Elliptic div-curl estiamtes in L 2 space

5.1 Energy and L 2 Elliptic Estimates

5.1.2 Elliptic div-curl estiamtes in L 2 space

Cauchy-Schwarz inequality alongΣτand (5.7), we get the desired estimate.

Lemma 5.6(Basic energy inequality for the transport equations). Letϕ be smooth on[0,T∗]×R³. Under the bootstrap assumptions of Section 3.6, the following inequality holds for t∈[0,T∗]:

kϕk²_L2

x(Σt).kϕk²_L2 x(Σ₀)+

Z t 0

∂∂∂~Ψ

L^∞_x(Σ_τ)kϕk²_L2

x(Σ_τ)dτ+ Z t

0

kϕk_L2

x(Σ_τ)kBϕk_L2

x(Σ_τ)dτ. (5.14) Proof of Lemma 5.6. LetJ^α:=ϕ²B^α, then∂αJ^α=2ϕBϕ+ (∂_αB^α)ϕ². We apply the divergence theorem on the space-time region[0,t]×R³ relative to the Cartesian coordinates. Note thatJ⁰=ϕ². By Cauchy- Schwarz inequality alongΣτ, we obtain the desired estimates.

Remark 5.7. We remark that for our implementation of the geometric energy method for wave equations, the timelike vectorfieldT(defined in Definition 2.11) plays the same role asB(Note thatB=∂t+v^a∂ain [9] is not the same asB=^vα

v⁰∂_α in this paper.) in [9, section 4.1]. All the arguments for geometric energy method for wave equations go through in the same fashion as in [9, section 4.1].

following equations hold onΣtfor vorticityωand entropy gradient S:

(G⁻¹)^ab∂a(ω_[)_b=F_ω, (G⁻¹)^ab∂aS_b=F_S, (5.17a)

∂a(ω_[)_b−∂b(ω_[)_a=^(ω)H_ab, ∂aS_b−∂_bS_a=^(S)H_ab, (5.17b)

where

F_ω=L(~Ψ, ~ω,~S)[∂∂∂~Ψ], F_S=L(~Ψ)D+L(~Ψ,~S)[∂∂∂~Ψ], (5.18a)

(ω)H_ab=L(~Ψ)C~+L(~Ψ, ~ω,~S)[∂∂∂~Ψ], ^(S)H_ab=L(~Ψ,~S)[∂∂∂~Ψ], (5.18b) (G⁻¹)^ab=δ^ab−v^av^b

(v⁰)². (5.18c)

For convenience, we write the above 2 div-curl systems as follows, where(η,F_η,^(η)H_ab)which¹is either (ω_[,F_ω,^(ω)H_ab)or(S,F_S,^(S)H_ab):

(G⁻¹)^ab∂_aη_b=F_η, (5.19a)

∂aη_b−∂bη_a=^(η)H_ab. (5.19b)

Proof of Prop 5.9. For the div part (5.19a), by the equations (2.12c) and (2.27a), we write

η₀=−η_bv^b

v⁰ . (5.20)

Also by using transport equation (2.25a) and (2.25b), we have

∂0(η^])^b=−v^a∂a(η^])^b

v⁰ +L(~Ψ, ~ω,S)[∂∂∂~Ψ]. (5.21) Using (2.26d) forη=ω_[, we write∂0ω⁰+∂aω^a=L(~ω)[∂∂∂~Ψ]. By lowering the index∂0ω⁰=−∂₀(ω_[)₀, equations (5.20) and (5.21), we prove (5.17a) forω. Similarly, forη=S, by definition ofD(2.14), we write

∂0(S^])⁰+∂a(S^])^a=L(~Ψ)D+L(~Ψ,~S)[∂∂∂~Ψ]. Using equations (5.20) and (5.21), we obtain the equation (5.17a) forS.

1We use the same notation throughout the remainder of the article.

