Paper Title: Regularization by noise for the inviscid primitive equations.



§ ABSTRACT 
    The deterministic inviscid primitive equations (also called the hydrostatic Euler equations) are known to be ill-posed in Sobolev spaces and in Gevrey classes of order strictly greater than 1, and some of their analytic solutions exist only locally in time and exhibit finite-time blowup. This work demonstrates that introducing suitable random noise can restore the local well-posedness and prevent finite-time blowups. Specifically, random diffusion addresses the ill-posedness in certain Gevrey classes, allowing us to establish the local well-posedness almost surely and the global existence of solutions with high probability. In the case of random damping (linear multiplicative noise), the noise prevents analytic solutions from forming singularities in finite time, resulting in globally existing solutions with high probability.



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    [
    August 5, 2024
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MSC Subject Classifications: 35Q35, 60H15, 60H50, 76M35


Keywords: inviscid primitive equations, hydrostatic Euler equations, pseudo-differential noise, Gevrey class, global well-posedness




§ INTRODUCTION


We are interested in the following three-dimensional stochastic inviscid primitive equations (PE) 
 

    d V + (V·∇_h V + w∂_z V +∇_h p ) dt = F(V)dW_t  , 
       ∂_z p  =0 , 
       ∇_h · V + ∂_z w =0,


which is also known as the stochastic hydrostatic Euler equations. Here, the horizontal velocity field V=(u, v), the vertical velocity w, and the pressure p are the unknown quantities. We employ the notations ∇_h = (∂_x, ∂_y) and ∇ = (∂_x, ∂_y, ∂_z) to distinguish between the horizontal and complete gradients. The Wiener process is denoted by W_t, and F(V)dW_t represents the driving noise, the form of which will be specified later. The domain of interest is the three-dimensional torus 𝕋^3 = ℝ^3/ℤ^3 with unit volume, and the boundary conditions are

    V  is periodic in   (x,y,z)  with period   1,       V   is even in   z   and   w   is odd in   z.
 
Observe that the space of periodic functions with respect to z with the symmetry condition (<ref>) is
invariant under the dynamics of system (<ref>).

The viscous PE is derived from the Navier-Stokes equations <cit.>, while the inviscid PE is derived from the Euler equations <cit.> by taking the hydrostatic limit. This model finds extensive application when the vertical-to-horizontal scale ratio is small. For example, it is particularly relevant in the study of large-scale oceanic and atmospheric dynamics, where the vertical scale (a few kilometers in oceans and 10-20 kilometers in the atmosphere) is significantly less than the horizontal scale, which spans thousands of kilometers.

Our work focuses on the well-posedness of the stochastic PE. Inspired by <cit.>,  we examine solutions to the PE perturbed by a specific type of noise, that is,  F(V)=ν|∇|^s V, where ν>0 and 0≤ s≤ 1 are constants, and |∇|= √(-Δ).
We further distinguish two different cases: random diffusion in which s∈ (0,1] and random damping in which s=0. Our primary focus is on the regularization effects these noises have on the system. Specifically, random diffusion helps to overcome the ill-posedness in Gevrey class of order 1/s, allowing for the local well-posedness almost surely and the 
global existence of solutions with high probability. In contrast, random damping prevents the finite-time blowup for analytic solutions with high probability. Following the spirit of previous works <cit.>, we interpret the equation (<ref>) with such a pseudo-differential noise as an equivalent random PDE (see (<ref>) below) through a transformation involving a geometric Brownian motion and the multiplier ν|∇|^s. The main results of our paper are summarized below. For the precise definition of the Gevrey class, see (<ref>) in the next section.  

    Consider system (<ref>) subject to random noise of the form F(V)=ν|∇|^s V.   
    
        
  * (Random diffusion, Theorem <ref>.) Let η, α>0, s∈(4/5,1] and σ∈(8/5s,2). With probability one, system (<ref>) is locally well-posed with solutions in the Gevrey class _η^σ,s for each deterministic initial data in _α+η^σ,s. Moreover, for any ε>0, there is a sufficiently large constant α=α(ε) such that for any given deterministic initial data in _α+η^σ,s, by choosing a sufficiently large diffusion intensity ν, the corresponding local solutions can be extended to global ones in _η^σ,s with probability at least 1-ε. 
        
  * (Random damping, Theorem <ref>.) Let s=0. The system (<ref>) is locally well-posed in the analytic class with probability one. Furthermore, given any ε>0 and deterministic analytic initial data, the local solution can be made global with probability at least 1-ε by taking sufficiently large damping intensity ν.
    


The primitive equations have been widely studied in the literature. For instance, the PE is globally well-posed in 3D when the system has full viscosity <cit.> in the deterministic setting. For the stochastic PE with full viscosity, local and global well-posedness were examined in <cit.>, and the existence and regularity of invariant measures were established in <cit.>. On the other hand, the deterministic inviscid PE are ill-posed in Sobolev spaces and Gevrey classes of order strictly greater than 1 <cit.>, and some smooth solutions can form singularities in finite time <cit.>. To obtain local well-posedness, one needs to assume certain special structures (local Rayleigh condition) on the initial data in 2D <cit.>, or real analyticity for general initial data in both 2D and 3D <cit.>. In the stochastic setting, the local well-posedness of the inviscid PE under the same initial assumptions was established in <cit.>, where the authors considered general multiplicative noise but did not explore the benefits of certain types of noise. 

In <cit.>, the authors considered adding random diffusion F(V)=ν|∇|^s V to the inviscid surface quasi-geostrophic equation and a large class of active scalar equations (such as Patlak–Keller–Segel model), and showed that these systems are globally well-posed in the Gevrey class of order 1/s with high probability. Inspired by these works, we investigate the effect of random diffusion in the inviscid PE and establish that the system not only overcomes the ill-posedness in the Gevrey class of order 1/s, but also prevents the blowup with high probability, in contrast to the blowup results in the deterministic case <cit.>. The special nonlinear structure of the PE, which involves a loss of horizontal derivative of V,  distinguishes it from those models considered in the previous works <cit.>.  To our best knowledge, this is the first study to show that random diffusion can tame an ill-posed system, resulting in local well-posedness and global solutions with high probability. 

In <cit.>, the authors considered the 3D stochastic Euler equations with random damping (linear multiplicative noise) F(V)=ν V and proved that regular solutions exist globally with high probability 1-ε when the initial condition meets certain size requirements depending on ν and ε. In light of recent breakthroughs on the blowup in the 3D deterministic Euler equations <cit.>, 
this demonstrates that random damping can help prevent the formation of finite-time singularities. This motivates us to consider random damping for the inviscid PE and prove the global existence of analytic solutions with high probability, provided that the damping intensity ν is large enough.

The rest of the paper is organized as follows. In Section <ref>, we introduce the notations, functional settings, and some preliminaries for the paper. Sections <ref> and <ref> are devoted to the study of the inviscid PE subject to random diffusion and random damping, respectively. 
Finally, we conclude in Section <ref>.




§ PRELIMINARIES



In this section, we introduce notations and some needed preliminary results. The universal constant C appearing below may change from line to line. When needed, we use subscripts to indicate the dependence of the constant on certain parameters, e.g., we write C_σ to emphasize that the constant depends on σ.

Let x:= (x',z) = (x_1, x_2, z)∈𝕋^3, where x' and z represent the horizontal and vertical variables, respectively. Define
f_L^2 := (∫_𝕋^3 |f(x)|^2 dx)^1/2,
associated with the inner product
⟨ f,g⟩ = ∫_𝕋^3 f(x)g(x) dx
for f,g ∈ L^2(𝕋^3). For a function f ∈ L^2(𝕋^3), let f̂_k be its Fourier coefficient such that

    f(x) = ∑_k∈ 2πℤ^3f̂_k e^ ik· x,     f̂_k = ∫_𝕋^3 e^- ik· x f(x) dx.

The L^2 norm of f can also be represented as 

    f_L^2 = (∑_k∈ 2πℤ^3 |f̂_k|^2)^1/2.

For s> 0 we define the following Sobolev H^s norm and Ḣ^s semi-norm

    -.1in
    f _H^s:= (∑_k∈ 2πℤ^3 (1+|k|^2s) |f̂_k|^2  )^1/2,     f _Ḣ^s:= (∑_k∈  2πℤ^3 |k|^2s |f̂_k|^2  )^1/2,

and we refer to <cit.> for more details about Sobolev spaces. When f has zero mean, i.e., f̂_0=0, these two norms are equivalent, and f_H^s≤√(2)f_Ḣ^s.

Let A=|∇| be the Fourier multiplier such that (Af)_k = |k|f̂_k. For s∈(0,1], σ>0, and a positive function ϕ(t), we define the Gevrey class with order 1/s≥ 1 and radius ϕ(t)>0 as

    𝒢_ϕ^σ,s: = {f∈ H^σ s: f__ϕ^σ,s<∞},

where the Gevrey norm is given by 

    f__ϕ^σ,s := (e^ϕ A^sf_Ḣ^σ s^2 + f_L^2^2 )^1/2= ( ∑_k∈  2πℤ^3 (1+ e^2ϕ(t)|k|^s |k|^2σ s) |f̂_k|^2)^1/2.

The norm above is equivalent to the homogeneous Gevrey norm 

    f__ϕ^σ,s := e^ϕ A^sf_Ḣ^σ s= ( ∑_k∈  2πℤ^3 e^2ϕ(t)|k|^s |k|^2σ s |f̂_k|^2)^1/2,

for f having zero mean,
and it holds that f__ϕ^σ,s≤√(2)f__ϕ^σ,s.
Note that we often write ϕ instead of ϕ(t) for simplicity. When s=1, the Gevrey class coincides with the space of analytic functions with radius ϕ. 

For a given function f, we define its barotropic part f̅ and baroclinic part f̃ as 

    f̅(x') = ∫_0^1 f(x',z) dz,   f̃(x) = f(x) - f̅(x').

Let 𝒫_h be the 2D Leray projection such that 𝒫_h φ= φ - ∇_h  Δ_h^-1∇_h ·φ. Here Δ_h^-1 represents the inverse of the Laplacian operator on 𝕋^2. Furthermore, let 𝒫 be the hydrostatic Leray projection such that 𝒫φ = 𝒫_h φ+ φ̃. As e^ϕ A^s and A^s are Fourier multipliers, these operators commute with 𝒫. Moreover, 𝒫 is a bounded operator in L^2, and consequently, one has

    𝒫 f_^σ,s_ϕ≤f_^σ,s_ϕ,   𝒫 f_^σ,s_ϕ≤f_^σ,s_ϕ.


From the divergence free condition (<ref>) and the boundary condition (<ref>), the vertical velocity w can be represented through V as

    w(V) = -∫_0^z ∇_h · V (x',z̃) dz̃.

In addition, we define the bilinear form Q(U,V) as

    Q(U,V) = U·∇_h V + w(U)∂_z V.

Using these notations and applying 𝒫 to (<ref>), system (<ref>) can be rewritten as

    dV + 𝒫Q(V,V)dt  = 𝒫 F(V) dW_t,      V(0)=V_0.


Since the noise has the form F(V)=ν |∇|^s V, integrating (<ref>) over 𝕋^3 yields ∫_𝕋^3 V(x,t) dx =0 for all t≥0, provided that ∫_𝕋^3 V_0 dx =0. Additionally, noting that V satisfies ∇_h ·V=0, we search solutions V in H for all t≥ 0, where

    H:= {f∈ L^2: ∫_𝕋^3 f(x,t) dx =0,   ∇_h·f̅=0 }.

For later use, we also define C_T_ϕ^σ,s for any given positive time T>0:

    C_T_ϕ^σ,s: ={f∈ C([0,T];H∩_ϕ^σ,s): f_C_T_ϕ^σ,s := sup_t∈[0,T]f(t)__ϕ(t)^σ,s<∞}.


To solve (<ref>), we introduce the multiplier Γ(t) = e^-ν W_t |∇|^s as in <cit.> and define U=Γ V. By applying Itô's formula to U=Γ V and using system (<ref>), we can derive the following random PDE for U: 

    dU+𝒫Γ Q(Γ^-1U,Γ^-1U)dt= -1/2ν^2|∇|^2sUdt,      U(0)= U_0 = V_0.

In the sequel, we will focus on this random equation and use its solution to construct the solution to (<ref>). Specifically, once we obtain solutions U to 
(<ref>), V=Γ^-1U will give the solutions to (<ref>). 

Regarding the stochastic setting, we consider 
the canonical realization of (Ω,ℱ,ℙ) where Ω={ω∈ C([0,∞);ℝ):ω(0)=0}
is the canonical Wiener space endowed with the compact-open topology, ℱ is the Borel σ-algebra, and ℙ is the Wiener measure. The Brownian motion is the coordinate process W_t(ω)=ω(t) so that each sample path is continuous in time. For global solutions, we will work on the following good subset of Ω:

    Ω_α,β,ν = {ω∈Ω: α+β t-ν W_t≥ 0  for all  t≥ 0}.

For α,β>0, this set has probability ℙ(Ω_α,β,ν)≥ 1-e^-αβ/ν^2 (cf. <cit.>). When applying this 
estimate below, we will be in a situation where β has the same order as ν^2, thus to make the probability high, we need sufficiently large α. See also <cit.> for relevant discussions. 





§ RANDOM DIFFUSION

In this section, we analyze the stochastic inviscid PE subject to random diffusion, where F(V)=ν|∇|^sV in system (<ref>). This is a special type of multiplicative noise defined through the Fourier multiplier, first considered in <cit.>. The equation then reads 

    dV + 𝒫Q(V,V)  dt = ν|∇|^sVdW_t,      V(0)=V_0,

where the differentiability index 0<s≤ 1 and the diffusivity ν>0 are constants. 

Recall the definition of Γ(t) = e^-ν W_t|∇|^s and U=Γ V. Itô's formula gives that 

    dΓ = -ν|∇|^sΓ dW_t + 1/2ν^2|∇|^2sΓ dt,

and U solves the random PDE:

    ∂_t U+𝒫Γ Q(Γ^-1U,Γ^-1U)= -1/2ν^2|∇|^2sU,      U(0)= U_0 = V_0,

provided that V is a solution to (<ref>), where we have used Γ(0)=1 to obtain U_0 = V_0. As mentioned above, we shall primarily work with this random PDE and use its solution U to construct the solution V to (<ref>) via V = Γ^-1U.


In the rest of this section, we set ϕ(t) = α+β t as the Gevrey radius, where  α>0 and  0<β<ν^2/2.  




 §.§ Local existence

We first provide the local existence of solutions to (<ref>). Compared to <cit.>, the special nonlinear structure of the PE requires a delicate analysis to obtain the optimal range of parameters σ and s.


      Let s∈(4/5,1] and σ∈(8/5s,2). For any initial data U_0∈ H∩_α^σ,s, there exists a random time T>0,  such that there is a unique local solution to (<ref>) in C_T_ϕ^σ,s for every ω∈Ω. Moreover, the random existence time T becomes deterministic over Ω_α,β,ν and depends only on the size of the initial data. 


    For any fixed ω∈Ω, define 
    
    T_ω :=inf{t≥ 0: ν W_t(ω)>ϕ(t)}.

    Note that T_ω>0 by the continuity of the Brownian path W_t(ω) and the fact that W_0(ω)=0 and ϕ(0)=α>0. Denote by 
    B(U,U) = Γ Q(Γ^-1U,Γ^-1U).

    Since Γ commutes with 𝒫, from (<ref>) we define the solution operator 
    
    Φ(U)(t):= e^-ν^2t/2A^2sU_0 - ∫_0^t e^-ν^2(t-r)/2A^2s𝒫B(U,U) dr
 
    for t∈[0,T_ω]. Note that since both U_0 and B(U,U) have zero means, so does Φ(U).
    
    We want to estimate the Gevrey norm of Φ(U). 
    For any j=(j_1,j_2,j_3)∈ℤ^3 we denote by j'=(j_1,j_2) and similar notations apply to k = (k_1, k_2, k_3). By our assumption β<ν^2/2 and the property (<ref>) of the hydrostatic Leray projection, we have
    
    Φ(U)(t)_^σ,s_ϕ(t)≤   √(2)Φ(U)(t)_^σ,s_ϕ(t) = √(2)e^ϕ(t) A^sΦ(U)(t)_Ḣ^σ s
    ≤   √(2)e^ϕ(t) A^s-ν^2t/2A^2sU_0_Ḣ^σ s + √(2)∫_0^te^ϕ(t)A^s-ν^2/2(t-r)A^2sB(U,U)_Ḣ^σ sdr 
    ≤   √(2)e^α A^sU_0_Ḣ^σ s + √(2)∫_0^te^ϕ(t)A^s-ν^2/2(t-r)A^2sB(U,U)_Ḣ^σ sdr.

    We next estimate the nonlinear term. A straightforward computation yields 
    
    B(U,U)(t,k)   = e^-ν W_t|k|^s∑_je^ν W_t|j|^sÛ_j· i(k'-j')e^ν W_t|k-j|^sÛ_k-j
          -e^-ν W_t|k|^s∑_je^ν W_t|j|^sj'·Û_j/j_3i(k_3-j_3)e^ν W_t|k-j|^sÛ_k-j
       :=B̂_1(t,k)+B̂_2(t,k).

    
    For the term B̂_2(t,k), using the facts that ϕ(t)-ν W_t(ω)≥ 0 for t∈[0,T_ω], and |k|^s≤ |k-j|^s+|j|^s when 0<s≤ 1, one has
    
    ∑_k||k|^σ se^ϕ(t)|k|^s-ν^2/2(t-r)|k|^2sB̂_2(r, k)|^2
        = ∑_k||k|^σ se^(ϕ(t)-ϕ(r))|k|^s-ν^2/2(t-r)|k|^2s∑_je^(ϕ(r)-ν W_r)(|k|^s-|j|^s-|k-j|^s)e^ϕ(r)|j|^sj'·Û_j/j_3(k_3-j_3)e^ϕ(r)|k-j|^sÛ_k-j|^2
       ≤∑_k||k|^σ se^(β |k|^s-ν^2/2|k|^2s)(t-r)∑_je^(ϕ(r)-ν W_r)(|k|^s-|j|^s-|k-j|^s)e^ϕ(r)|j|^sj'·Û_j/j_3(k_3-j_3)e^ϕ(r)|k-j|^sÛ_k-j|^2
       ≤∑_k|k|^2σ se^2(β |k|^s-ν^2/2|k|^2s)(t-r)(∑_j|e^ϕ(r)|j|^sj'·Û_j/j_3(k_3-j_3)e^ϕ(r)|k-j|^sÛ_k-j|)^2.

    Since β<ν^2/2, there exist a constant C>0 such that  
    |k|^2σ se^2(β |k|^s-ν^2/2|k|^2s)(t-r)≤ C(t-r)^-σ.
    By Young's inequality for convolution with 1/p+1/q=3/2 and 1< p,q <2, one has
    
    (∑_k(∑_j|e^ϕ(r)|j|^sj'·Û_j/j_3(k_3-j_3)e^ϕ(r)|k-j|^sÛ_k-j|)^2)^1/2
       ≤(∑_j|e^ϕ(r)|j|^s|j'|/|j_3|Û_j|^p)^1/p(∑_j||j|e^ϕ(r)|j|^sÛ_j|^q)^1/q.

    Using the Hölder inequality, the first term in the above inequality is bounded by
    
    (∑_j|e^ϕ(r)|j|^s|j'|/|j_3|Û_j|^p)^1/p = (∑_j_3≠ 0∑_j'≠ 0|e^ϕ(r)|j|^s|j'|/|j_3|Û_j|^p)^1/p
    ≤   (∑_j_3≠ 0(∑_j'≠ 0|e^pϕ(r)|j|^s|j|^pσ s|Û_j|^p|^2/p)^p/2(∑_j'≠ 0| |j_3|^-p |j|^p-pσ s|^2/2-p)^2-p/2)^1/p
    ≤   (∑_j_3≠ 0(∑_j'≠ 0e^2ϕ(r)|j|^s|j|^2σ s|Û_j|^2)^p/2 |j_3|^-p(∑_j'≠ 0 |j|^2p(1-σ s)/2-p)^2-p/2)^1/p
    ≤    C (∑_j_3≠ 0(∑_j'≠ 0e^2ϕ(r)|j|^s|j|^2σ s|Û_j|^2)^p/2 |j_3|^-p)^1/p
    ≤    C (∑_j_3≠ 0∑_j'≠ 0e^2ϕ(r)|j|^s|j|^2σ s|Û_j|^2)^1/2(∑_j_3≠ 0  |j_3|^-2p/2-p)^2-p/2p≤ C e^ϕ(r)A^sU_Ḣ^σ s,

    where in the third inequality we need the assumption that
    
    2p/2-p(σ s-1) >2,

    and we have used the fact that |j|^-2p/2-p(σ s-1)≤ |j'|^-2p/2-p(σ s-1), for j_3≠ 0.
    Similarly, for the second term we deduce
    
    (∑_j||j|e^ϕ(r)|j|^sÛ_j|^q)^1/q≤   (∑_je^2ϕ(r)|j|^s|j|^2σ s|Û_j|^2)^1/2(∑_j|j|^2q/2-q(1-σ s))^2-q/2q
    ≤   Ce^ϕ(r)A^sU_Ḣ^σ s,

    as long as assuming
    
    2q/2-q(σ s-1)>3.

    
    To ensure that both (<ref>) and (<ref>) hold, it is equivalent to guarantee that
     σ s-1> f(p,q), where 
    f(p,q)=max{2-p/p,3(2-q)/2q}.
    Since 1/p+1/q=3/2, we consider the function
    
    g(p)=f(p,1/(3/2-1/p))=max{2/p-1,3(1-1/p)},

    which has a minimum value of 3/5 at p=5/4.
    Thus, as long as σ s>8/5, both (<ref>) and (<ref>) will hold. 
    Therefore, we have 
    
    (∑_k||k|^σ se^ϕ(t)|k|^s-ν^2/2(t-r)|k|^2sB̂_2(r, k)|^2)^1/2≤ Ce^ϕ(r)A^sU_Ḣ^σ s^2/(t-r)^σ/2.

    
    Similarly, the same bound holds for the corresponding term involving B̂_1, with the condition (<ref>) replaced by 2p/2-pσ s >3, which automatically holds as long as (<ref>) holds since 1< p < 2. 
    
