰+

tr_g/θ−2 er L³(Σt)

kξk⁴_L6(Σt).

By bootstrap assumptions (9.5c) where we have tr_g/θ−²

er

L³(Σt)≤1, and Sobolev embeddingL⁶,→H¹for kξk⁴_L6(Σ_t), we have:

Z u u−^t−u

2

Z

S_t,u0

tr_g/θ|ϕξ|⁴

bdϖ_g/du⁰. Z u

u−^t−u

2

Z

S_t,u0

er⁻¹|ξ|⁴

dϖ_g/du⁰+kξk⁴_H1(Σt). (9.55)

Using Lemma 9.4, the H´older inequality and the Sobolev embeddingL⁶,→H¹, we have:

Z u u−^t−u₂

Z

S_t,u0

er⁻¹|ξ|⁴

dϖ_g/du⁰.kξk_H1(Σt)

|ξ|³

L²(Σt)

Z u u−^t−u₂

Z

S_t,u0er⁻¹(t,u⁰)du⁰

!1/2

.kξk⁴_H1(Σ_t). (9.56)

Proof of (9.31). Replaceξbyξ^1/2in (9.49). By the same approach, we derive the following:

kξk²

L²_g/(St,u).kξk_H1(Σ_t)kξk_L2(Σ_t). (9.57)

Integrating (9.57) with respect touonΣt, we obtain (9.31).

Proof of (9.32). We defineCu,ω:=CuT

{ω^A=ω}. For anyt∈[0,T_∗;(λ)], letφbe a smooth function verify- ing 0≤φ≤1,φ(t) =1 andsuppφ⊂[¹₂t,³₂t]. Since|L(φ)|.er⁻¹, by the Fundamental Theorem of Calcules, we have:

er^p|ξ|^2p_g/ φ^2p

(t) = Z t

0

L

er^p|ξ|^2p_g/

φ^2p+er^p|ξ|^2p_g/ L φ^2p

dτ (9.58)

. Z t

0er^p−1|ξ|^2p_g/ +er^p|ξ|^2p−2_g/ /D_Lξ·ξdτ.

Hence, er¹²ξ

2p

L^2p_ωL_t^∞(Cu). Z

S² Z _t

0er^p−1|ξ|^2p_g/ +er^p|ξ|^2p−2_g/ D/_Lξ·ξdτdϖ_/e (9.59) .

Z

S²

kξk_L2

t(Cu,ω)kξk_L2

t(Cu,ω)+kξk_L2

t(Cu,ω)kerD/_Lξk_L2 t(Cu,ω)

er^p−1ξ^2p−2

L_t^∞(Cu,ω)dϖ_/e .kξk_L∞

ωL²_t(Cu)

r¹²ξ

2p−2 L^2pωL^∞_t (Cu)

kerD/_Lξk_L^p

ωL²_t(Cu)+kξk_L^p

ωL²_t(Cu)

.

This yields the desired result.

Proof of (9.33b). We first consider the Euclidean sphere case, applying the Morrey’s inequality and the

Gagliardo–Nirenberg interpolation inequality with dimensionn=2, ans using Young’s inequality, we have:

kξk

C

0,1−2 ω Q(St,u)

.

∇_e/ξ

L^Qω(St,u)+kξk

L^Qω(St,u) (9.60)

.

∇_e/ξ

L^Qω(S_t,u)+

∇_e/ξ

1−_Q² L²_ω(St,u)kξk

2 Q

L²_ω(St,u)+kξk_L2 ω(St,u)

.

∇_e/ξ

L^Qω(S_t,u)+

∇_e/ξ

1−²

Q

L^Qω(St,u)kξk

2 Q

L²ω(St,u)+kξk_L2 ω(St,u)

.

∇_e/ξ

L^Qω(S_t,u)+kξk_L2 ω(St,u).