Now we consider the curl part, note that we have the facts (2.27d) and (2.27e):

∂γη_δ−∂_δη_γ=ε_{γ δ κ λ}v^κvort^λ(η)−(v^κ∂κη_δ)(v_[)_γ+v^κ(∂_δη_κ)(v_[)_γ (5.22) + (v^κ∂κη_γ)(v_[)_δ−v^κ(∂_δη_κ)(v_[)_δ.

Recall that

C^α=vort^α(ω_[) +L(~Ψ,~S)[∂∂∂~Ψ]. (5.23)

Hence forη=ω, the first term on the right-hand side of (5.22) forω is manifestly in^(ω)Hab. Next, using v^κ∂κ(ω_[)_δ =v⁰B(ω_[)_δ and (2.25a), as well as (2.27b), we have that the right-hand side of (5.22) forω is

(ω)Hab. Similarly, by (2.26c),v^κ∂κSγ=v⁰BS_γ, (2.25b) and (2.27c), we obtain the equation (5.17b) forS.

Remark 5.10. In terms of elliptic estimates, there is a major difference in Proposition 5.8 compared to the Hodge system of the non-relativistic 3D compressible Euler equations. In the non-relativistic case, the analogous elliptic equations are constant-coefficient div-curl equations along flat hypersurfaces of constant Cartesian time and, for example, the basic L²theory can be derived with the simple Hodge identity forΣt

vectorfields V∈H¹(R³;R³):

3

∑

a,b=1

∂_aV^b

2

L²_x(R³)=kdivVk²_L2

x(R³)+kcurlVk²_L2

x(R³). (5.24)

In contrast, the divergence equation (5.19a) has dynamic, solution-dependent coefficients.

Proposition 5.11(The top-order div-curl system on constant-time hypersurfaces). Using the same notation as in Proposition 5.19, we have

(G⁻¹)^ab∂a(∂η)_b=F_∂η, (5.25a)

∂a(∂η)_b−∂_b(∂η)_a=^(∂η)H_ab, (5.25b)

where

F_∂η=∂F_η−∂ n

(G⁻¹)^abo

∂aη, (5.26a)

(∂η)H_ab=∂^(η)H_ab, (5.26b)

Moreover, we have

(G⁻¹)^ab∂a∂P_νη_b=F_∂Pνη, (5.27a)

∂a∂P_νη_b−∂b∂P_νη_a=^(∂Pνη)H_ab, (5.27b)

where

F_∂Pνη=P_νF_∂η+ [P_ν,(G⁻¹)^ab]∂_a∂η_b, (5.28a)

(∂Pνη)H_ab=P_ν

(∂η)H_ab

. (5.28b)

Proof of Proposition 5.11. (5.25) is the direct result of taking one spatial derivative of(5.17). By commuting Littlewood-Paley projection operatorP_ν(defined in Section 3.2) with (5.25), we deduce (5.27).

For convenience, we write the above 3 div-curl systems (5.19), (5.25) and (5.27) as follows, where (X,F,H_ab)is(η,Fη,^(η)H_ab)or(∂η,F_∂η,^(∂η)H_ab)or(∂Pνη,F_∂Pνη,^(∂Pνη)H_ab):

(G⁻¹)^ab∂_aX_b=F, (5.29a)

∂aX_b−∂bX_a=H_ab. (5.29b)

In order to derive elliptic estimates and Schauder estimates from the div-curl system (5.19), we need Lemma 5.12 provided below, which allows us to do estimates via Littlewood-Paley theory. We provide a partition of unity before the Lemma 5.12.

We want to apply the Fourier transform to a localized version of the div-curl system in Proposition 5.8.

We consider the latticeA :=δ2Z³, whereδ2is assumed to be small and will be determined in future analysis.