    Therefore we have 
    
    ∫_0^te^ϕ(t)A^s-ν^2/2(t-r)A^2sB(U,U)_Ḣ^σ sdr
               ≤ C∫_0^te^ϕ(r)A^sU_Ḣ^σ s^2/(t-r)^σ/2 dr
       ≤ C∫_0^t1/(t-r)^σ/2 drsup_r∈[0,t]e^ϕ(r)A^sU_Ḣ^σ s^2
       ≤ Ct^1-σ/2sup_r∈[0,t]e^ϕ(r)A^sU_Ḣ^σ s^2,

    which requires σ<2. Thus, the valid ranges of σ and s are s∈(4/5,1] and σ∈(8/5s,2).
    Consequently, for any T∈[0,T_ω], we obtain 
    
    Φ(U)_C_T_ϕ^σ, s≤√(2)e^α A^sU_0_Ḣ^σ s+ √(2)CT^1-σ/2U_C_T_ϕ^σ, s^2.

    
    Choose a radius R≥2√(2)e^α A^sU_0_Ḣ^σ s and denote by B_R(0) the ball in C_T_ϕ^σ, s centered at the origin. Then for  0<T≤ T_ω satisfying √(2)CT^1-σ/2R  ≤1/2, the operator Φ maps B_R(0) to itself.  
    Since the nonlinear term B(U,U) is bilinear, we can use similar estimates as above to obtain
    
    Φ(U)-Φ(V)_C_T_ϕ^σ, s   ≤  CT^1-σ/2(U_C_T_ϕ^σ, s+V_C_T_ϕ^σ, s)U-V_C_T_ϕ^σ, s
       ≤ 2CT^1-σ/2RU-V_C_T_ϕ^σ, s

    for all U,V∈ B_R(0) with U(0)=V(0). As long as T is small enough such that 2CT^1-σ/2R≤1/2, Φ is a contraction on B_R(0), and its unique fixed point corresponds to the unique local solution of (<ref>) for the sample ω. The local solution of (<ref>) starting from U_0 is then obtained for every sample ω in Ω by the arbitrariness of ω∈Ω. 

Recall the definition of Ω_α,β,ν from (<ref>). On Ω_α,β,ν, one has ϕ(t)-ν W_t≥ 0 for all t≥ 0. Thus T_ω=∞ on Ω_α,β,ν. Consequently, the random existence time T can be made uniform for all ω∈Ω_α,β,ν, which means that the existence time T becomes deterministic over Ω_α,β,ν. 


Transforming back to the original system (<ref>) using V = Γ^-1U, we obtain the local well-posedness in the Gevrey class of order 1/s for the stochastic inviscid PE subject to random diffusion. The result is summarized as follows. 

    Let η>0, s∈(4/5,1] and σ∈(8/5s,2). For any V_0∈ H∩_α+η^σ,s, with probability one, there exists a random time T>0 such that (<ref>) has a unique solution V∈ C_T_η^σ,s. 


    Recall that U(t)=e^-ν W_t A^s V(t). Since U_0=V_0, we have U_0∈_α+η^σ,s. By Proposition <ref> we know that with probability one, the solution U to equation (<ref>) exists in C_T_ϕ+η^σ,s up to a random time T. Since ν W_t-ϕ(t)≤ 0 for all t∈[0,T], we have
    
    V(t)__η^σ,s≤√(2)e^η A^se^ν W_t A^sU(t)_Ḣ^σ s= √(2)e^(η+ϕ(t)) A^se^(ν W_t-ϕ(t)) A^sU_Ḣ^σ s≤√(2)U__ϕ+η^σ,s.

    Therefore, V∈ C_T_η^σ,s. 






 §.§ Global existence

We next investigate the global solutions to equation (<ref>) on the sample set Ω_α,β,ν.  To this end, we first derive  energy estimates that allows us to extend the local solutions. 

    Let s∈(4/5,1] and σ∈(8/5s,2). For T>0, assume that system (<ref>) has a unique solution U∈ C_T_ϕ^σ+1,s for all sample paths in Ω_α,β,ν. Then U satisfies
    
    d/dtU__ϕ^σ,s^2+(ν^2-2β-C^*U__ϕ^σ,s)U__ϕ^σ+1,s^2≤ 0,

    where the constant C^*>0 depends only on σ and s. In particular, if U_0__ϕ(0)^σ,s≤ν^2-2β/C^*, then U(t)__ϕ(t)^σ,s is decreasing in time t and
    U(t)__ϕ(t)^σ,s≤U_0__ϕ(0)^σ,s
    for all t∈[0,T].


    From equation (<ref>) and notation (<ref>), one has
    
    1/2d/dtU_L^2^2 = - ν^2/2A^s U_L^2^2

    due to the cancellation ⟨ B(U,U), U⟩=0, and
    
    1/2d/dte^ϕ(t)A^sU_Ḣ^σ s^2 =    βe^ϕ(t)A^s A^s/2U_Ḣ^σ s^2 - ν^2/2e^ϕ(t)A^s A^s U_Ḣ^σ s^2 
       -⟨ e^ϕ(t)A^sB(U,U),e^ϕ(t)A^sA^2σ sU⟩.

    Combining the above two equations together gives
    
    1/2d/dtU__ϕ^σ,s^2 = βU__ϕ^σ+1/2,s^2 - ν^2/2A^s U__ϕ^σ,s^2 
            -⟨ e^ϕ(t)A^sB(U,U),e^ϕ(t)A^sA^2σ sU⟩.

    In view of (<ref>), one has 
    
    |⟨ e^ϕ(t)A^sB(U,U),e^ϕ(t)A^sA^2σ sU⟩|
       ≤∑_k|e^ϕ(t)|k|^sB̂_1(t,k)e^ϕ(t)|k|^s|k|^2σ sÛ_k|+∑_k|e^ϕ(t)|k|^sB̂_2(t,k)e^ϕ(t)|k|^s|k|^2σ sÛ_k|:=B_1+B_2.

    Using the fact that  ϕ(t)-ν W_t≥ 0 for all t≥ 0 and ω∈Ω_α,β,ν, and |k|^s≤ |k-j|^s+|j|^s for 0<s≤ 1, one deduces, by the Cauchy–Schwarz inequality, that  
    
    B_2    = ∑_k|e^(ϕ(t)-ν W_t)|k|^s∑_je^ν W_t|j|^sj'·Û_j/j_3i(k_3-j_3)e^ν W_t|k-j|^sÛ_k-je^ϕ(t)|k|^s|k|^2σ sÛ_k|
        = ∑_k|∑_je^(ϕ(t)-ν W_t)(|k|^s-|k-j|^s-|j|^s)e^ϕ(t)|j|^sj'·Û_j/j_3i(k_3-j_3)e^ϕ(t)|k-j|^sÛ_k-je^ϕ(t)|k|^s|k|^2σ sÛ_k|
       ≤∑_k∑_j|e^ϕ(t)|j|^s|k|^(σ-1)sj'·Û_j/j_3i(k_3-j_3)e^ϕ(t)|k-j|^sÛ_k-je^ϕ(t)|k|^s|k|^(σ+1)sÛ_k|
       ≤e^ϕ(t)A^sU_Ḣ^(σ+1)s(∑_k(∑_j|e^ϕ(t)|j|^s|k|^(σ-1)sj'·Û_j/j_3i(k_3-j_3)e^ϕ(t)|k-j|^sÛ_k-j|)^2)^1/2.

    Using |k|^(σ-1)s≤ C( |k-j|^(σ-1)s+|j|^(σ-1)s) and the Minkowski inequality, the last term in the above inequality is controlled by 
    
    (∑_k(∑_j|e^ϕ(t)|j|^s|k|^(σ-1)sj'·Û_j/j_3i(k_3-j_3)e^ϕ(t)|k-j|^sÛ_k-j|)^2)^1/2
           ≤ C(∑_k(∑_j|e^ϕ(t)|j|^s|j|^(σ-1)sj'·Û_j/j_3(k_3-j_3)e^ϕ(t)|k-j|^sÛ_k-j|)^2)^1/2
               +C(∑_k(∑_j|e^ϕ(t)|j|^sj'·Û_j/j_3|k-j|^(σ-1)s(k_3-j_3)e^ϕ(t)|k-j|^sÛ_k-j|)^2)^1/2
            :=B_21+B_22.

    We first estimate the term B_21. By Young's inequality for convolution with 1/p+1/q=3/2 and 1< p,q< 2, one has
    
    B_21   ≤ C(∑_j_3≠ 0∑_j'≠ 0|e^ϕ(t)|j|^s|j|^(σ-1)s|j'|/|j_3|Û_j|^p)^1/p(∑_j||j|e^ϕ(t)|j|^sÛ_j|^q)^1/q.

Similar to the derivations of estimates (<ref>) and (<ref>),  Hölder inequality gives 

    (∑_j'≠ 0|e^ϕ(t)|j|^s|j|^(σ-1)s|j'|/|j_3|Û_j|^p)^1/p
           ≤   C(∑_j_3≠ 0(∑_j'≠ 0|e^pϕ(r)|j|^s|j|^p(σ+1) s|Û_j|^p|^2/p)^p/2(∑_j'≠ 0| |j_3|^-p |j|^p(1-2s)|^2/2-p)^2-p/2)^1/p
           ≤(∑_j_3≠ 0(∑_j'≠ 0e^2ϕ(r)|j|^s|j|^2(σ+1) s|Û_j|^2)^p/2 |j_3|^-p(∑_j'≠ 0 |j'|^2p(1-2 s)/2-p)^2-p/2)^1/p
           ≤ C e^ϕ(r)A^sU_Ḣ^(σ+1) s,

and
 
    (∑_j||j|e^ϕ(r)|j|^sÛ_j|^q)^1/q≤ Ce^ϕ(r)A^sU_Ḣ^σ s,

    provided that
    
    2p/2-p(2 s-1)>2,  and 2q/2-q(σ s-1)>3.

    The same bound for B_22 can be obtained similarly, provided there exist 1< p,q<2 with 1/p+1/q=3/2 such that
    
    2p/2-p(σ s-1)>2,  and 2q/2-q(2s-1)>3.

    Note that the above two conditions, (<ref>) and (<ref>), are implied by (<ref>) and (<ref>) since σ<2. As a result, the assumptions in Proposition <ref> on σ and s ensure the existence of such p,q, making the above inequalities hold. Thus we obtain 
    
    B_2≤e^ϕ(t)A^sU_Ḣ^(σ+1)s(B_21+B_22)≤ CU__ϕ^σ,sU__ϕ^σ+1,s^2,

    where C depends only on σ and s.
    The same bound also holds for the term B_1 defined in (<ref>) under the same conditions. Indeed, the estimate for B_1 is simpler than that for B_2 as there is no loss of horizontal derivative. 

    The proof is then complete by combining the bounds for B_1 and B_2 with (<ref>) and (<ref>). 




We are now ready to extend the local solutions globally. Note that the fixed point argument provides solutions in C_T_ϕ^σ,s, but we require solutions to be in C_T_ϕ^σ+1 when deriving the energy bound. To extend the local solutions, we apply a trick from <cit.>, taking advantage of Gevrey embeddings. 

    Let s∈(4/5,1] and σ∈(8/5s,2), and fix η>0. For α>0 and 0<β<ν^2/2, assume that the initial condition U_0∈ H∩_α+η^σ,s satisfies
    
    U_0_𝒢_α+η^σ,s<1/C^*(ν^2/2-β),

    where C^* is the constant appearing in Proposition <ref>. Then, system (<ref>) has a unique global solution in C([0, ∞) ; 𝒢_ϕ+η^σ,s) for every sample ω∈Ω_α, β, ν, where ϕ(t)=α+β t. Furthermore, 
    t→U(t)_𝒢_ϕ(t)+η^σ,s
    is decreasing for all t≥ 0.  


    For a given constant η>0, we fix some η'∈ [0,η). We will first show by induction that the solution exists globally in C([0,∞);_ϕ+η'^σ,s). To this end, let us define  
    
    ϕ_0=ϕ+η and ϕ_j=ϕ_j-1-2^-j(η-η')  for  j≥ 1.

    Since U_0∈𝒢_α+η^σ,s, by Proposition <ref>, there exists a deterministic T>0, depending only on U_0__α+η^σ,s and σ, such that system (<ref>) has a unique solution U∈ C([0, T]; 𝒢_ϕ_0^σ,s) for every sample  ω∈Ω_α, β, ν. 
    Our goal is to show that for all j≥ 1, the unique solution U satisfies 
    
    U ∈ C([0, jT]; _ϕ_j^σ,s),       t→U(t)__ϕ_j(t)^σ,s is decreasing,
       and U(t)__ϕ_j(t)^σ,s≤U_0__α+η^σ,s for  t∈[0,jT].

    
    Let us start with j=1.
    Since ϕ_0(t)-ϕ_1(t)=1/2(η-η')=:ϕ_0>0 is independent of time t, it follows that
    
    e^ϕ_1(t_1) A^s U(t_1) - e^ϕ_1(t_2) U(t_2) _Ḣ^(σ+1)s^2 = e^-ϕ_0 A^s(e^ϕ_0(t_1) A^s U(t_1) - e^ϕ_0(t_2) U(t_2)) _Ḣ^(σ+1)s^2 
    
            =    ∑_k |k|^2σ s |k|^2s e^-2ϕ_0 |k|^s| e^ϕ_0(t_1)|k|^sÛ(t_1,k) -  e^ϕ_0(t_2)|k|^sÛ(t_2,k) |^2 
    ≤    C_ϕ_0∑_k |k|^2σ s| e^ϕ_0(t_1)|k|^sÛ(t_1,k) -  e^ϕ_0(t_2)|k|^sÛ(t_2,k) |^2  
    
            =    C_ϕ_0e^ϕ_0(t_1) A^s U(t_1) - e^ϕ_0(t_2) U(t_2) _Ḣ^σ s^2,

    for any t_1, t_2 ∈ [0,T]. Consequently,
    the solution U∈ C([0, T]; 𝒢_ϕ_1^σ+1,s). From Proposition <ref> and the bound on the initial data (<ref>), it follows that U(t)_𝒢_ϕ_1(t)^σ,s is decreasing for t∈[0,T], and 
    
    U(t)_𝒢_ϕ_1(t)^σ,s≤U_0_𝒢_ϕ_1(0)^σ,s≤U_0__α+η^σ,s.

    
    We now iterate this process. Suppose at the j-th step we have (<ref>) holds. Specifically, it holds that U(jT)__ϕ_j(jT)^σ,s≤U_0__α+η^σ,s.
    Then, Proposition <ref> enables us to extend the solution to [jT, (j+1)T] such that U∈ C([0, (j+1)T]; 𝒢_ϕ_j^σ,s). Since ϕ_j(t) - ϕ_j+1(t) =2^-(j+1)(η-η')=:ϕ_j>0, one can follow the procedure similar to the one used for (<ref>) to conclude that
    U∈ C([0, (j+1)T]; 𝒢_ϕ_j+1^σ+1,s). It then follows from Proposition <ref> and the initial assumption (<ref>) that U(t)_𝒢_ϕ_j+1^σ,s is monotonically decreasing in t∈[0,(j+1)T] and 
    
    U(t)_𝒢_ϕ_j+1(t)^σ,s≤U_0_𝒢_ϕ_j+1(0)^σ,s≤U_0__α+η^σ,s,   t∈[0,(j+1)T].

    This shows that (<ref>) holds for the (j+1)-th step, thus completing the induction proof of (<ref>).
The construction of ϕ_j and the induction process give us the unique solution U∈ C([0, ∞); 𝒢_ϕ+η'^σ,s) such that 
    t→U(t)_𝒢_ϕ(t)+η'^σ,s
    is decreasing for all t≥ 0 and sup_t≥ 0U(t)_𝒢_ϕ(t)+η'^σ,s≤U_0__α+η^σ,s.

    The above argument holds for all η'∈[0,η), and the sequence of functions f^η'(t):= U(t)_𝒢_ϕ+η'^σ,s is increasing as η' →η for each fixed t≥ 0, one can apply the monotone convergence theorem to conclude that
    
    U(t)_𝒢_ϕ+η^σ,s=lim_η'→ηU(t)_𝒢_ϕ+η'^σ,s≤U_0__α+η^σ,s,

    which implies that U∈ L^∞(0, ∞; 𝒢_ϕ+η^σ,s). To see the continuity in time of U in 𝒢_ϕ+η^σ,s, we first apply Proposition <ref> to obtain that the solution U∈ C([0, T]; 𝒢_ϕ+η^σ,s) since U_0∈_α+η^σ,s. Thanks to (<ref>), we know that at the time t=T, U(T)_𝒢_ϕ(T)+η^σ,s≤U_0__α+η^σ,s. As the existence time T depends only on the size of the initial condition and ϕ(t) depends linearly on time, we can utilize Proposition <ref> again to conclude that U∈ C([0, 2T]; 𝒢_ϕ+η^σ,s). Repeating this process gives us that U∈ C([0, ∞); 𝒢_ϕ+η^σ,s). 
    
    Finally, the monotonicity of 
    t→U(t)_𝒢_ϕ(t)+η^σ,s
    also follows from the monotone convergence theorem since for t_1≤ t_2, 
    
    U(t_1)_𝒢_ϕ(t_1)+η^σ,s=lim_η'→ηU(t_1)_𝒢_ϕ(t_1)+η'^σ,s≥lim_η'→ηU(t_2)_𝒢_ϕ(t_2)+η'^σ,s=U(t_2)_𝒢_ϕ(t_2)+η^σ,s.



Finally, we are ready to present the following result regarding the original system (<ref>): the global well-posedness of V = Γ^-1U in the Gevrey class of order 1/s. 


   Let s∈(4/5,1] and σ∈(8/5s,2), and fix η>0. For any small ε∈ (0,1), there exists α=α(ε)>0 such that for any initial condition V_0∈ H∩_α+η^σ,s, there exists a unique global solution V∈ C([0, ∞) ; 𝒢_η^σ,s) with probability at least 1-ε, provided that ν is sufficiently large.


    For a given ε∈(0,1), we choose α=-4lnε so that e^-α/4=ε. For the initial condition V_0∈ H∩_α+η^σ,s, we take ν large enough such that ν^2 > 4C^* V_0__α+η^σ,s and β=ν^2/4. Since U_0=V_0, these choices of ν and β ensures that the assumption (<ref>) holds. By Proposition <ref>, we know that system (<ref>) has a unique global solution U∈ C([0, ∞) ; 𝒢_ϕ+η^σ,s) with ϕ(t)=α+β t for every sample ω∈Ω_α, β, ν. Now, one can define V by V = Γ^-1U and follow the proof of Corollary <ref> to obtain the unique global solution V∈ C([0, ∞) ; 𝒢_η^σ,s) for every sample ω∈Ω_α, β, ν, and thus with probability at least ℙ(Ω_α, β, ν) ≥ 1-e^-αβ/ν^2=1-ε.





§ RANDOM DAMPING

This section is dedicated to the analysis of the stochastic inviscid PE subject to random damping, i.e., the term F in (<ref>) takes the form F(V)=ν V. Hence, the equation reads 

    d V + 𝒫Q(V,V)dt =  ν V dW,      V(0)=V_0.

Recall Γ = e^-ν W_t |∇|^s and U = Γ V as defined in Section <ref>. Since s =0 when considering random damping, Γ now takes the form Γ(t) = e^-ν W_t and is independent of spatial variables. This gives a random PDE for U,

    ∂_tU + Γ^-1𝒫Q(U,U) =  -1/2ν^2 U,      U(0)=U_0.

In what follows we will focus on this equation. Once we obtain solutions U of (<ref>), we can obtain solutions V=e^ν WU of (<ref>) with the same regularity as U. Since there is no dissipation, the contraction mapping argument used in Section <ref> based on the mild formulation is no longer suitable. Moreover, system (<ref>) is still ill-posed in Gevrey class of order strictly greater than 1 due to the lack of dissipation <cit.>. Thus, one has to work in the analytic class, i.e., _ϕ^σ,s with s=1.

We shall solve the random equation (<ref>) in a pathwise manner on [0,T] for any fixed T>0. For each fixed ω∈Ω, the sample path t→ W_t(ω) is continuous, thus  

    M_ω:=sup_t∈[0,T]Γ^-1(t)<∞.
 
Let the analytic radius ϕ be such that ϕ(t)> 0 and ϕ'(t)≤ 0 for all t∈[0,T]. By using a Galerkin approximation scheme, one can obtain the local existence result as summarized in Proposition <ref> below. To this end, we first recall estimates on the nonlinear term (cf. <cit.>): for any ϕ≥ 0 and σ>2,

    |⟨ e^ϕ AA^σQ(U,U),e^ϕ AA^σU⟩|_L^2≤ C_σU__ϕ^σ,1U^2__ϕ^σ+1/2,1,

for any U∈ H∩_ϕ^σ+1/2,1.  


    Let σ>5/2 and ϕ_0>0 be constants. For any U_0∈_ϕ_0^σ, 1 and ω∈Ω, there exists a function ϕ∈ C^1(ℝ) (depending on ω) with ϕ(0)=ϕ_0, and a time 𝒯≤ T depending on U_0__ϕ_0^σ, 1, ϕ_0, and M_ω, such that there is a unique solution U to (<ref>) satisfying
    
    U∈ C([0,𝒯];_ϕ^σ,1)∩ L^2(0,𝒯;_ϕ^σ+1/2,1).
 
 

    The proof follows similarly to <cit.>, and we highlight two main differences. First, the additional damping term does not introduce any complications. Second, while <cit.> established that  U∈ L^∞(0,𝒯;_ϕ^σ,1)∩ L^2(0,𝒯;_ϕ^σ+1/2,1), we will prove here that U∈ C([0,𝒯];_ϕ^σ,1). From Lemma <ref>, the definition of the nonlinear term Q(U,U), and the fact that σ>5/2, one can infer that 
    
    e^ϕ A A^σ-1/2𝒫 Q(U,U)_L^2 = sup_ϕ_L^2 =1⟨ e^ϕ A A^σ-1/2𝒫 Q(U,U), ϕ⟩≤ C_σU__ϕ^σ,1U__ϕ^σ+1/2,1 .

    Since ∂_t ( e^ϕ A A^σ U ) = ϕ̇e^ϕ A A^σ+1 U + e^ϕ A A^σ∂_t U, using (<ref>) and (<ref>), we get
    
    ∂_t (e^ϕ A A^σ U) _H^-1/2≤    C (e^ϕ A A^σ+1 U _H^-1/2 +  e^ϕ A A^σ𝒫 Q(U,U)_H^-1/2 + e^ϕ A A^σ U _H^-1/2)
           
    ≤    C (U__ϕ^σ+1/2,1 +  U__ϕ^σ,1U__ϕ^σ+1/2,1).