Noticing that for ^m_n

tensorfieldξ, we have:

|ξ|_e/≈er^m−n|ξ|_g/. (9.61)

Thus, for any 0<p<∞, there holds:

∇_e/ξ e/

Lω^p(St,u)≈ er∇_e/ξ

Lω^p(S_t,u). (9.62)

Now we consider the difference between round metric/eand the geometric sphere metricg/. We have:

/∇ξ=∇_/eξ+ (ΓΓΓ−ΓΓΓ_(e/))ξ, (9.63)

whereΓΓΓ_(e/) are the Christoffel symbols ofe/relative to Euclidean sphere coordinatesω^A. By the bootstrap assumptions (9.3a)-(9.3b), we have:

er^m−n

(ΓΓΓ−ΓΓΓ_(e/))ξ /e

L^Qω(S_t,u).λ^−ε0kξk_L∞

ω(S_t,u). (9.64)

The RHS of (9.64) can be absorbed in the left hand side of (9.60).

Proof of (9.33a). We use the similar approach as in the proof of (9.33b), for the Euclidean sphere case, we have:

|ξ|_/e

L^Qω(St,u).

∇_/eξ e/

1−_Q² L²_ω(St,u)

|ξ|_e/

2 Q

L²_ω(St,u)+ |ξ|_/e

L²_ω(St,u). (9.65) The proof of (9.65) can be reduced, by a partition of unity, to the case whereξhas compact support in a local chartU⊂St,u. Then we can apply the Gagliardo–Nirenberg interpolation inequality with dimensionn=2.

Now we consider the difference between Euclidean round metric/eand the geometric sphere metricg/. We

have:

/∇ξ=∇/_eξ+ (ΓΓΓ−ΓΓΓ_(e/))ξ. (9.66)

whereΓΓΓ_(e/) are the Christoffel symbols ofe/relative to Euclidean sphere coordinatesω^A. By the bootstrap assumptions (9.3a)-(9.3b), by H¨older’s inequality, we have

er^m−n

(ΓΓΓ−ΓΓΓ_(e/))ξ e/

_L2

ω(St,u).

ΓΓΓ−ΓΓΓ_(e/)

Lω^p(St,u)kξk

L

2p p−2 ω (St,u)

.λ^−4ε0kξk_L∞ ω(St,u)er

p−2

p . (9.67)

Note thater.λ^1−8ε0. Hence, forpsufficiently close to 2, we have:

er^m−n

(ΓΓΓ−ΓΓΓ_(e/))ξ e/

_L2

ω(St,u).kξk_L∞

ω(St,u). (9.68)

(9.33a) is then obtained by applying (9.33b).

Proof of (9.34). The proof is similar to the proof of (9.32). We use a cut-off function ofu with support in u∈[−⁴₅T_∗;(λ),t]instead oft∈[0,T_∗;(λ)], and we consider that the derivative vectorfield ^∂

∂u alongΣ_t instead ofLalongCu. Also noticing that

/D_∂

∂u

ξ

g/.|(D/_N,/∇)ξ|_g/and:

er^12−Q1ξ

2

L^2Q_g/ L^∞_u(Σt)= er¹²ξ

2

L^2QωL^∞_u(Σt), (9.69)

we obtain the desired estimate.

Proof of (9.35a). We decompose f as follows:

f =

∑

ν>1

Pνf+P_≤1f. (9.70)

Using (9.33a), we have

erkP_νfk

L^Qω(St,u).

∇_/eP_νf

1−_Q²

L²_g/(St,u)kP_νfk

2 Q

L²_g/(St,u)+kP_νfk_L2 g

/(St,u). (9.71) Integrating (9.71) with respect toualongΣ_t, using the finite band property and bootstrap assumption (9.3a)- (9.3b), and summing overν>1, when 0<1−_Q² <N−2, we have:

₁₋2

For the low frequency terms, we use Bernstein’s inequality:

kerP≤1fk

L²_uL^Qω(Σt)≈ kP≤1fk

L²_uL_g^Q_/(Σt).kfk_L2(Σt). (9.73)

Lemma 9.5(Transport lemma). Let m be a constant, and letξandFbe St,u-tangent tensorfields such that the following transport equation holds along the null cone portionCu⊂M:

D

/_Lξ+mtr_g/χξ=F. (9.74)

Then we have the following identities, where[u]₊:=max{u,0}:

(υ^mξ)(t,u,ω) = lim

τ↓[u]+

(υ^mξ)(τ,u,ω) + Z t

[u]+

(υ^mF)(τ,u,ω)dτ, (9.75) (er^2mξ)(t,u,ω) = lim

τ↓[u]₊(er^2mξ)(τ,u,ω) (9.76)

+ Z _t

[u]+

(er^2mF)(τ,u,ω) +m

er^2m 2

er−tr_g/χ

ξ

(τ,u,ω)dτ.