Notice that{x_l}l∈N:=A ⊂Σthas points equally spread out, that is, for eachx_l, there are 6 points inA such that the distance betweenx_land any of them isδ2. We define the family of functions{ψ_l}l∈Nas follows:

ψ_l(x) =



















1 x∈B(x_l,¹₈δ2),

exp( ⁴

3δ₂²)exp( ¹

|x−x_l|²−(⁷₈δ2)²) x∈B(x_l,⁷₈δ2)−B(x_l,¹₈δ2),

0 x∈/B(x_l,⁷₈δ2),

(5.30)

and set

φl(x):= ψ_l(x)

∑

k

ψk(x). (5.31)

We note thatkφ_lk_L∞(Σt),k∂ φ_lk_L∞(Σt).1.

We have constructed cut-off functions {φ_l}l∈N ⊂C₀^∞(Σ_t) such that φl =1 in B(x_l,¹₈δ2), supp(φ_l)⊂ B(x_l,⁷₈δ₂), ∑

l

φ_l(x) =1 and for anyx_a,x_b∈A,φ_a(x) =φ_b(x−x_a+x_b). We want to apply Fourier trans- form on a localized region, where(G⁻¹)^ab(x_l)is a constant and(G⁻¹)^ab(x)−(G⁻¹)^ab(x_l)will be shown to be a controllable error term in the future analysis.

Lemma 5.12. Given the Proposition 5.9 and Proposition 5.11 with x_landφ_l, l∈N, defined as above, let X be the solution of equations (5.29). Then the following identity holds in frequency space for i=1,2,3:

(G⁻¹)^ab(x_l)ξ_aξb\(φ_lX_i) =Cξ_iFˆ^l+

∑

k6=i

C(G⁻¹)^ak(x_l)ξ_aHˆ^l_ki, (5.32)

where

F^l=(G⁻¹)^ab(x_l)∂_a(φ_lX_b) =φ_lF+ (G⁻¹)^ab(x_l)(∂_aφ_l)X_b (5.33a)

−h

(G⁻¹)^ab(x)−(G⁻¹)^ab(x_l)i

∂a(φ_lX_b)−h

(G⁻¹)^ab(x)−(G⁻¹)^ab(x_l)i

(∂_aφl)X_b,

H^l_ab=∂_a(φ_lX_b)−∂_b(φ_lX_a) = (∂_aφ_l)X_b−(∂_bφ_l)X_a+φ_lH_ab. (5.33b)

Proof of Lemma 5.12. By multiplying the div-curl system (5.19a) and (5.19b) byφ_l, we rewrite the system as follows:

n

(G⁻¹)^ab(x_l) + (G⁻¹)^ab(x)−(G⁻¹)^ab(x_l)o

{∂_a(φ_lX_b)−(∂_aφ_l)X_b}=φ_lF, (5.34a)

∂a(φ_lX_b)−(∂_aφl)X_b−∂b(φ_lX_a) + (∂_bφl)X_a=φlH_ab. (5.34b)

Taking the Fourier transform of (5.34a) and multiplying byξ₁, we have

ξ1

n

(G⁻¹)^a1(x_l)ξ_a(φ\_lX1) + (G⁻¹)^a2(x_l)ξ_a(φ\_lX2) + (G⁻¹)^a3(x_l)ξ_a(φ\_lX3)o

=Cξ1Fˆ^l, (5.35)

whereC=_2πi¹ is a constant from Fourier transform, and

F^l=(G⁻¹)^ab(x_l)∂_a(φ_lX_b) =φlF+ (G⁻¹)^ab(x_l)(∂_aφl)X_b (5.36)

−h

(G⁻¹)^ab(x)−(G⁻¹)^ab(x_l)i

∂a(φ_lX_b)−h

(G⁻¹)^ab(x)−(G⁻¹)^ab(x_l)i

(∂_aφl)X_b.