     This implies that ∂_t ( e^ϕ A A^σ U ) ∈ L^2(0,𝒯; H^-1/2). As e^ϕ A A^σ U ∈ L^2(0,𝒯; H^1/2) and H^-1/2 is the dual space of H^1/2, by the Lions–Magenes lemma, we infer that e^ϕ A A^σ U ∈ C([0,𝒯]; L^2), and thus U∈ C([0,𝒯];_ϕ^σ,1).



We next show that by appropriately choosing the analytic radius and utilizing the damping term, a global solution exists for small initial data with quantifiable high probability. Let α>0, 0<β<ν^2/2, and recall the sample set Ω_α,β,ν defined in (<ref>), which has probability 
 ℙ(Ω_α,β,ν)≥ 1-e^-αβ/ν^2. 


    Let σ>5/2 and ϕ_0>0. For the constant C_σ appearing in (<ref>), assume that
    
    (ν^2-2β)ϕ_0>4C_σ.

    For any ω∈Ω_α,β,ν and initial data U_0 satisfying 
    
    e^αU_0__ϕ_0^σ, 1≤(ν^2-2β)ϕ_0/4C_σ-1,

    there exists a unique global solution U∈ C([0,∞);_ϕ^σ,1) to system (<ref>) with analytic radius 
    
    ϕ=ϕ(t)=ϕ_0-4C_σ/ν^2-2β(e^αV_0__ϕ_0^σ, 1+1)(1-e^-(ν^2/2-β)t).



 
    From equation (<ref>), one has 
    
    1/2d/dtU__ϕ^σ,1^2 + ν^2/2U__ϕ^σ,1^2
                   = ϕ'(t)U__ϕ^σ+1/2,1^2+Γ^-1⟨ e^ϕ(t) AA^σQ(U,U),e^ϕ(t) AA^σU⟩
       ≤(ϕ'(t)+C_σΓ^-1U__ϕ^σ,1)U__ϕ^σ+1/2,1^2

    by the nonlinear estimate (<ref>). For ω∈Ω_ν,α,β, we have Γ^-1(t)=e^ν W_t≤ e^α+β t for all t. Thus, we derive that
    
    1/2d/dtU__ϕ^σ,1^2    + C_σ e^-(ν^2/2-β)tU__ϕ^σ+1/2,1^2+ν^2/2U__ϕ^σ,1^2
       ≤(ϕ'(t)+C_σΓ^-1U__ϕ^σ,1+C_σ e^-(ν^2/2-β)t)U__ϕ^σ+1/2,1^2
       ≤(ϕ'(t)+C_σ e^-(ν^2/2-β)t( e^α+ν^2/2tU__ϕ^σ,1+1))U__ϕ^σ+1/2,1^2
       =C_σe^-(ν^2/2-β)t(e^α+ν^2/2tU__ϕ^σ,1-2e^αU_0__ϕ_0^σ, 1-1)U__ϕ^σ+1/2,1^2.

    
    Define the stopping time T^∗
    
    T^* = min{t≥ 0: e^α+ν^2/2tU__ϕ(t)^σ,1-2e^αU_0__ϕ_0^σ, 1-1 ≥ 0 }.

    By time continuity, we know that T^*>0. We will prove that T^∗ = ∞ by contradiction. Assume that T^*<∞, then we know that 
    
    e^α+ν^2/2T^*U__ϕ(T^*)^σ,1-2e^αU_0__ϕ_0^σ, 1-1 = 0,

    and for t∈ [0, T^*] we have
    
    1/2d/dtU__ϕ^σ,1^2 + C_σ e^-(ν^2/2-β)tU__ϕ^σ+1/2,1^2+ν^2/2U__ϕ^σ,1^2 ≤ 0.

    Then one can apply the Gronwall inequality to conclude that
    
    e^α+ν^2/2tU__ϕ(t)^σ,1≤ e^αU_0__ϕ_0^σ, 1,

    for t∈[0,T^*]. In particular, the above holds for t=T^*, which leads to a contradiction to (<ref>). Consequently, T^*=∞. Thanks to condition (<ref>), one can infer that ϕ(t) >0 for all t≥ 0, thus the analytic solution U exists globally. 


Finally, the global existence of V with high probability will be established using the construction V = Γ^-1U. This result is summarized below.


    Let σ>5/2 and ϕ_0>0. For any small ε∈ (0,1) and initial condition V_0∈_ϕ_0^σ,1, there exists, with probability at least 1-ε, a unique global solution V∈ C([0,∞);_ϕ^σ,1) to system (<ref>) with analytic radius ϕ(t) defined in (<ref>), provided that the damping intensity ν is sufficiently large.


    For any ε∈(0,1) and V_0∈_ϕ_0^σ,1, set α=-4lnε so that e^-α/4=ε. Choosing β=ν^2/4 and ν sufficiently large such that such that 
    
    ν^2 ≥8C_σ/ϕ_0(e^αV_0__ϕ_0^σ,1 +1) = 8C_σ/ϕ_0(ε^-4V_0__ϕ_0^σ,1 +1),

    then both conditions (<ref>) and (<ref>) are satisfied. Since U_0=V_0, Proposition <ref> ensures that there exists a unique global solution U∈ C([0,∞);_ϕ^σ,1) to system (<ref>) with analytic radius ϕ(t) defined in (<ref>) for any ω∈Ω_α,β,ν. Consequently, using V=e^ν W_t U, one obtains a unique global solution V∈ C([0,∞);_ϕ^σ,1) to system (<ref>) for any ω∈Ω_α,β,ν, with probability at least 1-e^-αβ/ν^2 = 1-ε.



    Compared to the random diffusion case (Theorem <ref>), here we do not need to assume the initial analytic radius to be large. 


    By taking R=e^α, the event Ω_α,β,ν corresponds to the assertion that the geometric Brownian motion exp(ν W_t-β t) never exceeds R, as in <cit.>. This probability can be made arbitrarily close to 1 by choosing a sufficiently large α. 


 




§ CONCLUSION

It is well-known that the deterministic inviscid PE is ill-posed in Sobolev spaces and Gevrey class of order strictly greater than 1, and some analytic solutions can form singularities in finite time. Therefore, only local well-posedness in the analytic class is sought. In this work, we investigated the regularization effect of noise on the inviscid PE. 
We prove that appropriate random diffusion can restore local well-posedness in certain Gevrey classes with probability one. Moreover, global solutions can be obtained with high probability when the diffusion intensity and the Gevrey radius for the initial condition are sufficiently large. Additionaly, we establish that random damping can regularize the system, meaning that analytic solutions exist globally in time with high probability, provided that the damping intensity is sufficiently large.

It is interesting to observe how different types of random noise can regularize the system in various ways.
Random diffusion allows us to obtain solutions with a non-decreasing Gevrey radius. However, to achieve global solutions with high probability, we need to impose a sufficiently smooth condition on the initial data, specifically by ensuring it has a large enough Gevrey radius. On the other hand, in the case of random damping, we exploit the dissipation from a decreasing analytic radius to handle the nonlinearity, leading to solutions with a decreasing analytic radius. Nonetheless, a large damping intensity can slow the rate at which the analytic radius decreases, thereby enabling global solutions with high probability. 

It is intriguing to explore whether appropriate types of noise could restore local well-posedness for the inviscid PE in Sobolev spaces and Gevrey class of order ≤4/5. We leave this investigation as a topic for future work.




§ ACKNOWLEDGMENTS

Q.L. was partially supported by an AMS-Simons Travel Grant. R.H. was partially supported by a grant from the Simons Foundation (MP-TSM-00002783).



§ NONLINEAR ESTIMATES


    Given smooth periodic functions f, g, and h such that f and g have zero means over 𝕋^3, the following inequality holds for any r≥ 0, ϕ≥ 0, and η>0:
    
    |⟨ e^ϕ A A^r  (fg),   h  ⟩| ≤  C_r( e^ϕ A A^r  fe^ϕ A A^3/2+η  g  + e^ϕ A A^3/2+η  fe^ϕ A A^r  g)   h,

where C_r is a constant depending on r.


    We start with writing the Fourier representations of f, g and h:
    
    -.8in
         f(x) = ∑_j∈  2πℤ^3f̂_j e^ ij· x,   
        g(x) = ∑_k∈ 2πℤ^3ĝ_k e^ ik· x,   
        h(x) = ∑_l∈ 2πℤ^3ĥ_l e^ il· x.

    Then 
    
    |⟨ e^ϕ A A^r  (fg),  h  ⟩| = |⟨ fg,e^ϕ A A^r  h  ⟩| ≤∑_j+k+l=0 |f̂_j||ĝ_k||l|^r e^ϕ |l| |ĥ_l|.

Since |l| = |j+k| ≤ |j|+|k|, we have 
|l|^r≤ (|j|+|k|)^r≤ C_r(|j|^r + |k|^r)
and 
e^ϕ |l|≤ e^ϕ |j| e^ϕ |k|,
thus

    |⟨ A^r  (fg),  h  ⟩| ≤∑_j+k+l=0 C_r (|j|^r+|k|^r)e^ϕ |j||f̂_j| e^ϕ |k||ĝ_k||ĥ_l| := A_1 + A_2.

Regarding the term A_1, with g having zero mean, using  the Cauchy–Schwarz inequality, we deduce 

    A_1      = ∑_j+k+l=0 C_r |j|^r  e^ϕ |j| |f̂_j| e^ϕ |k||ĝ_k| |ĥ_l| 
    = C_r ∑_k∈ 2πℤ^3 
     k≠ 0  e^ϕ |k||ĝ_k|  ∑_j∈ 2πℤ^3 
     j≠ 0 |j|^r e^ϕ |j| |f̂_j|  |ĥ_-j-k| 
          ≤ C_r ( ∑_k∈ 2πℤ^3 
     k≠ 0 |k|^-3-2η)^1/2( ∑_k∈ 2πℤ^3 
     k≠ 0 |k|^3+2η  e^2ϕ |k| |ĝ_k|^2)^1/2
          1in ×sup_k∈ 2πℤ^3( ∑_j∈ 2πℤ^3 
     j≠ 0 |j|^2r e^2ϕ |j| |f̂_j|^2)^1/2( ∑_j∈ 2πℤ^3 
     j≠ 0 |ĥ_-j-k|^2)^1/2
          ≤ C_r e^ϕ A A^r  fe^ϕ A A^3/2+η  g  h,

for any η>0. 

For A_2, since f has zero mean, similarly we have

    -.8in 
    A_2 = ∑_j+k+l=0 C_r |k|^2r e^ϕ |j||f̂_j| e^ϕ |k||ĝ_k| |ĥ_l| 
    ≤ C_r e^ϕ A A^3/2+η  fe^ϕ A A^r  gh.






plain



 Paper Title: A non-autonomous p-Adic diffusion equation on time changing graphs.



Operator-based semantics for choice programs: is choosing losing? (full version)
    
Jesse Heyninck


Open Universiteit, the Netherlands

University of Cape Town, South-Africa

    August 5, 2024
================================================================================================



§ ABSTRACT 
Motivated by the recently proven presence of ultrametricity in physical models (certain spin glasses) and the very recent study of Turing patterns on locally ultrametric state spaces, first non-autonomous diffusion operators on such spaces, where finitely many compact p-adic spaces are joined by a graph structure, are studied, including their Dirichlet and van Neumann eigenvalues. Secondly, the Cauchy problem for the heat equation associated with these operators is solved, its solution approximated by Trotter products, and thirdly, the corresponding Feller property as well as the Markov property (a Hunt process) is established. 






§ INTRODUCTION


The Gibbs measure associated with certain mean-field spin glass models, including the Sherrington-Kirkpatrick model,  was conjectured to  have ultrametric support in the thermodynamic limit
<cit.>. This conjecture, known as the Parisi ultrametricity conjecture, was proved by Panchenko  under the condition that the so-called Ghirlanda-Guerra identities are satisfied <cit.>.
This conjecture (and now result) has motivated up to this day the study of random processes on ultrametric state spaces, the most studied type being  diffusion on the p-adic numbers, cf. e.g. <cit.> and many other research articles in the physical and mathematical literature.


Ultrametricity of a space means that it has a hierarchical structure, i.e. its points can be viewed as the boundary of a tree, and therefore its topology, as well as any physical process on it, are determined by this tree structure. The p-adic numbers form a convenient mathematical framework for studying ultrametric spaces by viewing those trees as part of the regular tree structure of the p-adic numbers—as long as the  vertex degrees do not exceed the bound given by the p-adic regularity.
However, if the state space is only locally ultrametric, then it seemed unclear how to mathematically model diffusion on such structures until Zúñiga-Galindo's idea of modelling interactions between those local ultrametric pieces appeared via an externally imposed graph structure on this ensemble of pieces which then can be viewed as disjoint compact open subsets of the p-adic number field ℚ_p <cit.>. He even argues that p-adic analysis is a natural tool for studying Turing patterns on mean-field models.
In his case, these local pieces were modelled as p-adic balls, and thus he was lead to define a p-adic Laplacian integral operator resembling the integral operator representation of the well-studied Vladimirov (or Vladimirov-Taibleson) operator, cf. <cit.>.


The study of non-autonomous diffusion on p-adic domains seems to have been given significantly less consideration in the literature.
This motivates the authors of this article to investigate diffusion processes on locally hierarchical structures whose interactions (given as weighted edges) are time-dependent. The vertices of the underlying time-dependent graph are fixed in this setting. After clarifying the construction of p-adic operators associated with finite graphs—the “dictionary” set up in <cit.> needs a correction—properties of a time-dependent version of the Zúñiga operators from <cit.> are studied, proving that the uniform continuity of the kernel function  implies the strong continuity of the new time-dependent operator (Theorem <ref>) which is necessary for proving the Feller property later. 
Dirichlet and van Neumann eigenvalues  are calculated (Theorem <ref>), whereby the corresponding boundary conditions are generalised from graph theory. Thereafter,
the non-autonomous Cauchy problem for  the heat equation associated with these operators is solved (Theorem <ref>), thereby approximating the solution using Trotter products, and then finding that the temporal discretisation error is in first order O(1/ne^1/n) (Corollary <ref>). In the end the Feller property (Theorem <ref>) and corresponding Markov property—a Hunt process—is established (Theorem <ref>).  






§ NON-AUTONOMOUS P-ADIC EQUATIONS FOR TIME-CHANGING GRAPHS





 §.§ Preliminaries


In order to fix notation, denote the field of p-adic numbers as ℚ_p. The usual p-adic absolute value on ℚ_p is written as ·_p. The additive group of ℚ_p being a locally compact abelian group means that it has a Haar measure, denoted as dx (if the current variable is x). It is invariant under translation by adding any fixed element of ℚ_p, used for integrating functions, and is normalised here by setting

    ∫_ℤ_pdx=1

where ℤ_p is the subring of ℚ_p consisting of elements with p-adic absolute value at most 1, i.e. ℤ_p is the p-adic unit ball.


    
    For each t∈ [0,∞), we take a matrix A(t)∈ℂ^n× n. If for every t∈ [0,∞) we assume that A(t) is a symmetric matrix, then we can view this matrix as an adjacency matrix of a weighted graph 𝒢_t with n nodes in a natural way. For every t, the vertex set of 𝒢_t can be embedded into ℚ_p. The fact that 𝒢_t has an unchanging set of nodes allows to fix an embedding for all t>0. In other words the only difference between 𝒢_t and 𝒢_s for s≠ t are the edge weights. The set of vertices being constant, allows to write: V(𝒢):=V(𝒢_t), for any t>0. 
Following <cit.>,    
    we fix the embedding 

    V(𝒢)→ℚ_p,

and fix N such that each vertex I∈ V(G) lies in a distinct p-adic ball of radius r=p^-N

    B_r(I)={x∈ℚ_p: |x-I|≤ r}

sufficiently small such that all these balls are disjoint in ℚ_p. Let 

    K_N =_I∈ V(𝒢)B_p^-N(I).

Each vertex of 𝒢 can be assumed to have the form 

    I=I_0+I_1 p+...+I_N-1p^N-1,

where each I_i∈{0,…,p-1} for i=0,1,2,… is a p-adic digit.




 §.§ Basis of L^2(K_N,ℂ)


We are interested in the Hilbert space L^2(K_N,ℂ) with inner product  

    ⟨ f, g ⟩ =∫_K_N f(x)  g(x)   dx

Let Ω(p^r|x-a|_p) denote the indicator function of the ball B_-r(a). If r=N and a=I∈ V(G), we write 

    Ω(p^N|x-I|_p)=:Ω_I,N(x)

The space of functions X_N=_ℂ{Ω_I,N(x)}≅ℂ^n is an n-domensional vector space and it will be denoted by X_N. 


The functions 

    Ψ_j,n,r(x)=p^r/2χ_p(p^-r-1jx)Ω(p^r|x-p^-rn|_p)

where r∈ℤ, j∈{1,...,p-1}, and n runs through a fixed set of representatives of ℚ_p / ℤ_p are called Kozyrev functions.  
ion. If Supp (Ψ_j,n,r)=B_-r(p^-rn) ⊂ B_-N(I), then |p^-rn|_p=|I|_p, with N≤ r. 

Proof. We have that p^-r≤ p^-N, and 

    |p^-rn-I|_p<p^-N |p^-N-rn-p^-NI|_p<1,

where p^-NI∈ℚ_p/ℤ_p, and therefore |p^-NI|_p>1. Hence, if p^-N-rn≠ p^-NI, we have

    1>|p^-N-rn-p^-NI|_p= max{|p^-N-rn|_p,|p^-NI|_p}>1.

This shows that we must have |p^-N-rn|_p=|p^-NI|_p, implying the assertion. 

Therefore an orthonormal basis of 

    ℒ_0^2(K_N)=f∈ L^2(K_N)|∫_K_Nf(x) dx=0

is the set of functions of the form 

    Ψ_j,I,r(x)=p^r/2χ_p(p^-r-1jx)Ω(p^r|x-I|_p),
 with r≥ N, j=0,...p-1 and I∈ V(𝒢). 




We have L^2(K_N,ℂ)=X_N ⊕ℒ_0^2(K_N).


Proof. 
This is shown in <cit.>, where orthogonality uses the well-known result:






    ⟨Ω_I,N,Ψ_j,J,r⟩=0,

for every r≥ N, j=0,...,p-1 and I,J∈ V(𝒢), cf. e.g. <cit.> or <cit.>. 

By the above proposition, the set of functions consisting of the functions p^-N/2Ω_I,N(x) and the Kozyrev functions of the form (<ref>) form an orthonormal basis of L^2(K_N,ℂ). 




 §.§ Two operators over L^2(K_N,ℂ)

For every matrix A∈ℂ^n× n we can define a bounded linear operator in the following way. We define 

    A(x,y)=p^N∑_I,J∈ V(𝒢)A_I,JΩ_I,N(x)Ω_J,N(y).

This function is a test function on K_N × K_N, and hence, A(x,y)∈ L^2(K_N× K_N, ℂ). This function gives rise to an integral operator in the usual way: 

    𝒜 u(x)=∫_K_N A(x,y)u(y)dy.

This operator is a bounded linear operator from L^2(K_N,ℂ) to itself. 

Let A∈ℂ^n× n and 𝒜 its integral operator of the form (<ref>). Then X_N^⊥⊂ Ker(𝒜), where X_N^⊥ is the orthogonal complement of X_N.
  
Proof. Note that 

    𝒜u(x)=p^N∑_I,J∈ V(𝒢) A_I,J⟨ u,Ω_J,N⟩Ω_I,N(x)∈ X_N.

If u∈ X_N^⊥, then ⟨ u, Ω_I,N⟩=0, therefore by the above equality 𝒜u=0.   

For every matrix A∈ℂ^n× n, and every u∈ℒ_0^2(K_N), 𝒜u=0. In particular the set Kozyrev functions of the form (<ref>) are contained in Ker (𝒜). 

Proof. By Proposition <ref> and equality (<ref>), the result follows. 

Recall that the Laplacian matrix L is defined as 

    L=A-(γ_Iδ_IJ)

where A is the adjacency matrix of a graph 𝒢, γ_I=∑_JA_I,J is the degree of the vertex I, and δ_I,J is the identity matrix.  
We can associate to L an operator of the form (<ref>) denoted by ℒ. By Corollary <ref>,
the operator ℒ only "acts" on the space X_N, that is the operator ℒ
defined
similarly as in (<ref>):

    ℒu(x)=∫_K_NL(x,y)u(y) dy

with

    L(x,y)=p^N∑_I,J∈ V(𝒢)(A_IJ-γ_Iδ_IJ)Ω_I,N(x)Ω_J,N(x)

is just a rewrite of the Laplacian matrix in a space of dimension n, namely X_N≅ℂ^n and does not give an outcome of functions in ℒ_0^2(K_N). In particular, the Kozyrev functions of the form (<ref>) are not associated to any non zero eigenvalue of the operator ℒ. Therefore the operator ℒ is not an adequate candidate for the p-adic Laplacian, because it gives the same information as the Laplacian matrix, but not more. 


This is important in order to define an analogous operator to the Laplacian operator in the p-adic case. In <cit.>  it is stated that the operator ℒ coincides with the operator defined by the integral

    𝕃u(x)=∫_K_NA(x,y) (u(y)-u(x))dy

in the space of test functions, but this is not true as the following computations shows: 
Note that

    𝕃u(x)
       =∫_K_NA(x,y) (u(y)-u(x))dy
    
       =p^N∑_I,JA_I,J⟨ u,Ω_J,N⟩Ω_I,N(x)-u(x)γ(x),

where

    γ(x)=∫_K_NA(x,y)dy=∑_I,JA_I,JΩ_I,N(x).

Then let f=Ω(p^r|x-a|_p) be the indicator function of a ball B_-r(a)⊂ B_-N(J_0), for some J_0∈ V(𝒢). Then 

    ℒf(x)   =p^N∑_I,JL_I,J⟨ f, Ω_J,N⟩Ω_I,N(x)
       =p^N∑_I,JL_I,Jp^-rδ_J_0,JΩ_I,N(x)
       =p^N-r∑_IL_I,J_0Ω_I,N(x)
       =p^N-r∑_IA_I,J_0Ω_I,N(x)-p^N-rγ_J_0Ω_J_0,N(x),

On the other hand, we have 

    𝕃f(x)   =∫_K_NA(x,y) (f(y)-f(x))dy
            
       =p^N∑_I,JA_I,J⟨ f(y),Ω_J,N(y)⟩Ω_I,N(x)-f(x)γ(x),
       =p^N∑_I,JA_I,Jp^-rδ_J_0,JΩ_I,N(x)-f(x)γ(x)
       =p^N-r∑_IA_I,J_0Ω_I,N(x)-Ω(p^r|x-a|_p)γ(x),

If B_-r(a)⊂ B_-N(J_0) is a proper inclusion, then we can take an x_0∈ B_-N(J_0) ∖ B_-r(a). Then

    ℒf(x_0)=-p^N-rγ_J_0,

and

    𝕃f(x_0)=0.