Similarly, ifξ,FandGare S_t,u-tangent tensorfields such that the following transport equation holds:

D /_Lξ+2m

er ξ=G·ξ+F, (9.77)

and if

kGk_L∞

ωL¹_t(Cu)≤C, (9.78)

then under the bootstrap assumptions, the following estimates holds (where the implicit constants in the estimates below depend on the constant C on (9.78)):

er^2mξ

g/(t,u,ω). lim

τ↓[u]₊

er^2mξ

g/(τ,u,ω) + Z _t

[u]₊

er^2mF

g/(τ,u,ω)dτ. (9.79) Proof of (9.75). By (9.74), we have:

L(υ^mξ) =υ^m/D_Lξ+mυ^m−1(Lυ)ξ=υ^m /D_Lξ+mtr_g/χ

=υ^mF. (9.80)

Integratingυ^malong null geodesics, we get the desired equation. Proof of (9.76) is by the same process.

Proof of (9.79). By (9.77), we have:

L er^2mξ

_g/

=er^2m/D_Lξ· ξ

|ξ|_g/+2mer^2m−1|ξ|_g/ (9.81)

=er^2m

−2m

er ξ+G·ξ+F

· ξ

|ξ|_g/+2mer^2m−1|ξ|_g/

=er^2m G|ξ|_g/+F· ξ

|ξ|_g/

! .

Integrating er^2mξ

g/along null geodesics and then using the Grownwall’s inequality, we get the desired esti- mate.

Proposition 9.6(Estimates for the fluid variables). ~Ψ, ~ω,~S, ~C,Dare rescaled variables defined in Definition 6.3. Under the bootstrap assumptions, for any2≤Q≤p, where0<δ₀<1−²_p<N−2, the following estimates hold onM:

∂∂∂(~Ψ, ~ω,~S), ~C,D _L2

uL_ω^p(Σt), er^1/2

∂

∂

∂(~Ψ, ~ω,~S), ~C,D _L∞

uL²_ω^p(Σt).λ^−1/2, (9.82a)

er^1−Q2

∂

∂

∂²(~Ψ, ~ω,~S),∂∂∂(C~,D) _L2

uL^Q_g/(Σt).λ^−1/2, (9.82b)

∂∂∂(~Ψ, ~ω,~S), ~C,D _L2

tL^∞_ω(Cu).λ^{−1/2−4ε0}, (9.82c)

∂∂∂(~Ψ, ~ω,~S), ~C,D

L²_tL^pω(Cu).λ^{−1/2−4ε0}, (9.82d)

er

∂

∂

∂(~Ψ, ~ω,~S), ~C,D

L²_tL^∞_ω(Cu).λ^1/2−12ε0, (9.82e)

(∇/ ,/D_L)∂∂∂~Ψ L²(Cu),

∂∂∂(C~,D) L²(Cu),

er^1−2p(∇/ ,/D_L)

∂

∂∂~Ψ, ~C,D

L²_tL^p_g/(Cu).λ^−1/2, (9.82f)

er^1/2∂∂∂

~Ψ, ~ω,~S

L²_uL_t^∞L^2pω(M).λ^−4ε0. (9.82g) Moreover, for any smooth functionf, we have:

er(∇/ ,/D_L) f

~Ψ, ~ω,~S,~L ∂∂∂~Ψ, ~C,D _L2

tL^Q_ω(Cu).λ^−1/2, (9.83a)

er∂∂∂

f

~Ψ, ~ω,~S,~L ∂∂∂(~Ψ, ~ω,~S), ~C,D

L²_uL^Qω(Σ_t).λ^−1/2, (9.83b)

er^1/2f

~Ψ,~L

∂

∂∂

~Ψ, ~ω,~S

L²_uL^∞_tL^2pω(M).λ^−4ε0. (9.83c)

Proof of (9.82b). By using (9.35a), and rescaling the top order energy estimates (5.1), we have:

er^1−Q2

∂∂∂²(~Ψ, ~ω,~S),∂∂∂(C~,D)

L_u²L^Q_g/(Σt)≈ er

∂∂∂²(~Ψ, ~ω,~S),∂∂∂(C~,D)

L²_uL^Qω(Σt) (9.84) .