Similarly, taking the Fourier transform of (5.34b) and multiplying by(G⁻¹)^a2(x_l)ξ_aand(G⁻¹)^a3(x_l)ξ_a, we have

(G⁻¹)^a2(x_l)ξ_an

ξ₂(φ\_lX₁)−ξ₁(φ\_lX₂)o

=C(G⁻¹)^a2(x_l)ξ_aHˆ^l₂₁, (5.37a) (G⁻¹)^a3(x_l)ξ_an

ξ3(φ\_lX1)−ξ1(φ\_lX3)o

=C(G⁻¹)^a3(x_l)ξ_aHˆ^l₃₁, (5.37b)

whereC=_2πi¹ is a constant from Fourier transform, and

H^l_ab=∂a(φ_lX_b)−∂_b(φ_lX_a) = (∂_aφl)X_b−(∂_bφl)X_a+φ_lH_ab. (5.38)

Adding (5.35), (5.37a) and (5.37b), we obtain

(G⁻¹)^ab(x_l)ξ_aξ_b(φ\_lX₁) =Cξ₁Fˆ^l+C(G⁻¹)^a2(x_l)ξ_aHˆ^l₂₁+C(G⁻¹)^a3(x_l)ξ_aHˆ^l₃₁. (5.39)

We use the same argument forX2andX3. Hence fori=1,2,3, we obtain

(G⁻¹)^ab(x_l)ξ_aξb\(φ_lX_i) =Cξ_iFˆ^l+

∑

k6=i

C(G⁻¹)^ak(x_l)ξ_aHˆ^l_ki. (5.40)

Lemma 5.13 (Positive definiteness of G⁻¹). For any Σ_t-tangent one-form ξ, that is, ξ₀=0, we denote

|ξ|²:= ∑

i=1,2,3

ξ_i². Then the following estimate holds for any x_l∈Σ_t, where G is defined in Proposition 5.9:

C|ξ|²≤(G⁻¹)^ab(x_l)ξ_aξ_b≤ |ξ|², (5.41)

where0<C<1is a constant depends only onkvk_L∞

x(Σt), which is in turn controlled by the bootstrap as- sumption (3.24a).

Proof of Lemma 5.13. Using the definition of(G⁻¹)^ab, we have

(G⁻¹)^ab(x_l)ξ_aξb=

δ^ab− v^av^b (v⁰)²

ξaξb=|ξ|²− vⁱξ_i

v⁰ 2

. (5.42)

Hence by the normalization(v_[)_αv^α=−1 in Section 2.2.1, we have

C|ξ|²≤(G⁻¹)^ab(x_l)ξ_aξ_b≤ |ξ|², (5.43)

where 0<C<1 is a constant depends only onkvk_L∞ x(Σt).

Lemma 5.14. For G defined in Proposition 5.9, the following inequality holds

(G⁻¹)^ab(x)−(G⁻¹)^ab(x_l)

≤C₃kvk⁴

C^0,δ₀ ⁰(Σ_t)|x−x_l|^δ0≤C₂⁴C₃|x−x_l|^δ0, (5.44) where C₂,C₃are constants (independent of l,x,a,b,δ₀).

Proof of Lemma 5.14. For H¨older continuous functionf,g∈C₀^0,δ0(Σ_t),

|f(x)g(x)−f(y)g(y)|

|x−y|^δ0 =|f(x)g(x)−f(x)g(y)|+|f(x)g(y)−f(y)g(y)|

|x−y|^δ0 (5.45)

≤ kfk_L∞ x(Σ_t)kgk

C˙^0,δ₀ ⁰(Σ_t)+kgk_L∞ x(Σ_t)kfk

C˙^0,δ₀ ⁰(Σ_t)

≤2kfk

C₀^0,δ0(Σt)kgk

C₀^0,δ0(Σt).