Here we supposed that A_I,I=0 for all I, i.e. that 𝒢 is a simple graph. It is clear that, if r=N and a=I, then equality holds, that is, the two operators coincide in the space X_N. Moreover, since  ℒ and 𝒜 (the operator associated to the adjacency matrix of 𝒢) are zero for all u∈ X_N^⊥, we have for such u that

    ℒu(x)=0,

and

    𝕃u(x)=𝒜u(x)-u(x)γ(x)=-u(x)γ(x).

There is another fundamental difference between these two operators, as the following results shows. 



The operator ℒ is compact.


Proof. It is clear that ℒ is a Hilbert-Schmidt operator, since it is zero for all the Kozyrev functions supported in K_N. It is known that all Hilbert-Schmidt operators are compact. 


The operator 𝕃:L^2(K_N,ℂ)→ L^2(K_N,ℂ) defined by

    𝕃u(x)=𝒜u(x)-u(x)γ(x),

where 

    γ(x)=∫_K_NA(x,y)dy=∑_I∈ V(𝒢)γ_I Ω_I,N(x),

is a bounded linear operator, and all the Kozyrev functions of the form (<ref>) are eigenfunctions.


Proof. 
In <cit.> it is shown that 𝕃 is compact. The eigenfunction property and corresponding eigenvalues of the Kozyrev wavelets are shown in <cit.>.
Notice that the claim in loc. cit. about
𝕃 being compact is false, cf. Corollary <ref> below.








𝕃 is not compact.



This is an immediate consequence of Proposition <ref>.


Notice that the non-compactness of 𝕃 was also observed in
<cit.>.



The operator 𝒜 is called the p-adic Zúñiga operator associated with the finite graph 𝒢, and
𝕃  is called the corresponding p-adic Zúñiga Laplacian. 





 §.§ Non-autonomous p-adic Zúñiga operators

For a time-changing graph 𝒢_t we define its p-adic Laplacian as the function t↦𝕃(t), where

    𝕃(t)=𝒜(t)-Γ(t)

where

    𝒜(t)u(x)=∫_K_NA(x,y,t)u(y)dy,

and 

    Γ(t)u(x)=u(x)γ(t)(x),

where the kernel function A(x,y,t) and the degree function γ(t)(x) are given by 

    A(x,y,t)=p^N∑_I,J∈ V(𝒢) A_IJ(t)Ω_I,N(x)Ω_J,N(y),

and

    γ(t)(x)=∑_I∈ V(𝒢)γ_I(t)Ω_I,N(x)

with 
γ_I(t)=∑_J∈ V(𝒢)A_IJ(t).


The operators 𝒜(t) and 𝕃(t) are called non-autonomous p-adic Zúñiga operators. Operator 𝕃(t) is also called non-autonomous p-adic Zúñiga Laplacian. They are associated with the time-changing graph 𝒢_t.



If t↦ A_I,J(t) is uniformly continuous for each I,J∈ V(𝒢) then the p-adic Laplacian t↦𝕃(t) of 𝒢_t  is strongly continuous, that is t↦𝕃(t)u is a continuous function from [0,∞) to L^2(K_N,ℂ) for every u∈ L^2(K_N,ℂ).

Proof. Let u∈ L^2(K_N,ℂ)  then

    ||𝒜(t)u-𝒜(s)u||_L^2^2=∫_K_N| ∫_K_N(A(x,y,t)-A(x,y,s))u(y)dy|^2dx.

For ε>0, there is a positive real number δ>0 such that 

    |t-s|<δ|A_IJ(t)-A_IJ(s)|<ε/n^2.

Hence

    |A(x,y,t)-A(x,y,s)|   ≤∑_IJ|A_IJ(t)-A_IJ(t)|
       ≤ε.

Therefore

    ||𝒜(t)u-𝒜(s)u||_L^2^2   ≤∫_K_N(∫_K_N|A(x,y,t)-A(x,y,s)||u(y)|dy)^2dx
       ≤ε^2 ∫_K_N(∫_K_N|u(y)|dy)^2dx
       ≤ε^2(K_N)||u||_L^2^2,

where the last inequality is due to Hölder's inequality. 
On the other hand, 

    |γ(t)(x)-γ(s)(x)|   =|∑_I∈
    V(ℐ)(γ_I(t)-γ_I(s))Ω_I,N(x)|
    
       ≤∑_I|∑_JA_IJ(t)-A_IJ(s)|
    
       ≤ε

Hence, 

    ||Γ(t)u-Γ(s) u||_L^2^2   ≤∫_K_N|γ(t)(x)-γ(t)(x)|^2|u(y)|^2dx
       ≤ε^2 ||u||_L^2^2

Finally, we have 

    ||𝕃(t)u-𝕃(s)u||_L^2   ≤ ||𝒜(t)u-𝒜(s)u||_L^2+||Γ(t)u-Γ(s) u||_L^2
       ≤ε||u||_L^2((K_N)^1/2+1)

 This ends the proof. In particular the proof of Theorem <ref> shows that t↦𝕃(t) is continuous in the uniform operator topology. 


For each t>0, 𝕃(t) is a generator of a contraction semigroup {e^s𝕃(t)}, associated to the Cauchy problem of the heat equation: 

    ∂ u/∂ s (s,x)=ℒ(t)u(s,x)
    
          u(0,x)=u_0(x)∈ L^2(K_n,ℂ)
         .

with t≥ 0. 
Now we present the initial value problem called the evolution system or the non autonomous abstract Cauchy problem

    ∂ u/∂ t (t,x)=𝕃(t)u(t,x)
    
          u(s,x)=u_0(x)∈ L^2(K_n,ℂ)
         .

In the next section we present some results of the theory of semigroups and non autonomous differential equations. 




 §.§ Dirichlet and von Neumann eigenvalues


Assume that the kernel function A(x,y,t) is symmetric in (x,y), and here also that it is an analytic function for t≥0. Define for fixed t≥0 the p-adic edge set of 𝒢(t) as:

    Ω_E(t)=(x,y)∈ K_N^2| A(x,y,t)≠ 0,

as well as the p-adic vertex boundary:

    δ_NS(t)=
    x∈ K_N∖ S|∃ y∈ S A(x,y,t)≠0

as well as the p-adic edge boundary of 𝒢(t):

    ∂_NS(t)=(x,y)∈Ω_E(t)| x∈ S∧ y∉ S

which could also be called the vertex and edge boundaries of the operator 𝒜(t) defined in Section <ref>. Further, define the compact open set

    Ω_S^*(t)=(Ω_E(t)∩ S× S)∪∂_NS(t)⊂ℚ_p^2

Now, let S⊂ K_N be a compact open subset. Then define:

    V(S):=I∈ V| B_p^-N(I)∩ S≠∅.

The space of functions satisfying the Dirichlet boundary condition for S is

    D_S^*(t)=
    f∈ L^2(K_N)|f(x)=0 for all x∈δ_NS(t),

and the space of functions satisfying the von Neumann boundary condition for S is

    N_S^*(t)=   
       
    f∈ L^2(K_N)|∀ x∈δ_NS(t)∫_(x,y)∈∂_NS(t)A(x,y,t)(f(x)-f(y)) dy=0
    ,

again with fixed t≥0.


In view of <cit.>, the von Neumann boundary condition can be formulated as

    D_∂ S(t),+f(x)=0

where 

    D_∂ S(t),+u(x)=
    ∫_(x,y)∈∂_NS(t)
    A(x,y,t)(f(x)-f(y)) dy

is an advection operator defined similarly as D_+^α from <cit.>, and imitating the normal derivative w.r.t. the boundary of S in the classical case of Riemann manifolds.


The first Dirichlet eigenvalue of 𝕃(t) (with respect to S) is defined as:

    λ_1,𝕃^(D)(t)
    =inf_f≠ 0 f∈ D_S^*(t)p^N∫_Ω_S^*(t)A(x,y,t)f(x)-f(y)^2 dx dy/∫_Sf(x)^2γ(x)(t) dx

The von Neumann eigenvalue of 𝕃(t) with respect to S is defined as:

    λ_S,𝕃^(N)(t)
    =inf_f≠ 0∫_Sf(x)γ(x) dx=0p^N∫_Ω_S^*(t)A(x,y,t)f(x)-f(y)^2 dx dy/∫_Sf(x)^2γ(x)(t) dx

where it is assumed that δ_NS(t)≠∅ for this instance of t≥0.


The Dirichlet and von Neumann eigenvalues of 𝒢(t) w.r.t. S will be denoted as

    λ_1,𝒢^(D)(t)  and  λ_S,𝒢^(N)(t),

respectively.



Let ψ S→ℂ be a Kozyrev wavelet. Then ψ∈ N_S^*(t), i.e. ψ satisfies the von Neumann boundary condition.



Assume that the Kozyrev wavelet ψ is supported in a ball B⊂ B_p^-N(I) for some vertex I∈ V(S). Then

    ∫_(x,y)∈∂_NS(t)   A(x,y,t)(ψ(x)-ψ(y)) dy
    
       =∫_(x,y)∈∂_NS(t)A(x,y,t) dy ψ(x)
    
       -∫_(x,y)∈∂_NS(t)A(x,y,t)ψ(y) dy
    
       =0-0=0

where the left term vanishes, because x∈δ_NS(t), and
the right integral vanishes, because ψ is a Kozyrev wavelet, and   from Kozyrev's theorem that

    ∫_K_Nψ(x) dx=0,

cf. e.g. <cit.> or <cit.>.



Let S⊆ K_N be compact open, and assume that
δ S(t) is non-empty for t∈ [0,a) with a>0. Then the following statements hold true:


  * λ_1,𝕃^(D)(t)≤minλ_1,𝒢^(D)(t),γ̂_S(t)≤1,

  * λ_S,𝕃^(N)(t)≤minλ_S,𝒢^(N)(t),γ̂_S(t),

where

    γ̂_S(t)=min_I∈ V(S)_∂ V(S)(t)(I)/∫_Sγ(t)(x) dx

and _∂ V(S)(t)(I)
is the degree of vertex I in the edge boundary ∂ V(S)(t), viewed as a subgraph of 𝒢(t),  for t∈[0,a).



The inequalities λ_1,𝕃^(D)(t)≤λ_1,𝒢^(D)(t) and λ_S,𝕃^(N)(t)≤λ_S,𝒢^(N)(t) are obvious. Now, let ψ be a Kozyrev wavelet supported in B_p^-r(I)⊂ B_p^-N(I)∩ S for some vertex I∈ V(S). 
It clearly satisfies the Dirichlet boundary condition,  and, by Lemma <ref>, also the von Neumann boundary condition.
Hence, the Dirichlet and the von Neumann eigenvalues are both less than or equal to:

    ∑_J∈δ V(S)(t)A_IJ/∫_Sγ(x)(t) dx
    =_∂ V(S)(t)(I)/∫_Sγ(x)(t) dx

where δ V(S)(t) is the vertex boundary of the set V(S)⊂ V in 𝒢(t). This now shows the first inequality in 1., and that of 2.
The remaining inequality in 1. is given in <cit.>. This proves the theorem.


The following examples show that the inequalities can be strict:


Let 𝒢(t) be the complete graph on three vertices a,b,c, and let S=B_p^-N(a)⊔ B_p^-N(b).
Then, according to <cit.>, λ_1,𝒢^(D)(t) is an eigenvalue of the submatrix of the graph-Laplacian of  𝒢(t) spanned by the vertices a,b.
This is the matrix

    L_S=[    1 -1/2; -1/2    1 ]

with eigenvalues 3/2 and 1/2.
Hence, λ_1,𝒢^(D)(t)=1/2. But since the degree of each vertex of 𝒢(t) equals 2, it follows that

    λ_1,𝕃^(D)(t)≤γ̂_S(t)=1/4<1/2=λ_1,𝒢^(D)(t)

for given t≥0.



Let 𝒢(t) be the complete graph on two vertices a,b, and let S=B_p^-N(a). Then any non-trivial function f constant on the vertex balls B_p^-N(a) and B_p^-N(b) satisfying the von Neumann boundary condition, must satisfy
f(a)= 0 and thus

    ∫_Sf(x)^2γ(x)(t) dx=0

This means that 

    λ_S,𝒢(t)^(N)(t)=∞

and consequently

    λ_S,𝕃^(N)(t)<λ_S,𝒢^(N)(t)

because γ̂_S(t)<∞.








§ NON AUTONOMOUS CAUCHY PROBLEM


Consider the p-adic non autonomous Cauchy problem for a finite graph.
Most of the underlying theory for this section is taken from
<cit.>.




 §.§ Review of the abstract Cauchy problem





Let A(t) be a family of operators with t∈ℝ on a Banach space X. The following problem is addressed:


    ∂ u/∂ t (t)=A(t)u(t)
    
          u(s)=x∈ X

with t,s ∈ℝ and t≥ s. The system (<ref>) is called (nACP).


A continuous function u:[s,∞)→ X is called a (strict) solution of (<ref>), if u∈ C^1([s,∞),X), u(t)∈ D(A(t)) for all t≥ s, u(s)=x, and ∂ u/∂ t=A(t)u for t≥ 0.

The solution of non-autonomous Cauchy problem on some Banach space X can be given by a so called evolution family which can be defined as follows, cf. <cit.>:

A family (U(t,s))_t≥ s of linear operators on a Banach space X is called an evolution family for (<ref>), if 

    
  * U(t,r)U(r,s)=U(t,s), U(t,t)=id, t≥ s ∈ℝ.
    
  * The mapping (s,t)↦ U(t,s) is strongly continuous. 
  * ||U(t,s)||≤ M e^ω(t-s) for some M≥ 1, ω∈ℝ, t≥ s ∈ℝ.
    
  * For s∈ℝ the regularity space
    
    Y_s:={y∈ X:[s,∞) ∋ t→ U(t,s)y  solves  (nACP)}


is dense in X, where U(t,s)y is the unique solution of (nACP). 

If there exists a solving evolution family, then (<ref>) is called well-posed. We have the following Theorem.




Let X a Banach space and for every t>0 let A(t) be a bounded linear operator on X. If the function t↦ A(t) is strongly continuous for 0≤ t<T
(that is, t↦ A(t)u is continuous for each u∈ X), then for every x∈ X the problem (<ref>) is well-posed.



<cit.>.


We can apply the above Theorem to the problem (<ref>) for time-changing graphs.

The system (<ref>) is well-posed.

Proof. This is a direct consequence of Theorem
<ref>. 


If we let U(t,s)x=u(t), where u(t) is the solution of (<ref>), then the following holds  true: 

    
  * ||U(t,s)||≤ e^∫_s^t ||𝕃(τ)||dτ.
    
  * (t,s)↦ U(t,s) is continuous in the uniform operator topology for s≤ t.
    
  * ∂ U(t,s)/∂ t= ℒ(t) U(t,s).
    
  * ∂ U(t,s)/∂ s=-U(t,s)ℒ(s). 

This is due to the fact that t↦𝕃(t) is continuous in the uniform norm topology, cf. <cit.>.

In general, the solution does not have a closed form, but in special cases it does, as the following example shows.

 Under the hypothesis of Theorem <ref>, assume that A(t) and ∫_s^t A(τ) dτ commute for each s,t. Then 

    U(t,s)=e^∫_s^tA(τ) dτ

with the integral ∫ A(τ) dτ being interpreted "elementwise". 



<cit.>.





 §.§ Reviewing  the autonomous case for graphs



First, we review how to solve the heat equation in the autonomous case:


    ∂ u/∂ t (x,t)=𝕃u(x,t)
    
          u(0,x)=u_0(x)∈ L^2(K_n,ℂ)
         ,

where 𝕃 is interpreted as 𝕃=𝕃(t_0) for some t_0≥0. The following Theorem allows us to apply the Fourier Method (or the separation of variables method) to solve the system (<ref>). 


[Zúñiga-Galindo, 2020]
The elements of the set: 

    (L) {-γ_I: I∈ V(𝒢)}

 are the eigenvalues of of 𝕃. The corresponding eigenfunctions are given by the following infinite set 

    {φ_μ/||φ_μ||_2: μ∈(L)}{Ψ_j,I,r:I∈ V(𝒢), j∈{1,...,p-1}, r∈ℤ, r≥ N},

where the functions φ_μ(x) are defined by: 

    φ_μ(x)=∑_J∈ V(𝒢)φ_μ(J)Ω(p^-N|x-J|_p)

and the vector (φ_μ(J))_J∈ V(𝒢) is an eigenvector of the (negative semi-definite) Laplacian matrix L associated with adjacency matrix A, corresponding to μ∈(L), and Ψ_r,I,j are the Kozyrev functions of the form (<ref>). Furthermore we have the following orthogonal decomposition 

    L^2(K_N,ℂ)=X_N
        
        ⊕ℒ_0^2(K_N).





<cit.>.


According to <cit.>, the Cauchy problem of the heat equation (<ref>), with t≥ 0 has a solution which is unique. The solution is given by the semigroup {e^t𝕃} generated by 𝕃. By the decomposition (<ref>), we propose a solution of the form 

    u(x,t)=∑_μ∈(L) c_μ(t)φ_μ(x)+∑_(Ψ_j,I,r) ⊂ K_N c_r,I,j(t)ψ_j,I,r(x),

where the infinite set {Ψ_j,I,r|(Ψ_j,I,r) ⊂ K_N} is described in Theorem <ref>. Hence, the solution is an infinite sum, and the convergence takes place in L^2(K_N). After inserting this into (<ref>) we obtain 

    ∂ c_μ/∂ t =μ c_μ
    ∂ c_r,I,j/∂ t=-γ_Ic_j,I,r .

where μ and γ_I are the eigenvalues associated to φ_μ and ψ_j,I,r, respectively.


This proves that 

    e^t𝕃u_0(x)=u(x,t)=∑_μ∈(L) C_μe^μ tφ_μ(x)+∑_(Ψ_j,I,r) ⊂ K_N C_j,I,re^-γ_I tψ_j,I,r(x).

with C_I and C_j,I,r is uniquely determined by the initial conditions 

    u_0(x)=u(x,0)=∑_μ∈(L) C_μφ_μ(x)+∑_(Ψ_j,I,r) ⊂ K_N C_j,I,rψ_j,I,r(x),

and solution (<ref>) is an infinite convergent sum.





 §.§ Approximating the non-autonomous case for graphs


We now want to use the Trotter product formula to approximate the operator U(t,s), i.e., the solutions of (<ref>). As approximating operators we choose the product 

    ∏_k=1^n e^t/nℒ(s+kt/n)=e^t/nℒ(s+t)· e^t/nℒ(s+t-t/n)… e^t/nℒ(s+t/n).

We have the following proposition:


Let {A(t)}_t∈ℝ be a family of bounded and strongly continuous operators on a Banach space X, and let U(t,s) be its respective evolution family. Then 

    lim_n→∞∏_k=1^ne^t/nA(s+kt/n)x=U(s+t,s)x

for all x∈ X and uniformly for s and t in compact intervals of ℝ and ℝ_+, respectively. 



<cit.>.



Now we outline how to apply Proposition <ref> to approximate the solution of (<ref>). For 0<b, and ε>0, we have 

    ||e^t/n𝕃(t)· e^t/n𝕃(t-t/n)… e^t/n𝕃(t/n)u_0(x)-U(t,0)u_0(x)||_L^2<ε

with 0 ≤ t<b, and some n∈ℕ. Take 0≤ t_0 <b fixed, hence, in order to compute the approximate (respect to the L^2 norm) solution we have to solve (<ref>) by iterating the initial conditions, or in other words we have to compute e^t_0/n𝕃(t_0)· e^t_0/n𝕃(t_0-t_0/n)… e^t_0/n𝕃(t_0/n)u_0(x). We can view this approximation like snapshots of the system at n≥0 discrete time points. 
We call this the temporal discretisation, and we fix some notation:


    
  * Let μ(t_0) be an eigenvalue of the matrix L(t_0) and φ_μ(x,t_0) the associated eigenfunction.
    
  * Let {e^t𝕃(t_0)}_t≥ 0 be the semigroup generated by the operator 𝕃(t_0) for some t_0≥ 0. 
    
  * Let γ_I(t_0)=∑_JA_IJ(t_0)

Therefore, for a fixed t_0, the solution of the Cauchy problem in the autonomous case (<ref>) associated to the operator 𝕃(t_0), with initial condition u(x,0)=u_0(x) as in (<ref>), is given by 

    e^t𝕃(t_0)   u_0(x)=u(x,t)
    
       =∑_μ=μ(t_0)∈(L(t_0)) C_μ(t_0)e^μ(t_0) tφ_μ(x,t_0)
    
       +∑_r,I,j C_j,I,r(t_0)e^-γ_I(t_0) tψ_j,I,r(x).

Now it is clear that 

    e^t_0/n𝕃(t_0/n)u_0(x)
            =∑_μ=μ(t_0)∈(L(t_0)) C_μ(t_0/n)e^μ(t_0/n) t_0/nφ_μ(x,t_0/n)
         
            +∑_j,I,r C_r,I,j(t_0/n)e^-γ_I(t_0/n) t_0/nψ_j,I,r(x).

The next step is to compute e^t_0/n𝕃(2t_0/n)e^t_0/n𝕃(t_0/n)u_0(x). For this we need to find the solution to the Cauchy problem (<ref>) associated to the operator 𝕃(2t_0/n) with initial condition u_1(x)=e^t_0/n𝕃(t_0/n)u_0(x), and evaluate the solution t↦ e^t𝕃(2t_0/n)e^t_0/n𝕃(t_0/n)u_0(x) in t=t_0/n.


Define for a given adjacency matrix A(t) the corresponding weighted graph G_A(t) whose adjacency matrix is A(t). The corresponding L^2-space will  be denoted as L^2(G_A(t),ℂ). It is isomorphic to ℂ^V(𝒢).


To find the solution of the form (<ref>) we need to express the initial condition u_1(x)=e^t_0/n𝕃(t_0/n)u_0(x) as an element of L^2(G_A(2t_0/n),ℂ)⊕ℒ_0^2(K_N), 
that is 

    e^t_0/n𝕃(t_0/n)u_0(x)=∑_μ∈(L(t_0)) B_μφ_μ(x,2t_0/n)+∑_j,I,r B_j,I,rψ_r,I,j(x).