∂∂∂²(~Ψ, ~ω,~S),∂∂∂(C~,D)

H^N−2(Σt).λ^−1/2.

Proof of (9.82a). By (9.31), (9.33a), (9.35a), (9.82b) and the energy estimates (5.1), we have:

∂∂∂(~Ψ, ~ω,~S), ~C,D _L2

uL_ω^p(Σt). er∇/

∂

∂∂(~Ψ, ~ω,~S), ~C,D

1−²_p L_u²L²_ω(Σt)

∂∂∂(~Ψ, ~ω,~S), ~C,D

2 p

L²_uL²_ω(Σt) (9.85) +

∂∂∂(~Ψ, ~ω,~S), ~C,D _L2

uL²_ω(Σt)

.

∂∂∂(~Ψ, ~ω,~S), ~C,D _H1(Σ

t).λ^−1/2.

For estimate of the second norm, by (9.34), (9.33b), (9.35a), (9.31), (9.82b) and the energy estimates (5.1), we have:

er^1/2

∂∂

∂(~Ψ, ~ω,~S), ~C,D

2

L^2pωL^∞_u(Σt).

∂∂∂(~Ψ, ~ω,~S), ~C,D

L²_uL^∞_ω(Σt)· (9.86)

er(D/_N,/∇)

∂

∂∂(~Ψ, ~ω,~S), ~C,D

L^QωL²_u(Σ_t)+

∂∂∂(~Ψ, ~ω,~S), ~C,D L^QωL²_u(Σ_t)

.

er∇/

∂

∂∂(~Ψ, ~ω,~S), ~C,D _L2

uL^Q_ω(Σ_t)+

∂

∂

∂(~Ψ, ~ω,~S), ~C,D _L2

uL_ω²(Σ_t)

·

er(D/_N,/∇)

∂

∂∂(~Ψ, ~ω,~S), ~C,D _LQ

ωL²_u(Σt)+

∂∂∂(~Ψ, ~ω,~S), ~C,D _LQ

ωL²_u(Σt)

.

∂∂∂(~Ψ, ~ω,~S), ~C,D

_HN−1(Σt).λ⁻¹.

Proof of (9.82g). Using the fact thatu.λ^1−8ε0 inM and (9.82a), we deduce:

er^1/2f

~Ψ,~L

∂∂∂

~Ψ, ~ω,~S

L²_uL^∞_tL²ω^p(M).λ^1/2−4ε0 er^1/2f

~Ψ,~L

∂∂∂

~Ψ, ~ω,~S

L^∞_uL^∞_tL²ω^p(M) (9.87) .λ^−4ε0.

Proof of other estimates in Proposition 9.6. (9.82c)-(9.82e) are direct results of rescaled bootstrap assump- tions (9.2).

The first and second estimates of (9.82f) are direct results of energy estimates along acoustic null hypersur- faces (8.3)-(8.4). The last estimates of (9.82f) is [36, Lemma 5.5], we use the Sobolev inequality (9.33a) and the following inequality in [34, Proposition 2.7]:

∑

l>1

l^N−2/P_l(D/_L,/∇)f

2

L²_tL²_g/(Cu).

∑

ν>1

ν^2(N−2)

F(wave)[P_νf;Cu] +kPνfk_H1(Σ_t)

(9.88) +F(wave)[P_νf;Cu] +kfk²_H1(Σ_t),

where/P_l is the Littlewood-Paley projection operator onS_t,u. We obtain the results by the energy estimates along constant-time hypersurfaces (5.1) and along acoustic null hypersurfaces (8.3)-(8.4).

(9.83a)-(9.83c) are the results of (9.82f), (9.82b) and (9.82g) respectively.

Proposition 9.7(Needed estimates to recover and improve the bootstrap assumption). Under the bootstrap assumptions, the following estimates hold whenever q>2is sufficiently close to 2, where p is the same as in the previous proposition.