Therefore, by definition ofG⁻¹in (5.18c) and the fact thatv⁰≥1, substitute(f,g)in (5.45) by(v^a,v^b)and again by(v^av^b, ¹

(v⁰)²), we have the estimate

(G⁻¹)^ab(x)−(G⁻¹)^ab(x_l)

≤C₃kvk⁴

C^0,δ₀ ⁰(Σt)|x−x_l|^δ0 (5.46) By Fundamental Theorem of Calculus, bootstrap assumptions (3.24) and (3.8), we have

kvkCx^0,δ0(Σ_t)≤ ∂∂∂~Ψ

L¹_tC^0,δx ⁰(Σ_t)+kvk

C^0,δx ⁰(Σ₀). ∂∂∂~Ψ

L¹_tC^0,δx ⁰(Σ_t)+1≤C₂, (5.47) Combining (5.46) and (5.47), we have the desired result.

Lemma 5.15(Commutator estimates). For scalar function F, G,δ1defined as in Section 3.4, Littlewood-

Paley projection operator P_νdefined in Section 3.2, we have the following estimates:

k[P_ν,G]Fk_L2(Σt).ν^−δ1kGk

C^0,δx ¹(Σt)kFk_L2(Σt). (5.48) Proof of Lemma 5.15. Forψdefined as in Section 3.2, we defineM(x)as follows:

Mν(x):=F⁻¹(ψ(ν⁻¹ξ)) = Z

e^ixξψ(ν⁻¹ξ)dξ. (5.49)

Then we have

[P_ν,G]F= Z

M_ν(x−y) (G(y)−G(x))F(y)dy (5.50)

≤ kGk

Cx^0,δ1(Σt) Z

M_ν(x−y)|x−y|^δ1F(y)dy.

By Young’s inequality, we have

k[P_ν,G]Fk_L2(Σt).kGk

Cx^0,δ1(Σt)kFk_L2(Σt)

M_ν(x)|x|^δ1

L¹(Σt). (5.51) By definition of Fourier transform andψin Section 3.2, we have

Z

Mν(x)|x|^δ1dx= Z

ν³ψ(−νx)ˆ |x|^δ1dx.ν^−δ1. (5.52)

Combining above equations, we obtain the desired result.

Lemma 5.16(Control of the inhomogeneous terms). For F_η,^(η)H,G⁻¹ defined as in Proposition 5.9, and F_∂η,F_∂Pνη,^(∂η)H,^(∂Pνη)H defined as in Proposition 5.11, we have the following estimates:

kF_ηk_L2(Σt),

(η)H

L²(Σt)≤C

∂∂∂~Ψ, ~C,D

L²(Σt), (5.53)

F_{∂η L}2(Σt),

(∂η)H

L²(Σ_t)≤C

∂∂∂²~Ψ,∂C~,∂D

L²(Σ_t)+Cα⁻¹ ∂~Ψ

2

H¹(Σ_t)k∂ηk_L2(Σ_t)+Cα

∂²η _L2(Σt),

(5.54) ν^N−2F_∂Pνη

_l2 νL²(Σt),

ν^N−2·^(∂Pνη)H

l_ν²L²(Σt).

∂²~Ψ

L²(Σt)+1

∂²η _L2(Σt)+

∂∂∂²~Ψ,∂C~,∂D

H^N−2(Σt), (5.55) where C,αare constants, C is independent ofα, andα>0(small), which will be determined later.

Proof of Lemma 5.16. (5.53) is the direct result of takingL²(Σ_t)for (5.18).

TakingL²(Σ_t)for (5.26), we have

k∂F_η,k_L2(Σ_t), ∂^(η)H

L²(Σt)≤C

∂∂∂²~Ψ,∂C~,∂D

L²(Σt), (5.56)

∂

n

(G⁻¹)^abo

∂_aη

L²(Σt)≤C ∂~Ψ

H¹(Σt)k∂ηk¹²

L²(Σt)

∂²η

1 2

L²(Σ_t) (5.57)

≤Cα⁻¹ ∂~Ψ

2

H¹(Σt)k∂ηk_L2(Σ_t)+Cα

∂²η _L2(Σt),

where for the first inequality in (5.57), we used the fact thatkF·Gk_L2(Σ_t)≤CkFk¹²

L²(Σt)kFk¹²

H¹(Σt)kGk_H1(Σ_t)

(see [9, (79b)]), for the second inequality in (5.57), we used Young’s inequality.