Then, after computing the coefficients B_μ and B_j,I,r the desired solution will be given by 

    e^t𝕃(2t_0/n)   e^t_0/n𝕃(t_0/n)u_0(x)= u_1(x,t)
            
       =∑_μ∈(L(t_0)) B_μe^μ(2t_0/n) tφ_μ(x,2t_0/n)+∑_j,I,r B_j,I,re^-γ_I(2t_0/n) tψ_j,I,r(x).

Comparing (<ref>) and (<ref>), it is clear that B_j,I,r=C_j,I,re^-γ_I(t_0/n)t_0/n. This is due to the fact that in decomposition (<ref>) the space ℒ_0^2(K_N) remains constant in time. Now we need to compute the constants B_μ. For this, note that 

    ω:=∑_μ∈(L(t_0)) B_μφ_μ(x,2t_0/n)=∑_μ∈(L(t_0)) C_μe^μ(t_0/n) t_0/nφ_μ(x,t_0/n)

Note that  for each s≥ 0, the set 

    φ_μ(x,s)=φ_μ(s)(x)|μ(s)∈(L(s))

is a basis for the finite dimensional space X_N. Hence, the coefficients B_μ are given by the basis change matrix from the basis {φ_μ(x,t_0/n)} to the basis {φ_μ(x,2t_0/n)}. The basis change matrix M=[M_λ,μ] is given by the following equalities: 

    φ_μ(x,2t_0/n)=∑_λ∈ V(𝒢)M_λ,μφ_λ(x,t_0/n).

In general, the basis change matrix from the basis {φ_μ(x,s)} to the basis {φ_μ(x,t)} M will be given by 

    M=M(s)^-1M(t),

where the matrices M(τ) are the modal matrices of the Laplacian matrix, that is, the μ-column of M(τ) is given by the eigenvector [φ_μ(K,τ)]_K∈ V(𝒢). Then, it is clear that the coefficients B_μ are given by

    [B_μ]_μ=M^-1[C_μe^μ(t_0/n) t_0/n]_μ.

Thus,

    [B_μ]_μ=M(2t_0/n)^-1M(t_0/n)(e^μ(t_0/n) t_0/n)[C_μ]_μ

and 

    B_j,I,r=C_j,I,re^-γ_I(t_0/n)t_0/n.

The process continues by solving the Cauchy problem for the operator 𝕃(3t_0/n) with initial condition e^t_0/n𝕃(2t_0/n)e^t/n𝕃(t_0/n)u_0(x). In order to do so, we repeat the process described above. That is we take as initial condition 

    u_1(x,t_0/n)   =∑_μ∈(L(t_0)) B_μe^μ(2t_0/n) t_0/nφ_μ(x,2t_0/n)
    
       +∑_j,I,r C_j,I,re^-γ_I(t_0/n)t_0/n-γ_I(2t_0/n) tψ_j,I,r(x)

and express it as an element of L^2(G_A(3t_0/n),ℂ)⊕ℒ_0^2(K_N).
Now,

    u_1(x,t_0/n)=∑_μ∈(L(t_0)) D_μφ_μ(x,3t_0/n)+∑_j,I,r D_j,I,rψ_r,I,j(x).

Again, we obtain the coefficients D_μ and D_j,I,r by:

    [D_μ]_μ=M(3t_0/n)^-1M(2t_0/n)(e^μ(2t_0/n)t_0/n)[B_μ]_μ

and 

    D_j,I,r=B_j,I,re^-γ_I(2t_0/n)t_0/n.

We can simplify these expressions by putting the value of B_μ and B_j,I,r in the above equalities, and we get 

    [D_μ]_μ=M(3t_0/n)^-1M(t_0/n)(e^μ(t_0/n)+μ_I(2t_0/n))[C_μ]_μ

and 

    D_j,I,r=C_j,I,re^-γ_I(t_0/n)t_0/n-γ_I(2t_0/n) t.

In the end, we obtain the function e^t_0/n𝕃(t_0)… e^t_0/n𝕃(t_0/n)u_0(x)∈ L^2(K_N), of the form 

    e^t_0/n𝕃(t_0)   · e^t_0/n𝕃(t_0-t_0/n)… e^t_0/n𝕃(t_0/n)u_0(x)=∑_μ∈(L(t_0))Q_μ(n,t_0)φ_μ(x,t_0)
       +∑_(Ψ_j,I,r)⊂ K_NC_r,I,j(t_0/n)e^-t_0/n∑_k=1^nγ_I(kt_0/n)Ψ_j,I,r(x),

where the coefficients Q_μ(n,t_0) are given by the equation

    [Q_μ(n,t_0)]_μ=M(t_0)^-1M(t_0/n)(e^t_0/n∑_k=1^nμ(kt_0/n))[C_μ(t_0/n)]_μ

For simplicity, we rename the approximation above: 

    U_n(t,0)u_0(x)   =∑_μ∈(L(t_0))Q_μ(n,t)φ_μ(x,t)
    
       +∑_(Ψ_j,I,r)⊂ K_NC_r,I,j(t/n)e^-t/n∑_k=1^nγ_I(kt/n)Ψ_j,I,r(x)

Then, for a general initial condition s, we obtain: 

    U_n(t,s)u_0(x)   =∑_μ∈(L(t_0))Q_μ(n,t,s)φ_μ(x,t_0)
    
       +∑_(Ψ_j,I,r)⊂ K_NC_r,I,j(s+t-s/n)e^-t-s/n∑_k=1^nγ_I(s+kt-s/n)Ψ_j,I,r(x),

where the coefficients Q_μ(n,t,s) are computed by the following equation: 

    [Q_μ(n,t,s)]=M(t)^-1M(s+t-s/n) (e^t-s/n∑_k=1^nμ(s+kt-s/n))[C_μ(s+t-s/n)]_μ

where the numbers C_μ(s+t-s/n) are the Fourier coefficients of the initial condition at time s+t-s/n. Note that in L^2(K_N) we have the following convergence:

    lim_n→∞U_n(t,s)u_0(x)=U(t,s)u_0(x),

and we can expand the function U(t,s)u_0(x)∈ L^2(K_N) by its Fourier series using the decomposition L^2(G_A(t_0),ℂ)⊕ℒ_0^2(K_N):

    U(t,s)u_0(x)=∑_μ∈(L(t_0))Q_μ(t,s)φ_μ(x,t_0)+∑_(Ψ_j,I,r)⊂ K_NQ_r,I,jΨ_j,I,r(x).

Hence, by Parseval's equality we obtain:

    lim_n→∞ Q_μ(n,t,s)=Q_μ(t,s)

and 

    lim_n→∞C_r,I,j(s+t-s/n)e^-t-s/n∑_k=1^nγ_I(s+kt-s/n)=C_r,I,j(s) e^-∫_s^tγ_I(τ)dτ,

where the last equality occurs by the continuity of t↦γ_I(t). On the other hand, the first limit  is more complicated, because it depends on the continuity of the matrix function t↦ M(t), more specifically, on the continuity of the eigenvectors: t↦φ_μ(t), where the vector φ_μ=[φ_μ(K,t)]_K is the eigenvector of the Laplacian matrix L(t) associated to μ(t)∈(L(t)). 

Let t↦ B(t) be a  matrix function. If B(t) is real symmetric, then we can find an orthonormal basis {φ_μ(t)}_μ  of ℂ^N formed by eigenvectors of B(t). It can be shown that if the matrix B(t) depends analytically on t inside a real interval, then we can chose an orthonormal basis {φ_μ(t)}_μ that depends analytically on t
<cit.>, 
and in particular, continuously on t. If  the analyticity assumption is deleted, the eigenvectors may not depend continuously on t. Therefore, if we suppose that the adjacency matrix A(t) depends analytically on t, then we have the following:


    lim_n→∞ [Q_μ(n,t,s)]_μ   =lim_n→∞M(t)^-1M(s+t-s/n) 
            
       ·(e^t-s/n∑_k=1^nμ(s+kt-s/n))[C_I(s+t-s/n)]_I
       =M(t)^-1M(s)(e^∫_s^tμ(τ)dτ)[C_μ(s)]_μ=[Q_μ(t,s)]_μ.

But, in order to have the above limit, it is sufficient to suppose that the eigenvectors φ_μ(t) are right continuous at the initial condition s≥ t_0. 


This proves the following:



If the eigenvectors of the Laplacian matrices L(t) are right continuous at some s≥0, then the evolution family of the Cauchy problem (<ref>) (with initial condition at s) is given by 

    U(t,s)u_0(x)   =∑_μ∈(L(t_0))Q_μ(t,s)φ_μ(x,t)
    
       +∑_(Ψ_j,I,r)⊂ K_N C_r,I,j(s) e^-∫_s^tγ_I(τ)dτΨ_j,I,r(x),

where t≥ s,

    [Q_μ(t,s)]=M(t)^-1M(s)(e^∫_s^tμ(τ)dτ)[C_μ(s)]_μ ,

and the coefficients C_μ(s) and C_r,I,j(s) are uniquely determined by the initial condition

    u_0(x)=∑_μ∈(L(t_0))C_μ(s)φ_μ(x,s)+∑_(Ψ_j,I,r)⊂ K_N C_r,I,j(s)  Ψ_j,I,r(x).

In particular, this result holds for all s≥ 0 if the adjacency matrix A(t) depends analytically on t.



Note that this is consistent with the autonomous problem, that is, when L(t)=L is constant. In this case, the matrices M(t) and the functions t↦μ(t), t↦φ_μ(t) are constant, thus the solution is given by 

    U(t,s)u_0(x)   =∑_μ∈(L)C_μ(s)e^(t-s)μ
       +∑_(Ψ_j,I,r)⊂ K_N C_r,I,j(s) e^-(t-s)γ_IΨ_j,I,r(x)

for t≥ s≥0.



Assuming analyticity of L(t), the first-order approximation error in the temporal discretisation is O(1/ne^1/n).



For the temporal discretisation, observe that by replacing each analytic function with its first order approximation leads to 
an error of O(1/n) in each entry of the eigenvectors and eigenvalues of L(t), and an error of O(e^1/n)
in each exponential integral, because the temporal discretisation is in this case a Riemann sum. Hence, the
error in each coefficient of the solution is given by the 
product error, and this is in first order O(1/n e^1/n), which proves the assertion. 



































§ INHOMOGENEOUS MARKOV PROCESSES AND FELLER EVOLUTION.





 §.§ Reviewing inhomogeneous Markov processes

In this section we review some definitions and results necessary to prove that the non-autonomous abstract Cauchy problem (<ref>) has attached to it an inhomogeneous Markov process. 


In the sequel E denotes a separable complete metrizable topological Hausdorff space. In other words E is a Polish space. The space C_b(E) is the space of all complex valued bounded continuous functions. The space C_b(E) is equipped with the uniform norm: ||·||_∞, and the strict topology 𝒯_β.

It is considered as a subspace of the bounded Borel measurable functions L^∞(E), also endowed with the supremum norm. 

Let (Ω,ℱ) be a measurable space. A filtration ℱ_t, t∈ [0,T], is a family of sub-σ-algebras of ℱ such that if 0≤ t_1 ≤ t_2 ≤ T, then ℱ_t_1⊂ℱ_t_2. A two-parameter filtration ℱ_t^τ, 0≤τ≤ t ≤ T, is a family of sub-σ-algebras of ℱ that is increasing in t and decreasing in τ. Let X_s, s∈ [0,T] be a stochastic process, with state space (E, ℰ). The σ-algebra σ(X_s:0≤ s≤ t) is the smallest σ-algebra such that all the state variables X_s with 0≤ s≤ t are measurable.

The process X_s, s∈ [0,T], generates the following two-parameter filtration: 

    ℱ_t^τ=σ(X_s:τ≤ s ≤ t)

with 0≤τ≤ t ≤ T. 


We recall some definitions:



A transition probability funcition P(r,x;s,A), where 0≤ r < s ≤ T, x∈ E, and A∈ℰ, is a nonnegative function for which the following condition hold: 

    
  * For fixed r,s and A, P is a nonnegative Borel measurable function on E. 
    
  * For fixed r,s and x, P is a Borel measure on ℰ. 
    
  * P(r,x;s,E)=1 for all r,s and x. 
    
  * P(r,x;s,A)=∫_EP(r,x;y,dy)P(u,y;s,A) for all r<u<s, and A. 


The number P(r,x;s,A) can be interpreted as the probability of the following event: The random system located at x∈ E at time r hits the target A⊂ E at time s. 



The process

    {
    (Ω,ℱ_T^τ,ℙ_τ,x), (X(t):T≥ t≥ 0),(E,ℰ)}

is called a time-inhomogeneous Markov process if 

    𝔼_τ,x[f(X(t))|ℱ_s^τ]=𝔼_s,X(s)[f(X(t))]

ℙ_τ,x-almost surely. Here f is a bounded Borel measurable function defined on the state space E and τ≤ s ≤ t ≤ T. 


Let P be a transition probability function. It is possible to construct a filtered measurable space (Ω, ℱ, ℱ_t^τ), a family of probability measures ℙ_τ,x, x∈ E, τ∈ [0,T], on the space (Ω, ℱ_T^τ), and a Markov process X_t, t∈ [0,T] such that 

    ℱ_t^τ=σ(X_s:τ≤ s ≤ t)

with 0≤τ≤ t ≤ T and 

    ℙ_τ,x(X_t∈ A)=P(τ,x;t,A)

with 0≤τ≤ t ≤ T, A∈ℰ.






A family {P(s,t):0≤ s ≤ t ≤ T} of operators defined on L^∞(E) is called a Feller Evolution or Feller propagator on C_b(E) if it possesses the following properties: 

    
  * It leaves C_b(E) invariant: P(s,t)C_b(E)⊂ C_b(E) for 0≤ s≤ t ≤ T;
    
  * It is an evolution: P(τ,t)=P(τ,s)P(s,τ) for all τ, s, t for which 0≤τ≤ s ≤ t and P(t,t)=I, t∈ [0,T];
    
  * If 0≤ f ≤ 1, f∈ C_b(E), then 0≤ P(s,t)f ≤ 1, for 0≤ s ≤ t≤ T. 
    
  * If the function (s,t,x)↦ P(s,t)f(x) is continuous on the space Λ:={(s,t,x)∈ [0,T]×[0,T]× E:s≤ t}. 


In the following theorem we see that with a Feller evolution a strong Markov process can be associated in such a way that the one-dimensional distributions or marginals are determined by the operators f↦ P(τ,t)f, f∈ C_b(E). In fact every operator P(τ,t) can be written as 

    P(τ,t)f(x)=∫ P(τ,x;t,dy)f(y)

for f∈ C_b(E). Where the mapping 

    (τ,x,t,B)↦ P(τ,x;t,B)

with (τ,x,t,B)∈ [0,T]× E× [0,T]×ℰ, τ≤ t, is a sub-probability transition function. 




Let {P(τ,t):τ≤ t ≤ T} be a Feller evolution in C_b(E). Then there exists a strong Markov process (in fact a Hunt Process) such that [P(τ,t)f](X)=𝔼_τ,x[f(X(t))], f∈ C_b(E), t≥0. Moreover, this Markov process possesses the properties in <cit.>. 



<cit.>.





 §.§ Stochastic process attached to a non-autonomous equation


In this section we prove that the system 

    ∂ u/∂ t (x,t)=𝕃(t)u(x,t)
    
          u(x,s)=u_s(x)∈ C(K_n,ℂ)

generates a time-inhomogeneous Markov process, where the operators 𝕃(t) satisfies the hypothesis of Theorem <ref>. In more generality, we study the following Cauchy problem 

    ∂ u/∂ t (x,t)=A(t)u(x,t)
    
          u(x,s)=u_s(x)∈ C(E) ,

where E is a compact Polish space. Thus, C_b(E)=C(E). Also, we suppose that t↦ A(t) is continuous in the uniform norm topology (In particular A(t) is bounded for all t≥ 0). Moreover, we suppose that for every τ≥ 0, A(τ) generates a strongly continuous, positive, contraction semigroup {e^tA(τ)}_t ≥ 0 on C(E). By Theorem <ref>,
the system (<ref>) 
is well-posed for every initial condition u_s(x)∈ C(E). Therefore, there exists an evolution family U(t,s), such that U(t,s)u_s(x) is the unique solution of the Cauchy problem (<ref>). 
Moreover, U(t,s) satisfies the properties in Remark <ref>.


The evolution U(t,s) described above is a Feller evolution.


Proof. 
We verify the properties of Definition <ref>.
The verification of 1. and 2. is straightforward.
We now prove condition 3. Let f∈ C(E) such that 0≤ f ≤ 1. By Proposition <ref>, 
we have 

    lim_n→∞∏_k=1^ne^(t-s)/n A(s+k(t-s)/n)f(x)=U(t,s)f(x)

in the supremum norm, and with 0≤ s ≤ t ≤ T. Therefore, we also have point-wise convergence. Since every semigroup {e^tA(τ)}_t≥ 0 is a strongly continuous, positive, contraction semigroup we have 

    0≤∏_k=1^n e^(t-s)/n A(s+k(t-s)/n)f(x)≤ 1.

We obtain the desire result by taking n→∞. For condition 4., note that, since the space E is Polish, we can prove the continuity in terms of sequences. Let (t,s,x)∈Λ, with s≤ t and x∈ E. And let (t_n,s_n,x_n) a convergent sequence, with s_n≤ t_n and x_n∈ E, such that (t_n,s_n,x_n)→(t,s,x) as n→∞. We have


    |U(t_n,s_n)f(x_n)-U(t,s)f(x)|   ≤ |U(t_n,s_n)f(x_n)-U(t,s)f(x_n)|
       +|U(t,s)f(x_n)-U(t,s)f(x) |  
    ≤   ||U(t_n,s_n)-U(t,s)||_∞||f||_∞
       +|U(t,s)f(x_n)-U(t,s)f(x) |

Since (t,s)↦ U(t,s) is continuous in the uniform norm topology (Remark 
<ref>.2.), 
and the function x↦ U(t,s)f(x) is continuous in E, by taking n→∞ the right hand side of the above inequality tends to zero. This now proves that U(t,s) is a Feller evolution. 
Let {A(t)}_t≥ 0 be a set of bounded operators on C(E). Suppose that t↦ A(t) is continuous in the uniform norm topology and that each A(t) generates a strongly continuous, positive, contraction semi-group. Then the Cauchy problem (<ref>)
is well-posed and its evolution family U(t,s) generates a strong Markov process with the properties listed in Theorem <ref>. 


Proof. This is a consequence of Lemma <ref>
and Theorem <ref>. 



The literature does not seem to contain a result proving that a non-autonomous Cauchy problem has attached to it a Markov process. That is why  the above result was obtained using the techniques learned above in order to find an aproximate solution (Proposition <ref>). 
In particular, it is important to note that evolution families are the two-parameter generalizations of semigroups, and analogously, Feller evolution families are the two parameter generalizations of Feller semigroups. 



There exists a probabilty transition function P(t,x;s,·), where (t,x,s)∈ [0,T]× K_N × [0,T], and s≤ t, on the Borel σ-algebra of K_N, such that the Cauchy problem (<ref>) 
has a unique solution of the form 

    u(x,t)=∫_K_N u_s(x) P(t,x;s,dy).

In addition, P(t,x;s,·) is the transition function of a Markov process with the properties listed in Theorem <ref>. 

Proof. We have to proove that t↦𝕃(t) satisfies the hypothesis in Theorem <ref>. 
By   <cit.>, we know that 𝕃(t) generates a strongly continuous, positive, contraction semigroup on C(K_N) for each t≥0. hence, we only have to show that t↦𝕃(t) is continuous in the uniform norm topology. Recall from the proof of Theorem
<ref> that 

    ||𝕃(t)u-𝕃(s)u||_∞≤ ||𝒜(t)u-𝒜(s)u||_∞+||Γ(t)u-Γ(s)u||_∞.

Let ε>0, and δ>0 such that 

    |t-s|<δ|A(x,y,t)-A(x,y,s)|<ε and   |γ(t)(x)-γ(s)(x)|< ε

We have that

    ||𝒜(t)u-𝒜(s)u||_∞   =sup_x∈ K_N|∫_K_N (A(x,y,t)-A(x,y,s))u(y)dy|
       ≤ε||u||_∞(K_N)

and 

    ||Γ(t)u-Γ(s)u||_∞ =sup_x∈ K_N|(γ(t)(x)-γ(s)(x))||u(x)|≤ε ||u||_∞ .

If ||u||_∞≤ 1, then

    ||𝕃(t)u-𝕃(s)u||_∞≤ε((K_N)+1).

This inequality implies the continuity in the uniform norm topology. We now apply Theorem <ref> and Theorem <ref> to obtain the result. 




§ ACKNOWLEDGEMENTS


The first named author is indebted to Norbert Paul for coming up  with questions on how to study topological questions in the setting of finite spaces a long time ago, which gave a long-lasting impulse to eventually combine these with ultrametric analysis and mathematical physics.
The second named author is indebted to
Wilson Zúñiga-Galindo for sharing valuable insights into p-adic analysis when he was under his supervision as an undergraduate student. Martin Breunig, Markus Jahn and David Weisbart are warmly thanked for fruitful discussions.

This work is supported by the Deutsche Forschungsgemeinschaft
under project number 469999674.



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 Paper Title: Global smooth solutions of 2D quasilinear wave equations with higher order null conditions and short pulse initial data.


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§ 

	
	
	claimClaim[section]
	theoremTheorem[section]
	lemmaLemma[section]
	propositionProposition[section]
	definitionDefinition[section]
	remarkRemark[section]
	corollaryCorollary[section]
	exampleExample[section]





Global smooth solutions of 2D quasilinear wave equations
with higher order null conditions
and short pulse initial data
    Ding
Bingbing^1,*,   Xin Zhouping^2,*,   Yin
Huicheng^1,[Ding Bingbing (13851929236@163.com, bbding@njnu.edu.cn) and Yin Huicheng (huicheng@nju.edu.cn, 05407@njnu.edu.cn) were
supported by the NSFC (No.12331007, No.12071223). Xin Zhouping(zpxin@ims.cuhk.edu.hk)
is partially supported by the Zheng Ge Ru Foundation, Hong Kong RGC Earmarked Research Grants
CUHK-14301421, CUHK-14300819, CUHK-14302819, CUHK-14301023, Basic and Applied Basic Research Foundations of Guangdong Province 2020131515310002, and the Key Projects of National Nature Science Foundation of China (No.12131010, No.11931013).]