Estimates for time-integrated terms

λ⁻¹ er⁻¹

Z t [u]+

er(C~,D)

g/dτ

L²_tL^∞_x(M)

.λ^{−1/2−12ε0}, (9.89a) λ⁻¹

er⁻²

Z t [u]+

er²(C~,D) g/dτ

L²_tL^∞_x(M)

.λ^{−1/2−12ε0}, (9.89b) λ⁻¹

er⁻²

Z _t [u]+

er²(C~,D) _g/dτ

L

q t2L^∞_x(M)

.λ

2 q−1−4ε₀

4 q+2

, (9.89c)

λ⁻¹ er^−1/2

Z t [u]+

er(C~,D)

g/dτ

L^∞_tL^∞_uLω^p(M)

.λ^{−1/2−12ε0}, (9.89d) λ⁻¹

er^−3/2

Z t [u]+

er²(C~,D) g/dτ

L^∞_tL^∞_uL_ω^p(M)

.λ^{−1/2−12ε0}, (9.89e) λ⁻¹

er⁻¹

Z _t [u]+

er²(C~,D) _g/dτ

L^∞(M)

.λ^−16ε0, (9.89f)

λ⁻¹ er⁻³²

Z t [u]+

er²(C~,D) g/dτ

L^∞(M)

.λ^{−1/2−12ε0}, (9.89g) λ⁻¹

er⁻²

Z _t [u]₊

er²(C~,D) _g/dτ

L^∞(M)

.λ^−1−8ε0. (9.89h)

λ⁻¹ er^−3/2

Z t [u]+

er³/∇(C~,D) g/dτ

L_t^∞L^∞_uLω^p(M)

.λ^{−1/2−8ε0}, (9.90a) _t

λ⁻¹ er^−3/2

Z t [u]+

er³(∂∂∂~Ψ,∂∂∂~ω,∂∂∂~S)·

∂∂∂~Ψ,tr

ge

/χ˜^(Small),χ,ˆ ζ,er⁻¹ g/dτ

L^∞_tL^∞_uL^pω(M)

.λ^{−1/2−12ε0}, (9.91a) λ⁻¹

er⁻²

Z _t [u]+

er³(∂∂∂~Ψ,∂∂∂~ω,∂∂∂~S)·

∂∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ˆ ζ,er⁻¹

g/dτ

L²_tL^∞_uL^p_ω(M)

.λ^{−1/2−12ε0}, (9.91b) λ⁻¹

er⁻¹

Z _t [u]₊

er²(∂∂∂~Ψ,∂∂∂~ω,∂∂∂~S)·

∂∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ˆ ζ,er⁻¹ g/dτ

L²_uL²_tL^pω(M)

.λ^−16ε0, (9.91c) λ⁻¹

er^−1/2

Z t [u]+

er²(∂∂∂~Ψ,∂∂∂~ω,∂∂∂~S)·

∂∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ˆ ζ,er⁻¹ g/dτ

L²_uL^∞_t L^pω(M)

.λ^−16ε0. (9.91d)

λ⁻¹ er⁻¹

Z _t [u]+

er²(∂∂∂C~,∂∂∂D) _g/dτ

L²_uL²_tL_ω^p(M)

.λ^−12ε0, (9.92a)

λ⁻¹ er^−1/2

Z t [u]+

er²(∂∂∂C~,∂∂∂D) g/dτ

L²_uL^∞_tLω^p(M)

.λ^−12ε0. (9.92b)

space-time norm estimates

λ⁻¹

er(C~,D)

L_t²L^∞_uLω^p(M).^{−1/2−8ε0}, (9.93a) λ⁻¹

er(C~,D) L

q

t2L^∞_uLω^p(M).⁻

2 q−1−4ε₀

4 q+1

, (9.93b)

λ⁻¹

er(C~,D)

L²_uL²_tLω^p(M).^−12ε0, (9.93c) λ⁻¹

er∂∂∂(C~,D)

L²_uL¹_tLω^p(M).^−12−8ε0, (9.93d) λ⁻¹

er

∂∂∂~Ψ,∂∂∂~ω,∂∂∂~S

·

∂∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ˆ ζ,er⁻¹

L²_uL¹_tLω^p(M).^{−12−10ε0} . (9.93e)