Now we consider the proof of (5.55). Takingl_ν²L²(Σ_t)norm of (5.28), we have

ν^N−2P_νF_∂η,

l_ν²L²(Σ_t),

ν^N−2P_ν

(∂η)H

l_ν²L²(Σt)≤C

∂∂∂²~Ψ,∂C~,∂D

H^N−2(Σt)+ ∂²~Ψ

L²(Σt)

∂²η L²(Σ_t),

(5.58) where the second term on the RHS of (5.58) is from the fact that (see [9, (81b)])

ν^N−2P_ν

∂ n

(G⁻¹)^abo

∂aη

l_ν²L²(Σt). ∂~Ψ

H^N−32(Σt)k∂ηk_H1(Σt)+k∂ηk

H^N−32(Σ_t)

∂~Ψ

H¹(Σt). (5.59) What remains to be controlled is the commutator term[P_ν,(G⁻¹)^ab]∂_a∂η_b. By Lemma 5.15, whereG:=

(G⁻¹)^abandF:=∂_a∂η_b, and (5.47), we have

ν^N−2[P_ν,(G⁻¹)^ab]∂_a∂η_b

lν²L²(Σ_t).

∂²η

L²(Σt) (5.60)

Combining above 3 estimates, we obtain (5.55).

Lemma 5.17. Let G⁻¹be defined as in (5.18c). F,H defined as in (5.29). For X the solution of the div-curl system (5.29), we have the following estimate:

k∂Xk_L2

x(Σt)≤CkX,F,Hk_L2

x(Σt), (5.61)

where C is a constant, which is independent of X,F,H.

Proof of Lemma 5.17. Throughout,X,F,G,Hare the same as in (5.29), andF,Hare defined in Lemma 5.12.

Since|ξ| '2νon support ofPν\(φ_lη_i), by Littlewood-Paley estimate (3.7), (5.41) and (5.32), we remindψis

defined in Section 3.2

k∂{φ_lX_i}k²_L2

x(Σt)≤C₁

∑

ν>1

kνP_ν(φ_lX_i)k²_L2

x(Σt) (5.62)

≤C1

∑

ν>1

F⁻¹ ψ |ξ|

ν

|ν|⁻¹ (

ξiFˆ^l+

∑

k6=i

(G⁻¹)^jk(x_l)ξ_jHˆ^l_ki )!

2

L²_x(Σt)

≤C₁

F^l

2 L²_x(Σt)+

H^l

2 L²_x(Σt)

,

whereC₁is a constant andF⁻¹is the Fourier inverse transform. By (5.33a), we have:

F^l

2

L²_x(Σ_t)≤ kφ_lFk²_L2 x(Σt)+

(G⁻¹)

2

L^∞_x(Σt)k(∂ φ_l)Xk²_L2

x(Σt) (5.63)

+ sup

x∈B(x_l,δ₂)

(G⁻¹)^ab(x)−(G⁻¹)^ab(x_l)

2k∂(φ_lX)k²_L2 x(Σt).

By (5.33b),

H^l

2

L²_x(Σt)≤ kφ_lHk²_L2

x(Σ_t)+2k(∂ φ_l)Xk²_L2

x(Σ_t). (5.64)

LetC⁰:=C₁C₂⁴C₃, whereC₁,C₂,C₃are constants from (5.62), (5.47), (5.44) respectively, and letδ2be small such that for allx∈B(x_l,δ2),

C⁰|x−x_l|^δ0 <1

4. (5.65)

Hence by (5.62), (5.63), (5.64), (5.65), soaking the last term in the RHS of (5.63) to the left of (5.62) (by Lemma 5.14), we have:

k∂{φ_lX_i}k²_L2

x(Σt)≤C₄ kφ_lFk²_L2

x(Σt)+kφ_lHk²_L2

x(Σt)+k(∂ φ_l)Xk²_L2 x(Σ_t)

, (5.66)

whereC₄is a constant.