1.  School of Mathematical Sciences and Mathematical Institute, Nanjing Normal University,

Nanjing, 210023, China.




2. Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, NT, Hong Kong.
    
======================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================





Abstract 0.3 true cm


For the short pulse initial data
with a first order outgoing constraint condition and optimal orders of smallness, we establish the global existence of smooth solutions
to 2D quasilinear wave equations with higher order null conditions. Such kinds of wave equations
include 2D relativistic membrane equations, 2D membrane equations,
and some 2D quasilinear equations which come
from the nonlinear Maxwell equations in electromagnetic theory or
from the corresponding Lagrangian functionals as perturbations of the Lagrangian densities of linear wave operators. The main ingredients of the analysis here include looking for a new good unknown, finding some key identities based on the higher order null conditions and the resulting null frames, as well as overcoming the difficulties due to the slow decay of solutions to the 2-D wave equation, so that the solutions can be estimated precisely.



0.2 true cm
Keywords: Quasilinear wave equations, short pulse initial data, higher order null condition,

          inverse foliation density, good unknown, null frame


0.2 true cm Mathematical Subject Classification: 35L05, 35L72




0.4 true cm





§ INTRODUCTION




 §.§ The problem and main results


Consider the 2D quasilinear wave equation

    ∑_,β=0^2g^β(ϕ)_β^2ϕ=0

with the short pulse initial data

    ϕ|_t=1=δ^2-ε_0ϕ_0(r-1δ,ω),_tϕ|_t=1=δ^1-ε_0ϕ_1(r-1δ,ω),

where (x^0, x)=(t, x^1, x^2)∈ [1,∞)×^2,
=(_0,_1,_2)=(_x^0, _x^1, _x^2),
g^β(ϕ) =g^β(ϕ) are smooth
functions of their arguments, >0 is small, 0<_0<1 is a fixed constant, r=|x|=√((x^1)^2+(x^2)^2),
ø=(ø_1,ø_2)=xr∈𝕊^1, (ϕ_0,ϕ_1)(s,ø) are smooth functions defined
in ℝ×𝕊^1
with compact support in (-1,0) for the variable s. It is pointed out that when _0=12 in (<ref>),
such a class of initial data are first introduced by D. Christodoulou <cit.>
for studying the formation of trapped surfaces in the Einstein vacuum spacetime
(see also <cit.>).


In this paper, we assume that for fixed integer k≥ 2, it holds that

    g^β(ϕ)=m^β+∑_0≤γ_1, ⋯, γ_k≤2g^β,
    	γ_1,⋯,γ_k_γ_1ϕ⋯_γ_kϕ+h^β(ϕ)

for small ϕ,
where m^00=-1, m^ii=1 for 1≤ i≤ 2, m^β=0 for ≠β,
the constants g^β,
γ_1,⋯,γ_k are equal for all the permutations of (,β) and (γ_1,⋯,γ_k) respectively,
and h^β(ϕ)=O(|ϕ|^k+1).


A lot of physical or geometric models admit the form (<ref>),
which include:

(1) The 2D relativistic membrane equation (corresponding to k=2 in (<ref>))

    _t(_tϕ√(1-(_tϕ)^2+|_xϕ|^2))
    -div(∇_xϕ√(1-(_tϕ)^2+|_xϕ|^2))=0  with _xϕ=(_1ϕ,_2ϕ).

Note that (<ref>) is  the Euler-Lagrange equation of the area functional
∫_ℝ×ℝ^2√(1+|∇_x ϕ|^2-(_tϕ)^2)dtdx
for the embedding of (t,x)→ (t,x,ϕ(t,x)) in the Minkowski spacetime.


(2) The 2D nonlinear membrane equation (corresponding to k=2 in (<ref>))

    _t^2ϕ-div(∇_xϕ√(1+|_xϕ|^2))=0,

where ϕ(t,x) stands for the position of  membrane at point (t,x).

(3) The nonlinear wave equation (corresponding to k=p in (<ref>), p∈ℕ)

    -(1+(_tϕ)^p+1)_t^2ϕ+Δϕ=0,

which comes from the variation of the Lagrangian functional

    L(ϕ)=-12(_tϕ)^2+12|∇_xϕ|^2
    -(_tϕ)^p+3(p+3)(p+2).

Especially, (<ref>) with p=1 may be regarded as a model equation from the nonlinear
version of Maxwell equations in the electromagnetic theory (see <cit.>);


(4) The nonlinear wave equation (corresponding to k=2p in (<ref>), p∈ℕ)

    -_t^2ϕ+Δϕ-_t((-12(_tϕ)^2+12|∇_xϕ|^2)^p_tϕ)
    +div((-12(_tϕ)^2+12|∇_xϕ|^2)^p∇_xϕ)=0,

which is the Euler-Lagrangian equation of
L(ϕ)=-12(_tϕ)^2+12|∇_xϕ|^2
+1p+1(-12(_tϕ)^2+12|∇_xϕ|^2)^p+1, which can be thought as a
suitable perturbation of the Lagrangian density of the linear wave operator.



(5) As a generalization of (<ref>) and (<ref>), the nonlinear wave
equation  (corresponding to k=2p in (<ref>), p∈ℕ)

    -(1+(_tϕ)^2p+1)_t^2ϕ+Δϕ-_t((-12(_tϕ)^2+12|∇_xϕ|^2)^p_tϕ)
       +div((-12(_tϕ)^2+12|∇_xϕ|^2)^p∇_xϕ)=0

satisfies the (2p)^th null condition but violates the (2p+1)^th (see the definition of the k^th
null condition in (<ref>)). Note that
(<ref>) corresponds to the Euler-Lagrangian equation of

    L(ϕ)=-12(_tϕ)^2+12|∇_xϕ|^2
    +1p+1(-12(_tϕ)^2+12|∇_xϕ|^2)^p+1-(_tϕ)^2p+3(2p+3)(2p+2)  
    (p∈ℕ).





For the study of the global existence of smooth solutions  to (<ref>)
with (<ref>), it is often crucial to introduce the following definition of the k^th
null condition (k≥ 2, k∈ℕ)
for (<ref>):

    ∑_0≤,β,γ_1, ..., γ_k≤ 2g^β,γ_1,⋯,γ_kξ_ξ_βξ_γ_1⋯ξ_γ_k≡ 0  holds for ξ_0=-1 and
    	(ξ_1, ξ_2)∈𝕊^1.

From now on, the k^th null condition with k≥ 2 will be called higher order  null condition in this paper.


We emphasize that for small ϕ and general g^β(ϕ) with

    g^β(ϕ)=m^β+∑_0≤γ≤ 2 g^β,γ_γϕ+∑_0≤γ_1,_2≤ 2g^β,γ_1_2_γ_1ϕ__2ϕ+O(|ϕ|^3),

the  first and second null conditions have been introduced for (<ref>)
(see <cit.>), namely,
there hold that for ξ_0=-1 and (ξ_1,ξ_2)∈𝕊^1,

    ∑_0≤,β,γ≤ 2 g^β,γξ_ξ_βξ_γ≡0  and∑_0≤,β,γ_1,γ_2≤ 2 g^β,γ_1γ_2ξ_ξ_βξ_γ_1ξ_γ_2≡0,   respectively.

Moreover, it has been shown that for small smooth  initial data with compact supports

    ϕ|_t=1=δψ_0(x),_tϕ|_t=1=δψ_1(x), (ψ_0(x),ψ_1(x))≢0,

(<ref>) with (<ref>) and (<ref>) admits
a global smooth solution ϕ if and only if (<ref>) holds true,
otherwise, the solution ϕ may blow up in finite
time and further the shock is formed (see <cit.>).


For 2D potential equation of irrotational isentropic
Chaplygin gases

    ∑_,β=0^2g^β(ϕ)_β^2ϕ= -_t^2ϕ+ϕ-2∑_i=1^2_iϕ_t_iϕ
    +2_tϕϕ-∑_i,j=1^2_iϕ_jϕ_ij^2ϕ
    +|∇ϕ|^2ϕ=0,

whose coefficients fulfill (<ref>).
Under the short pulse initial data (<ref>) and the first order outgoing constraint condition

    (_t+_r)ϕ|_t=1=O(δ^2-ε_0),

it is shown in <cit.> that when 0<_0<12, (<ref>) has a global smooth solution ϕ
with |ϕ|≤ Cδ^1-ε_0t^-1/2 for all t≥ 1.
As illustrated in <cit.>, in order to keep the strict hyperbolicity of (<ref>) or (<ref>),
the smallness of ϕ is certainly required
and can be derived from (<ref>) and (<ref>).





Motivated by the results of <cit.> in 3D case, we expect to find the optimal order ε_k^*
of smallness in (<ref>) such that if (<ref>) satisfies the k^th null condition
but does not fulfill the (k+1)^th null condition, then the smooth solution of (<ref>) with (<ref>) and (<ref>)
exists globally when ε_0∈(0,ε_k^*) and may blow up in finite time when ε_0∈[ε_k^*,1).








Note that it follows easily from the forms of (<ref>) and (<ref>) that for 0≤ p≤ 1,

    (_t+_r)^pØ^j^ϕ|_t=1=O(δ^2-ε_0-||),

where Ø=x^1_2-x^2_1. Without loss of generality and for simplicity, it is assumed that in (<ref>),

    g^00(ϕ)=-1.


Our main result in the paper is:

	Let _k^*=kk+1 with k≥2.
Under the conditions (<ref>), (<ref>),  (<ref>) and (<ref>),
it holds that for small δ>0,
when _0∈(0,_k^*), (<ref>) with (<ref>)
admits a global smooth solution ϕ which satisfies

    ϕ∈ C^∞([1,+∞)×ℝ^2),   |ϕ|≤ Cδ^1-ε_0t^-1/2
    and |ϕ|≤ Cδ^1-ε_0t^1/2 for t≥ 1,

where C>0 is a uniform constant independent of  and _0.



	As a direct application of Theorem <ref>, (<ref>) with p=k≥ 2 has a global solution when the initial data are identical to those in Theorem <ref> and ε_0 ∈ (0, ε_k^*). Conversely,
for certain initial short pulse data,	
it follows from <cit.> that the solution of (<ref>) with p=k≥ 1  may blow up
in finite time when ε_0∈[ε_k^*,1).
As previously mentioned, (<ref>)  with p=k is a special equation satisfying the k^th
null condition but violates the (k+1)^th null condition. Therefore, we conjecture that, for ε_0∈[ε_k^*,1) with k≥2, if
	h^β(ϕ)=∑_0≤γ_1, ⋯, γ_k+1≤2h^β,_1,⋯,_k+1(__1ϕ)⋯(__k+1ϕ)+O(|ϕ|^k+2) in (<ref>)
and ∑_0≤γ_1, ..., γ_k+1≤2h^β,γ_1,⋯,γ_k+1ξ_ξ_βξ_γ_1⋯ξ_γ_k+1≢0 for ξ_0=-1 and (ξ_1, ξ_2)∈𝕊^1, then there exist a class of initial data specified by (<ref>) such that the smooth solution of (<ref>) will undergo finite-time blowup.
This intriguing phenomenon will be one of the topics we intend to explore in our future research.



On the other hand, it is easy to verify that (<ref>) satisfies the second order null condition but (<ref>) does not,
(<ref>) satisfies the (2p)^th null condition, (<ref>) fulfills the (2p)^th null condition
but does not fulfills the (2p+1)^th null condition.
Collecting Theorem <ref> and the analogous arguments in  <cit.>
as for (<ref>) with (<ref>) and (<ref>), we have




Under the conditions (<ref>) and (<ref>),

(1) (<ref>) has a global smooth solution ϕ when 0<_0<_2^*=23.

(2)  the smooth solution ϕ of (<ref>) exists globally when 0<_0<_2p^*=2p2p+1.

(3) assume that there exists a point (s_0,ø_0)∈(-1,0)×𝕊^1 such that

    _sϕ_0(s_0,ø_0)_s^2ϕ_0(s_0,ø_0)>1    for _1^*=12<_0<1,
       _sϕ_0(s_0,ø_0)_s^2ϕ_0(s_0,ø_0)>2    for _0=_1^*,

the smooth solution ϕ of (<ref>) will blow up in finite time and further the shock is formed.

(4) when 0<_0<_2p^*=2p2p+1, the smooth solution ϕ of (<ref>) exists globally;
when _2p^*≤_0<1, if there exists a point (s_0,ø_0)∈(-1,0)×𝕊^1 such that

    ϕ_1^2p(s_0,ø_0)_sϕ_1(s_0,ø_0)>22p+1    for _2p^*<_0<1,
       ϕ_1^2p(s_0,ø_0)_sϕ_1(s_0,ø_0)>(p+1)2^2p+1(2^2p-1)(2p+1)    for _0=_2p^*,

then the smooth solution ϕ of (<ref>) can blow up in finite time and further the shock is formed.











 §.§ Some remarks


Note that
(<ref>) can be easily fulfilled for any given smooth function ϕ_0
and the choice of ϕ_1(s,ø)=-_sϕ_0(s,ø).








Under the outgoing constraint conditions (<ref>) with large enough p and some number _0, for the short pulse initial data,
the authors in <cit.> have
established  the existence of global smooth solutions
for the 3D semi-linear wave system satisfying the first null condition
and for n-dimensional (n=2,3) relativistic membrane equations, respectively.	
Note that, as illustrated in <cit.>, the largeness of  p and some suitable _0 for (<ref>)
play the  key roles in the arguments of <cit.>.
Note that the equation of the form (<ref>) studied in <cit.> is a special case with k=2
and h^β(ϕ)≡0 in (<ref>). In addition, the particular divergence structure in <cit.> makes it possible to introduce an elaborate vector field which approximates _t+_r effectively and to obtain the global existence of the solution with the help of the method in <cit.>. However, it seems difficult to adopt such an approach in <cit.> to treat the general (<ref>) with the short pulse initial data (<ref>) under the weaker outgoing constraint condition (<ref>) and the optimal scope of ε_0. On the other hand, some recent progress on the global large data solutions of (<ref>)
can be referred to <cit.> and <cit.>.



 Our analysis is strongly motivated by the geometric approach pioneered by D. Christodoulou <cit.>, whose original purpose was to study the formation of shocks in multi-dimensional hyperbolic systems with genuinely nonlinear conditions, and subsequently for second order wave equations that do not satisfy the corresponding null conditions. See also <cit.> for further details.



 Compared to <cit.> and <cit.>, the main new difficulty in this paper is how to get the precise estimates of related quantities which makes possible to get global solution when _0 approaches _k^*. To this end, near the outgoing light cone surface, by identifying new suitable unknown variables (e.g. (<ref>), (<ref>), (<ref>)) and utilizing some fundamental identities (e.g. (<ref>), (<ref>)) that arise under the higher-order null conditions and the corresponding null frames, we are able to estimate accurately the solution and determine the optimal smallness exponent for the short pulse initial data (for example, the higher order null conditions help us rewrite (_φ_γg^β) L_ L_β L_γ in (<ref>) as (<ref>), and the resulting good unknown 𝒜 is defined in (<ref>). The dedicate estimate of 𝒜 in (<ref>) yields (<ref>) with the aid of (<ref>), which is crucial for closing energy estimates in (<ref>) and (<ref>) when _0 approaches _k^*). Within the interior of the outgoing light cone, based on precise boundary estimates of the solution on  C̃_2δ={(t,x):t≥ t_0,t-r=2δ} (see Proposition <ref>) due to the condition 0<_0<_k^*, we derive global spacetime weighted energy estimates for the related Goursat problem.





 §.§ Sketch of the proof and some notations


We now sketch the proof of Theorem <ref>.
As in <cit.>, the proof of Theorem 1.1 relies on some a priori uniform energy estimates and the continuous induction
argument. Let A_2δ={(t,x):t≥ 1+2δ,0≤ t-r≤ 2δ}
and B_2={(t, x): t≥ 1+2, t-r≥ 2}, which are near the outermost outgoing conic surface  C_0={(t, x): t≥ 1+2, t=r}
and inside the light cone {(t,x): t≥ 1+2, t≥ r}, respectively.
Based on the local existence of solution ϕ with some desired properties
for t∈[1,1+2δ], both the global uniform energy estimates
in A_2δ and B_2 will be established.

We start with estimates of the solution ϕ of (<ref>) in A_2δ.
Using the geometric framework of D. Christodoulou in <cit.>,
we shall prove that the outgoing characteristic conic surfaces never intersect as in  <cit.>.
To this end,
as in <cit.>, the “optical function" u of (<ref>) can be defined as

    {   ∑_,β=0^2g^β(ϕ)_u_βu=0,
       u(1+2δ,x)=1+2δ-r,
    .

and subsequently  the inverse foliation density is

    μ=
    -(∑_=0^2g^0_ u)^-1,

where g^β=g^β(ϕ). Set the metric g=∑_,β=0^2g_βdx^ dx^β
with (g_β) being the inverse matrix of (g^β) and

    L=-μ∑_,β=0^2g^β_ u_β.

Extend the local coordinate þ on the standard circle 𝕊^1 by

    {    Lϑ=0,
       ϑ|_t=1+2=þ.
    .

As in <cit.>, perform the change of coordinates:
(t, x^1, x^2)⟶ (s, u, ϑ) near C_0 with

    [ s=t,   u=u(t,x),  ϑ=ϑ(t,x).;                             ]

Then X:=ϑ is a tangent vector on the curved circle S_s,u:={(s',u',ϑ):s'=s,u'=u}.
Under the suitable bootstrap assumptions on the smallness and time decay rate of  ϕ (see (⋆)
in Section <ref>), we will show that the inverse foliation density
μ satisfies | Lμ|≲δ^k(1-ε_0)s^-(k+2)/2μ+δ^k-(k+1)ε_0s^-(k+1)/2.
By μ(1+2δ,x)∼ 1 and through the integration along integral curves of L, μ∼ 1 can
be derived for small >0 and _0∈ (0,_k^*). The positivity of μ implies that the outgoing characteristic conic surfaces never
intersect as long as the smooth solution ϕ to (<ref>) with (<ref>)
exists. Set φ=(φ_0, φ_1, φ_2):=(_tϕ, _1ϕ, _2ϕ).
Then φ solves a quasilinear wave system:

    μ_gφ_γ=F_γ(φ, φ),  γ=0,1,2,

where _g=1√( g)_(√( g)g^β_β), and F_γ are smooth functions
in their arguments. Let Ψ^m+1_=Z^m+1φ_,
where Z stands for one of some chosen first order vector fields. It follows from (<ref>)
and direct computation on the commutator [μ_g, Z^m+1] that

    μ_gΨ_^m+1=Φ_^m+1

with Φ_^m+1 containing the (m+2)-th order derivatives of φ. In fact,
both ∇ Z^mtrχ and ∇^2Z^mμ appear in the expression of Φ_^m+1,
where χ_XX=g(𝒟_X L, X) is the second fundamental form of S_s,u
with 𝒟 being the Levi-Civita connection of g, and trχ is the trace of χ on S_s,u.
Following the analogous procedures in Section 3-Section 11 of <cit.> or
in Section 2-Section 9 of <cit.>, we can derive the energy estimates for (<ref>)
to obtain the appropriate smallness orders of  and time decay rates of Ψ_^m+1. However, in order
to look for the optimal exponent _k^* related to the k-th null condition, it is necessary to derive
the optimal smallness order of  and precise time decay rate for all related quantities.
This gives rise to certain essential difficulties.
Some of these key difficulties and our new strategies to overcome them are sketched as follows (which
are different from those in <cit.>):

∙  Control Z^m+1μ_L^2 precisely by the corresponding  energies and fluxes so that the
highest (m+1)-order  energies of φ
admit the required optimal smallness orders of  and further the bootstrap assumptions can be closed.
In this process, note that Z^m+1μ_L^2
can be estimated from  Z^m+1 Lμ_L^2 by integrating along integral curves of L.
In addition, due to the k-th null condition, Lμ can be composed of Lϕ
and dϕ with  d being a differential on S_s,u. Thus,
in order to estimate Z^m+1 Lμ_L^2, one needs to treat both Z^m+1 Lϕ_L^2
and dZ^m+1ϕ_L^2. Unfortunately, although it has been found that Z^m+1 Lϕ_L^2
admits a better time decay rate than dZ^m+1ϕ_L^2 (see Lemma <ref>), the optimal smallness order
of  cannot be obtained. Thus more careful analysis on the structure of Lμ is needed. A key observation here is that
𝒜
=g^β, a_3⋯_kø_ø_βø_(m_ab d^Xx^b d_Xϕ)ø__3⋯ø__k is a good unknown
since Z^m+1𝒜_L^2 admits the higher order smallness  of  and better time decay rate than dZ^m+1ϕ_L^2.
Note that Z^m+1 Lμ consists of mainly Z^m+1𝒜 and Z^m+1 Lϕ,
moreover, Z^m+1𝒜_L^2 and Z^m+1 Lϕ_L^2 are bounded by R^mμ_L^2, where R is a specific vectorfield of Z which is projected by Ø on S_s,u and  is a Laplace operator of S_s,u.
Thus the remaining task is to treat R^mμ_L^2.

∙ Deal with R^mμ_L^2.
Note that Φ_^m+1 encompasses both dZ^mtrχ and ∇^2Z^mμ. Consequently, as demonstrated in <cit.>, we can estimate the L^2 norms of dZ^mtrχ and Z^mμ simultaneously, not solely R^mμ_L^2.
By deriving the transport equations for μ and trχ along L and making full use of the k-th null condition, the estimates of dZ^mtrχ_L^2
and Z^mμ_L^2 can be precisely controlled by the energies E_i,≤ m+2 (i=1,2) and the fluxes  F_1,≤ m+2
(see (<ref>)-(<ref>) and (<ref>)-(<ref>)).
In the process, one of the main observations is that ∑_j=1^3 dφ_j· dx^j may be represented by L (see (<ref>)). This leads to that the estimate on the basic terms dZ^m+1φ_j_L^2 in
dZ^mtrχ_L^2 can be bounded by  ∫_0^uF_1,m+2(s,u')du' (see (<ref>)-(<ref>)).
Then all required quantities are eventually estimated through the optimal smallness orders of 
and better time decay rates, which are the keys to treat the case when _0 is near  _k^* and further to
close the arguments of bootstrap assumptions on .