Now we consider thek∂X_ik²_L2

x(Σ_t). By (5.66), we have:

k∂X_ik²_L2 x(Σ_t)=

∑

l

∂{φ_lX_i}

2

L²_x(Σt)

≤C₅

∑

l

k∂{φ_lX_i}k²_L2

x(Σ_t)≤C₄C₅

∑

l

kφ_l(F,H)k²_L2

x(Σ_t)+k(∂ φ_l)Xk²_L2 x(Σ_t)

! .

(5.67)

For the first term on the RHS of (5.67), by (5.53), we have :

∑

l

kφ_l(F,H)k²_L2

x(Σt)=

∑

l Z

Σt

(φ_l)²(F²+H²)dx (5.68)

≤

∑

l

kφ_lk²_L∞(Σt) Z

Σ_t^T{∂ φ_l6=0}

(F²+H²)dx

≤C₆kF,Hk²_L2 x(Σ_t).

For the second term on the RHS of (5.67), we have:

∑

l

k(∂ φ_l)Xk²_L2

x(Σt)=

∑

l Z

Σt

(∂ φ_l)²X²dx≤C₇

∑

l

k∂ φ_lk²_L∞(Σ_t) Z

ΣtT{∂ φ_l6=0}X²dx≤C₈kXk²_L2

x(Σt) (5.69) where the last inequality holds for both (5.68) and (5.69) since for eachx∈Σ_t, there are finite many (at most 8)φ_lsuch thatφ_l(x)6=0.

Combining (5.67)-(5.69) and lettingC:=C4C5(C₆+C₈), we conclude the desired estimate.

Proof of Proposition 5.8. (5.15) is a direct result by substituting(X,F,H)in (5.61) by(η,F_η,^(η)H)and using the schematic definitionη=∂~Ψ, estimate (5.53).

Using (3.7), (5.16) can be proved in a similar fashion by using the following interpolation estimate:

k∂η_ik²_HN−1(Σt)≤

∂²η_i

2

L²_x(Σt)+C₁

∑

ν

ν^N−2P_ν(∂²η_i)

2

L²_x(Σt), (5.70) where we bound the first term on the RHS of (5.70) as follows:

∂²η_i

L²_x(Σt).

∂ ∂∂∂~Ψ,∂C,∂D _L2

x(Σt)+ ∂∂∂~Ψ

3

H¹(Σt), (5.71)

(5.71) is obtained by exactly the same method as in the proof of (5.15). That is, substituting(X,F,H,C,D) in (5.61) by(∂η,F_∂η,^(∂η)H,∂C,∂D)and using (5.54), we have

∂²η_{i L2}

x(Σ_t)≤C

∂∂∂²~Ψ,∂C~,∂D

L²(Σt)+Cα⁻¹ ∂~Ψ

2

H¹(Σt)k∂ηk_L2(Σt)+Cα

∂²η _L2(Σ

t). (5.72) We pickαsmall (Cα≤¹₂) such thatCα

∂²η

_L2(Σt)can be soaked into the LHS of (5.72).

Now we consider the second term on the RHS of (5.70).

Substituting(X,F,H)in (5.61) by(∂Pνη,F_∂Pνη,^(∂Pνη)H), we have:

Multiplying (5.73) byν^N−2and taking thel_ν²norm, we have:

ν^N−2P_ν(∂²η_i) _l2

νL²_x(Σt). ν^N−2

∂P_νη,F_∂Pνη,^(∂Pνη)H

l²_νL²_x(Σt). (5.74) By (5.74) and (5.55), we have:

ν^N−2Pν(∂²η_i)

l²_νL²_x(Σt).

∂²~Ψ

L²(Σ_t)+1

∂²η

L²(Σt)+

∂∂∂²~Ψ,∂C~,∂D

H^N−2(Σ_t). (5.75) By (5.70), (5.71) and (5.75), we have proved (5.16).