On the other hand, in order to study the global Goursat problem for (<ref>) and derive the a priori
uniform estimates of ϕ in B_2 by energy methods,
the properties for |^ L^qϕ|≲δ^2-ε_0t^-1/2-q (L=_t+_r) on C_2δ
with the higher order smallness O(δ^2-ε_0) and the time decay rate of t^-1/2-q
are crucial since the related boundary integrals on C_2 in the energy estimates require the
the smallness order of O(δ^2-ε_0).  However, all the estimates on ϕ in  A_2
are obtained under the modified frame { L, T, R},
where L, T, R are some suitable vectors approximating L, -_r=L-L2 (L=_t-_r),
Ø=x^1_2-x^2_1 respectively on the time t=1+2δ. To get the L^∞ estimates on ϕ under the
derivatives {L,L, Ø}
and further obtain the required smallness and time decay rate of Lϕ, we need to express the vector fields {L,L, Ø}
by { L, T, R}. As in <cit.>, the most noteworthy expression is L=c_L L+c_TT+c_ØØ,
where c_L, c_T and c_Ø are
smooth functions.  One expects that the desired derivative Lϕ and its derivatives can admit the smallness of higher
orders with respect to 
and faster time decay rate. Unfortunately, Tϕ and its derivatives have not enough smallness and time decay rate as needed,
it is hoped as in <cit.> that the coefficient c_T can make up for these deficiencies. The generality of equation (<ref>) and
the optimal scope of _0∈ (0, _k^*)
make it more difficult to analyze the smallness orders and the time decay rates of the related coefficients c_T, c_Ø and so on.
Thanks to the k-th null condition (<ref>), we can find
their governing differential equations along L and subsequently
get the desired estimates of  c_L, c_T and c_Ø.



The rest of the paper is organized as follows. In Section <ref>, first, some preliminaries on
the related differential geometry and definitions
are given. Based on the studies of the local solution to (<ref>), the crucial bootstrap assumptions
are given and further some important quantities are estimated in A_2.
In Section <ref>, we first carry out the energy estimates for the linearized equation μ_gΨ=Φ
and define some suitable higher order weighted energies and fluxes as in <cit.>. Subsequently,
the higher order L^2 estimates of some related quantities with the precise smallness orders of 
and time decay rates are obtained.
In Section <ref>, the top order L^2 estimates on the derivatives of χ and μ are established.
On the other hand, the estimates for the error terms are derived in Section <ref>.
In Section <ref> and Section <ref>, the global uniform estimates of the solution to (<ref>) in A_2
and B_2 are obtained, respectively. Therefore, the proof of Theorem <ref>  can be completed
by the continuous induction argument.
In addition, some computations on the deformation tensor and commutator relations are given in Appendix A.
The proof on the  local solution properties of (<ref>) for 1≤ t≤ 1+2 is given in Appendix B.



Through the whole paper, unless stated otherwise, Greek indices {, β, γ, ⋯} corresponding to the
time-space coordinates are chosen in {0, 1, 2}, Latin indices {i, j, a, b, c, ⋯} corresponding to the
spatial coordinates are {1, 2}, and the Einstein summation convention to sum over repeated upper
and lower indices is used. In addition, the convention f_1≲ f_2 means that there exists a generic positive constant
C independent of the parameter >0 in (<ref>) and the variables (t,x) such that f_1≤ Cf_2. The coefficients g^β(ϕ) are denoted as g^β by convenience.
Denote g=g_βdx^ dx^β by a Lorentz metric, where (g_β) is the inverse matrix of (g^β).

Finally, the following conventional notations will be used too:

    t_0:=1+2δ,
       ø^i:=ø_i=x^ir, i=1,2,  ø_⊥:=(-ø^2, ø^1),
       _x=∇_x=(_1, _2,),   L:=_t+_r,  L:=_t-_r,
       Ø:=ϵ_i^jx^i_j,   S:=t_t+r_r=t-r2L+t+r2L,
       H_i:=t_i+x^i_t=ø^i(r-t2L+t+r2L)+tø^i_⊥rØ,
       Σ_t:={(s,x): s=t, x∈ℝ^2},
       φ=(φ_0, φ_1, φ_2)=(_0ϕ, _1ϕ, _2ϕ),

where ϵ_1^2=1, ϵ_2^1=-1, ϵ_i^i=0 for i=1,2.






§ PRELIMINARIES AND SOME ESTIMATES UNDER BOOTSTRAP ASSUMPTIONS




 §.§ Geometry and basic equalities under null frames


We give some preliminaries on the
related geometry and definitions, which will be
utilized as basic tools to establish the a priori global estimates on the
smooth solution ϕ to (<ref>) in A_2. This part is completely similar to <cit.>,
so we just list the notations and state the facts without going into the details.

The definitions of the optical function u, the inverse foliation density μ and
the definition of ϑ have been
given in (<ref>), (<ref>) and (<ref>), respectively. Note that

    L̃=-g^β_ u_β

is a tangent vector field of the outgoing light cone { u=C} and L̃t=μ^-1 holds.

Set

    L=μL̃,  L=μ L+2T  with T=μT̃
    and T̃=-g^ 0_- L.

Then
L, L and  X=ϑ
form a  null frame with respect to the metric g.

In the new coordinate (s, u, ϑ) of (<ref>), the following subsets can be defined (see Figure <ref>):



    Σ_s^u:={(s',u',ϑ): s'=s,0≤ u'≤  u},   u∈ [0, 4δ],
       C_u:={(s',u',ϑ): s'≥ 1+2δ, u'=u},
       C_u^s:={(s',u',ϑ): 1+2δ≤ s'≤ s, u'=u},
       S_s, u:=Σ_s∩ C_u,
       D^s, u:={(s', u',ϑ): 1+2δ≤ s'<s, 0≤ u'≤ u}.



	


The following geometric notations are needed.
 For the metric g on the spacetime,
	
		
  * g=(g_ij) is defined as the induced metric of g on Σ_t, i.e., g(U,V)=g(U,V) for any tangent vectors U and V of Σ_t;
		
  * Π_^β:=δ_^β-δ_^0 L^β+ L_T̃^β is the
projection tensor field on S_s,u of type (1,1), where δ_^β is Kronecker delta;
		
  * define ξ:=Πξ as the tensor field on S_s,u for the (m,n)-type spacetime tensor field
ξ, whose components are 
    ξ^_1⋯_m_β_1⋯β_n:=(Πξ)^_1⋯_m_β_1⋯β_n
    =Π_β_1^β_1'⋯Π_β_n^β_n'Π__1'^_1⋯Π__m'^_mξ^_1'⋯_m'_β_1'⋯β_n'.

		Specially, g=( g_β) is the induced metric of g on S_s,u;
		
  * g^XX is defined as the reciprocal of g_XX with g_XX=g(X, X);
		
  * 𝒟 and ∇ denote the Levi-Civita connection of g and g, respectively;
		
  * _g:=g^β𝒟^2_β, := g^XX∇^2_X;
		
  * ℒ_Vξ denotes the Lie derivative of ξ with respect to V and ℒ_Vξ:=Π(ℒ_Vξ) for any tensor field ξ and vector V;
		
  * when ξ is a (m,n)-type spacetime tensor field, the square of its norm is defined as
		
    |ξ|^2:=g__1_1'⋯ g__m_m'g^β_1β_1'⋯ g^β_nβ_n'ξ_β_1⋯β_n^_1⋯_mξ_β_1'⋯β_n'^_1'⋯_m'.

	



In the null frame { L, L, X},
the second fundamental forms χ and σ are defined as

    χ_XX:=g(𝒟_X  L, X),  σ_XX:=g(𝒟_XT̃,X).

Meanwhile, define the one-form tensors ζ and η as

    ζ_X:=g(𝒟_X  L,T̃),  η_X:=-g(𝒟_X T,  L).

For any vector field V, denote its associate deformation tensor by

    ^(V)π_β:=g(𝒟_V,_β)+g(𝒟_βV,_).

For t≥ t_0, the following error terms are defined as

    Ľ^0:=0, Ľ^i:= L^i-x^iϱ, Ť^i:=T̃^i+x^iϱ,χ̌_XX:=χ_XX-1ϱ g_XX,

here and below ϱ=t-u.
Note that in the new coordinate system (s, u, ϑ), it holds that
L= s and  T= u-Ξ with Ξ=Ξ^XX for some smooth function Ξ^X.
In addition, <cit.> gives


In domain D^s, u, the Jacobian determinant of the map (s, u, ϑ)→
(x^0, x^1, x^2) is

    (x^0, x^1, x^2)(s, u, ϑ)=μ(g)^-1/2√( g_XX).



On 𝕊^1, the vector field Ω=x^1_2-x^2_1
is taken as the tangent derivative.
In order to project Ω on S_s, u, as in (3.39b) of <cit.>, we define

    R:=ΠΩ,   d:=Π d

to be the rotation vectorfield and the differential of S_s, u, respectively. Note that the explicit expression of R is

    R=(δ_j^i+g_ j L^T̃^i)Ω^j_i
    =Ω-g_ajT̃^aΩ^jT̃.

For brevity, one writes

    υ:=g_abT̃^aΩ^b=g_ajŤ^aϵ_i^jx^i-(g_aj-m_aj)ϵ_i^j x^ix^aϱ.

Then

    R=Ω-υT̃.



For any continuous function f and tensor field ξ, define

    ∫_S_s, uf:=∫_S_s, ufdν_ g=∫_𝕊f(s, u,ϑ)√( g_XX(s, u,ϑ))dϑ,
       ∫_ C^s_ uf:=∫_t_0^s∫_S_τ, uf(τ, u,ϑ) dν_ gdτ,  ∫_Σ_s^ uf:=∫_0^ u∫_S_s, u'f(s, u',ϑ) dν_ gd u',
       ∫_D^s, uf:=∫_t_0^s∫_0^ u∫_S_τ, u'f(τ, u',ϑ)dν_ gd u'dτ,  ξ_s,u:=√(∫_Σ_s^ u|ξ|^2).



Finally, for reader's convenience, we recall the  notations of contraction and trace as follows.

(1) If ξ is a (0,2)-type spacetime tensor, Λ is a 1-form, U and V are vector fields, then the contraction
of ξ with respect to U and V is defined as

    ξ_UV:=ξ_βU^V^β,

and the contraction of Λ with respect to U is

    Λ_U:=Λ_U^.

(2) If ξ is a (0,2)-type tensor on S_s,u, the trace of ξ is defined as
	
    trξ:= g^XXξ_XX.






Note that (g_β) is the inverse matrix of (g^β).
By g^βg^λκg_κβ=g^λ, it holds that

    g^βg^λκ(_φ_γ g_κβ)=-_φ_γ g^λ.


With the help of χ, σ, ζ and η
defined as before, one can derive some basic equalities in the frame { L, L, X}
or the frame {T, L, X}.

Let

    G_β^γ:=_φ_γ g_β.

For any vector fields U=U^_ and V=V^_, set G_UV^γ=G_β^γ U^ V^β. One has the following transport equation of μ along L, whose proof is exact similar to <cit.>.


μ satisfies
 
    Lμ=-12μ G_ L L^γ Lφ_γ-μ G_T̃ L^γ Lφ_γ+12G_ L L^γ Tφ_γ.




In the analysis later, special attention is needed for the terms containing Tφ_γ since  Tφ_γ has the worse
smallness and slower time decay rate than the ones of Z̅φ_, where Z̅∈{ L, R}.



As in <cit.>, the following explicit expressions hold:

    ζ_X=-12{G_T̃ L^γ d_Xφ_γ+G_T̃T̃^γ d_Xφ_γ-G_XT̃^γ Lφ_γ+G^γ_X LT̃φ_γ},
       σ_XX=-G_X L^γ d_Xφ_γ-G_XT̃^γ d_Xφ_γ+12G_XX^γ Lφ_γ+12G_XX^T̃φ_γ-χ_XX.


Note that the quantity “deformation tensor" defined in (<ref>)
 will occur in the forthcoming energy estimates. It is necessary  to compute the components
 of ^(V)π in the null frame { L, L,X}.
 This will be given in Appendix A. In addition, as in <cit.>,
it follows from (<ref>) and (<ref>)-(<ref>) that
the vector fields L and T appear frequently. Based on this, we need the equations for L^i and Ľ^i under the actions of the derivatives of null frame {T, L, X} as well as the connection coefficients of the frames.


	It holds that
	
    L L^i=12G^γ_ L L Lφ_γT̃^i
    -{G^γ_X L Lφ_γ-12G^γ_ L L d_Xφ_γ} d^Xx^i,
        L(ϱĽ^i)=ϱ L L^i,
        d_X L^i=trχ d_Xx^i-{G^γ_ LT̃ d_Xφ_γ
    +12G^γ_T̃T̃ d_Xφ_γ}T̃^i-12G_XX^γ Lφ_γ d^Xx^i,
        d_XĽ^i=trχ̌ d_Xx^i-{G^γ_ LT̃ d_Xφ_γ
    +12G^γ_T̃T̃ d_Xφ_γ}T̃^i-12G_XX^γ Lφ_γ d^Xx^i,
       TL^i={ d_Xμ-12μ G^γ_T̃T̃ d_Xφ_γ
    -G^γ_X LTφ_γ} d^Xx^i+{12μ G^γ_ L L Lφ_γ
             +μ G^γ_T̃ L Lφ_γ+12μ G^γ_T̃T̃ Lφ_γ} L^i+12G^γ_ L LTφ_γT̃^i

and

    𝒟_ L L=(μ^-1 Lμ) L,  𝒟_T L=- Lμ L+η^XX,  𝒟_X L=-ζ_X L+trχX,
       𝒟_ LT=- Lμ L-μζ^XX,  𝒟_TT=μ Lμ L
    +(μ^-1Tμ+ Lμ)T-μ( d^Xμ) X,
       𝒟_XT=μζ_X L+μ^-1η_XT+μtrσX,
       𝒟_XX=∇_XX+(σ_XX+χ_XX) L+μ^-1χ_XXT.

In addition, the covariant derivatives of the frame {L,  L, X} are

    𝒟_L L=- Lμ L+2η^XX,  𝒟_ LL=-2μζ^XX,
       𝒟_LL=(μ^-1Lμ+
     Lμ)L-(2μ d^Xμ) X.



The proofs are similar to these in Lemma 3.4-Lemma 3.6 in <cit.> so are omitted.





As in <cit.>, to estimate φ, one needs to
derive the equations for φ_γ under the action of the covariant wave operator _g=g^β𝒟^2_β=g^β_β^2-g^βΓ_β^λ_λ, where Γ_β^γ are Christoffel symbols with

    Γ_β^γ=12g^γκ(G_κβ^ν_φ_ν
    +G_κ^ν_βφ_ν-G_β^ν_κφ_ν).


Note that

    _gφ_γ=   _φ_νg^β(-_νφ_γ_βφ_+_νφ__βφ_γ)
    +12(g^β_φ_νg_β)g^λκ_κφ_ν_λφ_γ.

In terms of

    g^β=- L^ L^β-T̃^ L^β- L^T̃^β+( d_Xx^)( d^Xx^β),
       _=δ_^0 L-μ^-1 L_ T+g_ i( d^Xx^i)X,

then in the frame {T,  L, X}, (<ref>) can be rewritten as

    μ_g φ_γ=   12(g^βG_β^
    ν)(μ d^Xφ_ν) d_Xφ_γ+f_1(φ, L^1,  L^2)φ^k-1(
    [           Lφ; ( d_Xx) d^Xφ;              ])Tφ_γ
       +f_2(φ,  L^1,  L^2)φ^k-1(
    [            Tφ;          μ Lφ; μ( d_Xx) d^Xφ;               ])
    (
    [           Lφ_γ; ( d_Xx) d^Xφ_γ;                ]),

where f_1 and f_2 are the generic smooth functions with respect to their arguments, and (
[ A_1;   ⋮; A_n;     ])
(
[ B_1;   ⋮; B_m;     ]) stands for the terms which conclude  A_iB_j (1≤ i≤ n,1≤ j≤ m).
It can be checked that in the right hand side of (<ref>), the worst factor Tφ
is always accompanied by the “good" multipliers
such as Lφ and dφ.





 §.§ The bootstrap assumptions and some related estimates


As in <cit.>, the existence and properties of local in time solution to (<ref>) with (<ref>), (<ref>) and
(<ref>)-(<ref>) can be established easily in the following Theorem,
whose proof is given in Appendix B.


Under the assumptions (<ref>) and
(<ref>)-(<ref>), when δ>0 is small, equation (<ref>) with (<ref>)
admits a local smooth solution ϕ∈ C^∞([1, t_0]×ℝ^2). Moreover,
for m,n,p∈ℕ_0, κ∈ℕ_0^3, it holds that

	
  * 
    |L^m^κØ^pϕ(t_0,x)|≲δ^2-|κ|-ε_0,   r ∈[1-2δ, 1+2δ],
       |L^n^κØ^pϕ(t_0,x)|≲δ^2-|κ|-ε_0,   r∈ [1-3δ, 1+δ].

	
  * 
    |^κØ^pϕ(t_0,x)|≲δ^3-|κ|-ε_0,
    	   r∈ [1-3δ, 1+δ].

	
  * 
    |L^mL^nØ^pϕ(t_0,x)|≲δ^2-ε_0,   r∈[1-2δ,1+δ].





It should be pointed out that the special null condition structures of 2D Chaplygin gases are extensively
applied in <cit.> and simplify many identities compared to the general null conditions in the paper.
Therefore, although the remainder of this subsection is analogous to Section 3 in <cit.>,
the differential equation of μ and  the last term on the right hand side of
(<ref>)  require  to be analyzed more delicately.

To show the global existence of solution ϕ to (<ref>) with (<ref>) near C_0,
we will utilize the bootstrap argument and make the following bootstrap assumptions in D^s,u:

    δ^l LZ^φ_γ_L^∞(Σ_s^u)+δ^l dZ^φ_γ_L^∞(Σ_s^u)≤  Mδ^1-ε_0 s^-3/2,
       δ^lLZ^φ_γ_L^∞(Σ_s^u)
    +δ^l-1Z^φ_γ_L^∞(Σ_s^u)≤  Mδ^-ε_0 s^-1/2,
       ∇^2φ_γ_L^∞(Σ_s^u)+
    ∇^2ϕ_L^∞(Σ_s^u)≤  Mδ^2-ε_0 s^-5/2,
       δ^l LZ^ϕ_L^∞(Σ_s^u)+δ^l dZ^ϕ_L^∞(Σ_s^u)≤  Mδ^2-ε_0 s^-3/2,
       δ^lZ^ϕ_L^∞(Σ_s^u)≤ Mδ^2-ε_0s^-1/2,
    ⋆

where ||≤ N, N is a fixed large positive integer, M is some positive constant to be suitably chosen
(at least double bounds of the corresponding quantities on time t_0 in Theorem <ref>),
Z∈{ϱ L,T,R}, and l is the number of T included in Z^.

One can now derive a rough estimate of μ under the assumptions (⋆) as follows. By

    1=g_ijT̃^iT̃^j=(1+O(M^kδ^k(1-ε_0) s^-k/2))∑_i=1^2|T̃^i|^2,

then

    |T̃^i|, | L^i|≤ 1+O( M^kδ^k(1-ε_0) s^-k/2).

Moreover, by | dx^i|^2= g^ab d_ax^i d_bx^i=g^ii+(g^0i)^2-(T̃^i)^2,
it follows from (⋆) and (<ref>) that

    | dx^i|≲ 1.

In addition, | L(ϱĽ^i)|≲ M^kδ^k(1-ε_0)s^-k/2 holds
by (<ref>) and (⋆), which further gives that through integrating L(ϱĽ^i)
along integral curves of L and
using Ť^i=-g^0i-Ľ^i,
 
    |Ľ^i|+|Ť^i|≲ M^kδ^k(1-ε_0)s^-1ln s.

Due to g_ij(Ť^i-x^iϱ)(Ť^j-x^jϱ)=1, then
 
    (g_ijø^iø^j)r^2ϱ^2-(2g_ijŤ^iø^j) rϱ+g_ijŤ^iŤ^j-1=0.

 Thus
 
    ϱ̌:=rϱ-1=1-g_ijø^iø^j-g_ijŤ^iŤ^j+2g_ijŤ^iø^j√(g_ijø^iø^j-(g_ijø^iø^j)(g_abŤ^aŤ^b)+(g_ijŤ^iø^j)^2)
     +g_ijø^iø^j-g_ijŤ^iø^j.

This yields
 
    |ϱ̌|≲ M^kδ^k(1-ε_0)s^-1ln s,

 and hence  by the definition of υ in (<ref>),
 
    |υ|≲ M^kδ^k(1-ε_0)ln s.


To estimate μ, one needs to handle the last term in (<ref>).
It follows from (<ref>) and (<ref>) that

    G_ L L^γ Tφ_γ=   G_ L L^0T^i( Lφ_i)
    -G^γ_ L L L_γT̃^i Tφ_i+μ g_γ iG_ L L^γ ( d^Xx^i)T̃^j d_Xφ_j
    
    =   G_ L L^0T^i( Lφ_i)
    +(_φ_γg^β) L_ L_β L_γT̃^i Tφ_i
    +μ g_γ iG_ L L^γ ( d^Xx^i)T̃^j d_Xφ_j.

The null condition (<ref>) means that
the underline factor in (<ref>) can be written as

    -G_ L L^ L_=(_φ_γg^β) L_ L_β L_γ=kg^β,γγ_2⋯γ_kφ_γ_2⋯φ_γ_k L_ L_β L_γ+(_φ_γh^β) L_ L_β L_γ
    
    =   kg^β,γγ_2⋯γ_kφ_γ_2⋯φ_γ_km_'m_ββ'm_γγ'x̃^'ϱx̃^β'ϱx̃^γ'ϱ+f(φ, Ľ^1, Ľ^2,xϱ)(
    [      φ^k; Ľ^1φ^k-1; Ľ^2φ^k-1 ])
    
    =   kg^β,γγ_2⋯γ_kφ_γ_2⋯φ_γ_kø_ø_βø_γ+f(φ, Ľ^1, Ľ^2,xϱ,ϱ̌)(
    [      φ^k; Ľ^1φ^k-1; Ľ^2φ^k-1;  ϱ̌φ^k-1 ])
    
    =   f(φ, Ľ^1,Ľ^2,xϱ,ϱ̌)(
    [                 φ^k;            Ľ^1φ^k-1;            Ľ^2φ^k-1;             ϱ̌φ^k-1;          ( Lϕ)φ^k-2; ( d^Xx)( d_Xϕ)φ^k-2 ]),

where x^iϱ=ϱ̌ø_i+ø_i, φ_γ_j=δ_γ_j^0 Lϕ- L_γ_jT̃^aφ_a+g_γ_ja d^Xx^a d_Xϕ, (<ref>) and (<ref>) are used,
(x̃^0, x̃^1, x̃^2)=(ϱ, x^1, x^2)
and (ø_0, ø_1, ø_2)=(-1, ø^1, ø^2). Therefore,
|(_φ_γg^β) L_ L_β L_γ|≲ M^kδ^k(1-ε_0)s^-k/2
holds by (<ref>)-(<ref>), (<ref>) and (⋆).
Subsequently, it follows from the expression (<ref>) that

    |G_ L L^γ Tφ_γ|≲μ M^kδ^k(1-ε_0)s^-(k+2)/2+M^k+1δ^k-(k+1)ε_0s^-(k+1)/2.

Together with (⋆) and (<ref>),  it is easy to
obtain | Lμ|≲ M^kδ^k(1-ε_0)s^-(k+2)/2μ+M^k+1δ^k-(k+1)ε_0s^-(k+1)/2. For δ>0  suitably small, by
integrating Lμ along integral curves of L and noting μ=1√((g^0iø_i)^2+g^ijø_iø_j)
=1+O(δ^k(1-ε_0)) on Σ_t_0,
one directly has that due to _0<_k^*,

    μ=1+O( M^k+1δ^k-(k+1)ε_0)

which has a positive lower bound as long as (⋆) holds and  is small enough.


To improve the assumptions (⋆), one can rewrite equation (<ref>) in the new frame
{L,  L, X} as

    LLφ_γ+12ϱLφ_γ
    =μφ_γ+H_γ,

here one has used the fact μ_gφ_=-𝒟_ LL^2φ_
+μ g^XX𝒟_X^2φ_=- LLφ_-2μζ^X( d_Xφ_)+μφ_-μ(trσ+trχ) Lφ_
-trχ Tφ_
by (<ref>) and (<ref>). In addition, by (<ref>)-(<ref>),

    H_γ=   -(trχ̌)Tφ_γ+12ϱμ Lφ_γ+f_1(φ, L^1,  L^2)φ^k-1(
    [           Lφ; ( d_Xx) d^Xφ;              ])Tφ_γ
       +f_2(φ,  L^1,  L^2,  dx,  g)φ^k-1(
    [   Tφ; μ Lφ; μ dφ;      ])
    (
    [ Lφ_γ; dφ_γ;      ]).


As in <cit.>, it is noted from the expression of H_ that if there are the terms including the factor Tφ_
which admits the “bad" smallness or slow time decay rate,
then there always appear another product factor equipped with the “good"
smallness and fast time decay rate, e.g.
trχ̌,  Lφ and so on.


Unless stated otherwise, from now on, in this subsection, the pointwise estimates for the related quantities
are all carried out inside domain D^s,u.


It follows from the expression (<ref>) that the elaborate
estimate of Lφ_γ can be achieved by integrating  (<ref>)
along integral curves of L. To this end, one should estimate χ̌ and
other terms in H_.






To estimate χ̌, one needs the structure equation for χ.
Analogously to <cit.>, one has




	The second fundamental form χ and its “error" form χ̌, defined in (<ref>) and (<ref>)
respectively, satisfy the following structure equations:

    Lχ_XX=   -G_X L^γ( d_X Lφ_γ)+12G_ L L^γ∇_X^2φ_γ+12G_XX^γ( L^2φ_γ)
    -{12G_ L L^γ+G_T̃ L^γ} Lφ_γχ_XX
       +(G_X L^γ d_Xφ_γ)trχ+(trχ)χ_XX+f(φ,  dx, L^1,  L^2)φ^k-2(
    [ Lφ; dφ;    ])^2,


    ℒ_Tχ_XX=   ∇_X^2μ-12{μ G_T̃T̃^γ∇_X^2φ_γ+2G_X L^γ d_XTφ_γ-G_XX^γ LTφ_γ}-μ(trχ)^2 g_XX
       -{G_T̃ L^γ d_Xφ_γ+G_T̃T̃^γ d_Xφ_γ-G_XT̃^γ Lφ_γ}d_Xμ+Υ·χ_XX
       +f_1(φ,  dx,  L^1,  L^2)φ^k-2(
    [ φ d L^1; φ d L^2;      Lφ;      dφ;         ])(
    [ μ Lφ; μ dφ;   Tφ;      ])
    +f_2(φ,  L^1,  L^2)φ^k-1∇^2x(
    [ μ Lφ;   Tφ;      ]).



And hence,

    Lχ̌_XX=   -G_X L^γ( d_X Lφ_γ)+12G_ L L^γ∇_X^2φ_γ+12G_XX^γ( L^2φ_γ)-{12G_ L L^γ+G_T̃ L^γ} Lφ_γχ̌_XX
       - g_XXϱ{12G_ L L^γ+G_T̃ L^γ} Lφ_γ+(G_X L^γ d_Xφ_γ)trχ̌+1ϱ(G_X L^γ d_Xφ_γ)+(trχ̌)χ̌_XX
       +f(φ,  dx, L^1, L^2)φ^k-2(
    [ Lφ; dφ;    ])^2,


    ℒ_Tχ̌_XX=   ∇_X^2μ-12{μ G_T̃T̃^γ∇_X^2φ_γ+2G_X L^γ d_XTφ_γ-G_XX^γ LTφ_γ}-μ(trχ̌)^2 g_XX
       -{G_T̃ L^γ d_Xφ_γ+G_T̃T̃^γ d_Xφ_γ-G_XT̃^γ Lφ_γ}d_Xμ+Υ·χ̌_XX+12ϱΘ· g_XX
       +μ-1ϱ^2 g_XX+f_1(φ,  dx,  L^1, L^2)φ^k-2(
    [ φ d L^1; φ d L^2;      Lφ;      dφ;         ])(
    [ μ Lφ; μ dφ;   Tφ;      ])
       
    +f_2(φ,  L^1, L^2)φ^k-1∇^2x(
    [ μ Lφ;   Tφ;      ]),

where

    Υ=   12μ G_ L L^γ Lφ_γ+μ G_T̃ L^γ Lφ_γ-12 G_ L L^γ Tφ_γ-32μ G_X L^γ d^Xφ_γ-12μ G_XT̃^γ d^Xφ_γ
       +12 g^XXG_XX^γ Tφ_γ-12G_ LT̃^γ Tφ_γ+12μ g^XXG_XX^γ Lφ_γ,
    Θ=   μ G_ L L^γ Lφ_γ+2μ G_T̃ L^γ Lφ_γ-G_ L L^γ Tφ_γ+μ G_X L^γ d^Xφ_γ+3G_XT̃^γ d^Xφ_γ
       -μ g^XXG_XX^γ Lφ_γ- g^XXG_XX^γ Tφ_γ-G_ LT̃^γ Tφ_γ,

and f, f_1, f_2 are the generic smooth functions of their arguments.



Based on (<ref>), χ̌ can be estimated by integrating along integral curves of L.


	Under the assumptions (⋆), when δ>0 is small, it holds that
	
    |χ̌|=|trχ̌|≲ M^kδ^k(1-ε_0) s^-2ln s.

	Meanwhile,
	
    |χ|=1ϱ+O(M^kδ^k(1-ε_0) s^-2ln s).





	By trχ̌= g^XXχ̌_XX, then
	
    L(trχ̌)=-2(trχ̌)^2-2ϱtrχ̌+ g^XX Lχ̌_XX.

	Substituting (<ref>) into (<ref>), and using assumptions (⋆), (<ref>) and (<ref>) to estimate
	the right hand side of (<ref>) except χ̌ itself, one can get
	
    | L(ϱ^2trχ̌)|≲
    	M^kδ^k(1-ε_0)s^-k/2+M^kδ^k(1-ε_0)s^-(k+2)/2|ϱ^2trχ̌|
    	+ϱ^-2|ϱ^2trχ̌|^2.

	Thus, for small δ>0, one has
	
    |χ̌|≲ M^kδ^k(1-ε_0) s^-2ln s.



Additionally, it is noted that the operator R is just only the scaling operator r∇ under
the assumptions (⋆) and the estimate (<ref>), which are similar to the ones in <cit.>.



 Under the assumptions (⋆), when δ>0 is small,

	
  * if ξ is a 1-form on S_s, u, then
  
    (ξ_a R^a)^2∼ r^2|ξ|^2,
       |ℒ_Rξ|^2∼ r^2|∇ξ|^2+O(M^kδ^k(1-ε_0)s^-1ln s)|ξ|^2;


  * if ξ is a 2-form on S_s, u, then
  
    |ℒ_Rξ|^2∼ r^2|∇ξ|^2+O(M^kδ^k(1-ε_0)s^-1ln s)|ξ|^2

  and
  
    |∇^2ξ|≲ϱ^-2|ℒ_R^≤ 2ξ|.






To close the bootstrap
assumptions (⋆), one can improve the estimate of φ by (<ref>) to obtain the following proposition,
whose proofs are same as those in <cit.>.



	Under the assumptions (⋆), for any operate Z∈{ϱ L, T, R}, when δ>0 is small,
it holds that for m≤ N-1,

    |ℒ_ Z^m;l,pχ̌|≲ M^kδ^k(1-ε_0)-ls^-2ln s,  |ℒ_Z^m+1;l,p dx^j|≲δ^-l,  
       |Z^m+1;l,pĽ^j|+|ℒ_Z^m;l,p^(R)π|+|ℒ_Z^m;l,p^(R)π_ L|+|Z^m+1;l,pϱ̌|≲ M^kδ^k(1-ε_0)-ls^-1ln s,
        |ℒ_Z^m;l,pR|+|Z^m+1;l,pυ|≲ M^kδ^k(1-ε_0)-lln s,  |Z^m+1;l,pμ|≲ M^k+1δ^k(1-ε_0)-ε_0-l,
        |ℒ_Z^m;l,p^(T)π|≲ M^kδ^k(1-ε_0)-1-ls^-1+s^-l-1,   |ℒ_Z^m;l,p^(T)π_ L|≲ M^kδ^k(1-ε_0)-1-ls^-1,
       |ℒ_Z^m;l,p^(R)π_T|≲(M^kδ^k(1-_0)-l+M^2kδ^2k(1-_0)-1-l)s^-1ln s,

and

    |Z^m;l,pφ_γ(s, u, ϑ)|+|Z^m;l,pϕ(s, u, ϑ)|≲δ^2-l-ε_0s^-1/2,

 where (m;l,p) means the numbers of Z, T and ϱ L are m, l and p respectively.



Noted that as stated in the end of <cit.>, all the related constants
in Proposition <ref> can be independent of M. Therefore,
from now on, we may apply these estimates independent of M.







§ ENERGY ESTIMATES AND SOME HIGHER ORDER L^2 ESTIMATES


As in <cit.>, to close the bootstrap assumptions (⋆), one needs further refined estimates than those derived in Subsection <ref>.  Note that φ_γ satisfies the nonlinear equation (<ref>), and each
derivative of φ_ also fulfills similar equation with the same metric. Thus, one may focus on
the energy estimates for any smooth
function Ψ solving the following linear equation

    μ_gΨ=Φ,

for a given function Φ, where Ψ and its derivatives vanish on C_0^s.

As in <cit.>, we choose two multipliers V_1Ψ:=ϱ^2ι LΨ (0<ι<12)
and V_2Ψ:=LΨ, the energies E_i[Ψ](s, u) and fluxes F_i[Ψ](s, u)
are defined after performing integration by part to μ(_gΨ)(V_iΨ) over D^s,u (i=1,2),

    E_1[Ψ](s, u):=12∫_Σ_s^uμϱ^2ι{( LΨ)^2+| dΨ|^2},
       E_2[Ψ](s, u):=12∫_Σ_s^u{(LΨ)^2+μ^2| dΨ|^2},
       F_1[Ψ](s, u):=∫_C_u^sϱ^2ι( LΨ)^2,
       F_2[Ψ](s,u):=∫_C_u^sμ | dΨ|^2.

Thus,

    E_i[Ψ](s, u)-E_i[Ψ](t_0, u)+F_i[Ψ](s, u)=-∫_D^s, uΦ· V_iΨ-∫_D^s, u12μ Q_β[Ψ]^(V_i)π^β,

where
Q_β[Ψ]:=(_Ψ)(_βΨ)-12 g_βg^νλ(_νΨ)(_λΨ)
and ^(V_i)π is the deformation tensor with respect to the vector field V_i as defined in (<ref>). Since 12μ Q_β[Ψ]^(V_i)π^β (i=1,2) have the same expressions as (7.7) and (7.8) in <cit.>,
one then can derive that
 
    ∫_D^s, u-12 μ Q^β[Ψ]^(V_1)π_β
    ≲   δ^k(1-ε_0)-ε_0∫_t_0^sτ^-3/2 E_1[Ψ](τ,u)dτ+δ^-1∫_0^uF_1[Ψ](s,u')du'+δ∫_t_0^sτ^2ι-2E_2[Ψ](τ,u)dτ

and

    ∫_D^s, u-12 μ Q^β[Ψ]^(V_2)π_β
    ≲   δ^-1∫_0^uF_2[Ψ](s,u')du'+δ^-1∫_0^uF_1[Ψ](s,u')du'
    +∫_t_0^sτ^-2E_2[Ψ](τ,u)dτ.

And hence,

    δ E_2[Ψ](s, u)+δ F_2[Ψ](s, u)+E_1[Ψ](s, u)+F_1[Ψ](s, u)
    ≲   δ E_2[Ψ](t_0, u)+E_1[Ψ](t_0, u)+δ∫_D^s, u|Φ·LΨ|+∫_D^s, uϱ^2ι|Φ· LΨ|.

Choose Ψ=Ψ_γ^m+1=Z^m+1φ_γ and then
Φ=Φ_γ^m+1=μ_gΨ_γ^m+1 (m≤2N-6)
so that (<ref>) holds. Note that

    Φ_γ^m+1   =μ[_g,Z]Ψ_γ^m+Z(μ_gΨ_γ^m)-(Zμ)_gΨ_γ^m
       =μ𝒟^(^(Z)C_γ^m_,)+(Z+^(Z)λ)Φ_γ^m,

where

    ^(Z)C_γ^m_,=(^(Z)π_ν
    -12g_ν(g_κλ^(Z)π^κλ))g^νβ_βΨ_γ^m,
       ^(Z)λ=-μ^-1^(Z)π_ LT+12tr^(Z)π-μ^-1Zμ,
       Ψ_γ^0=φ_γ,  Φ_^0=μ_gφ_,

 and Φ_^0 equals
the right hand side of (<ref>).
Consequently, for Ψ_^m+1=Z_m+1Z_m⋯ Z_1φ_γ with Z_j∈{ϱ L, T, R},
then by (<ref>), the induction argument gives

    Φ_γ^m+1=   ∑_j=1^m(Z_m+1+^(Z_m+1)λ)…(Z_m+2-j
    +^(Z_m+2-j)λ)(μ𝒟^^(Z_m+1-j)C_γ^m-j_,)
       +μ𝒟^^(Z_m+1)C_γ^m_,+(Z_m+1+^(Z_m+1)λ)…(Z_1
    +^(Z_1)λ)Φ_γ^0
       =:J_1^m+1+J_2^m+1,     m≥ 1,
    Φ_γ^1=   (Z_1+^(Z_1)λ)Φ_γ^0+μ𝒟^^(Z_1)C_γ^0_,,

where J_1^m+1 and J_2^m+1 stand for the first and the second line in the right hand side of (<ref>), respectively.


By (<ref>)-(<ref>) in Appendix A, one has

    ^(T)λ=12tr^(T)π,  ^(ϱ L)λ=ϱtr_ gχ̌+2,  ^(R)λ=12tr^(R)π.

In addition, notice that in the null frame {L, L,X}, the
term μ𝒟^^(Z)C_γ^m_, can be written as

    μ𝒟^^(Z)C_γ^m_,=   -12 L(^(Z)C_γ^m_,L)-12L(^(Z)C_γ^m_, L)
    +∇^X(μ^(Z) C_^m_,X)
       -12( Lμ+μtrχ+tr^(T)π)^(Z)C_^m_, L-12trχ^(Z)C_^m_,L,

where

    ^(Z)C_^m_, L=^(Z)π_ LX( d^XΨ_^m)-12(tr^(Z)π) LΨ_^m,
       ^(Z)C_γ^m_,L=-2(^(Z)π_LT+μ^-1^(Z)π_TT)( LΨ_γ^m)+^(Z)π_LX( d^XΨ_γ^m)-12(tr^(Z)π)LΨ_γ^m,
       μ^(Z) C_γ^m_,X=-12^(Z)π_LX( LΨ_γ^m)-12^(Z)π_ LX(LΨ_γ^m)
    +^(Z)π_ LT( d_XΨ_γ^m)+12μtr^(Z)π d_XΨ_γ^m.

Substituting (<ref>) into (<ref>) directly would lead to a lengthy and tedious equation for μ𝒟^^(Z)C_γ^m_,. To overcome this default and handle the resulting terms more easily,
one can decompose μ𝒟^^(Z)C_γ^m_, into the following three parts as in <cit.>:

    μ𝒟^^(Z)C_γ^m_,=^(Z)D_γ,1^m+^(Z)D_γ,2^m+^(Z)D_γ,3^m,

where

    ^(Z)D_γ,1^m=   12tr^(Z)π( LLΨ_γ^m+12trχLΨ_γ^m)-^(Z)π_LX( d^X LΨ_γ^m)-^(Z)π_ LX( d^X LΨ_γ^m)
       +(^(Z)π_ LT+^(Z)π_T̃T)( L^2Ψ_γ^m)+12μtr^(Z)πΨ_γ^m+^(Z)π_ LTΨ_γ^m,


    ^(Z)D_γ,2^m=   { L(^(Z)π_ LT+^(Z)π_T̃T)-12∇^X^(Z)π_LX+14L(tr^(Z)π)} LΨ_γ^m+14 L(tr^(Z)π)LΨ_γ^m
       -{12ℒ_L^(Z)π_ LX- d_X^(Z)π_ LT-12 d_X(μtr^(Z)π)+12(ℒ_ L^(Z)π_LX)} d^XΨ_γ^m
       -12(∇^X^(Z)π_ LX)LΨ_γ^m,


    ^(Z)D_γ,3^m=   12{(tr^(Z)π)( d_Xμ+2μζ_X)-( Lμ-μtrχ-tr^(T)π)^(Z)π_ LX+trχ^(Z)π_LX} d^XΨ_γ^m
       +{trχ(^(Z)π_ LT+^(Z)π_T̃T)+14(μtrχ+tr^(T)π)tr^(Z)π+12 d^Xμ^(Z)π_ LX} LΨ_γ^m.

 Note that all the terms in ^(Z)D_γ,1^m are the products of the deformation tensor and the second
order derivatives of Ψ_^m, except the first term containing the factor of the form LLΨ_γ^m+12trχLΨ_γ^m (see (<ref>)). It should be emphasized here that such a structure is crucial in our analysis since Ψ_γ^m is the derivative of φ_γ and by (<ref>), LLφ_γ+12trχLφ_γ=H_+12trχ̌Lφ_ admits the better smallness and the faster time decay than those for LLφ_γ and 12trχLφ_γ separately. ^(Z)D_γ,2^m
collects all the products of the first order derivatives of the deformation
 tensor and the first order derivatives of Ψ_^m, and ^(Z)D_γ,3^m is the rest.

The explicit expressions of Φ_^m+1 given in (<ref>)-(<ref>) and (<ref>)-(<ref>) are important to estimate the last two integrals of (<ref>).
From the left side of (<ref>), it is natural to define the corresponding weighted energies and fluxes as in <cit.>:

    E_i,m+1(s , u)   =∑_γ=0^4∑_Z∈{ϱ L,T,R}δ^2lE_i[Z^mφ_γ](s, u),   i=1,2,
    
    F_i,m+1(s, u)   =∑_γ=0^4∑_Z∈{ϱ L,T,R}δ^2lF_i[Z^mφ_γ](s, u),   i=1,2,
    
    E_i,≤ m+1(s, u)   =∑_0≤ n≤ mE_i,n+1(s, u),   i=1,2,
    
    F_i,≤ m+1(s, u)   =∑_0≤ n≤ mF_i,n+1(s, u),   i=1,2,

where l is the number of T in Z^m. These weighted energies will be estimated in the subsequent sections.





Next, one derives the higher order L^2 estimates for some related quantities
so that the last two terms of (<ref>) can be absorbed by the left hand side, and hence the higher order energy estimates on (<ref>) can
be obtained. To this end, on needs two lemmas in <cit.>, where analogous results in 3D case
can be found in <cit.> and <cit.> respectively.



 For any function ψ∈ C^1(D^s, u) vanishing on C_0, one has that  for small δ>0,
 
    ∫_S_s, uψ^2   ≲δ∫_Σ_s^u|Tψ|^2≲δ∫_Σ_s^u(|Lψ|^2+μ^2| Lψ|^2),
    ∫_Σ_s^uψ^2   ≲δ^2∫_Σ_s^u|Tψ|^2≲δ^2∫_Σ_s^u(|Lψ|^2+μ^2| Lψ|^2).

Therefore,
 
    ∫_S_s, uψ^2   ≲δ(E_2[ψ](s, u)+ϱ^-2ιE_1[ψ](s, u)),
    ∫_Σ_s^uψ^2   ≲δ^2(E_2[ψ](s, u)+ϱ^-2ιE_1[ψ](s, u)).




	For any function f∈ C(D^s, u), set
	
    F(s,u,ϑ):=∫_t_0^sf(τ,u,ϑ)dτ.

	Under the assumptions (⋆), then for small δ>0,
	
    F_s,u≲ϱ(s,u)^1/2∫_t_0^s1ϱ(τ,u)^1/2f_τ,udτ,

	where F_s,u denotes the L^2 norm of F over Σ_s^u  which is defined in Definition <ref>.



One can now derive L^2 estimates for the quantities such as
χ̌, ^(Z)π_ L, ^(R)π, Ľ, υ, x^j
 (j=1,2) and R similarly as in <cit.>. However, here one has to pay more attentions to the
precise orders of smallness for  and the time decay or increasing rates in each involved
quantity, one can get





 Under the assumptions (⋆), when δ>0 is small, it holds that for m≤ 2N-6,
 
    δ^lZ^m+1μ_s,u≲δ^k(1-ε_0)-1/2s^1/2ln s+δ^(k-1)(1-ε_0) s^1/2(√(Ẽ_1,≤ m+2(s,u))
                     +ln s√(Ẽ_2,≤ m+2(s,u))+^k(1-_0)-1√(∫_0^u F̃_1(s,u')du')),
       δ^lℒ_Z^mχ̌_s,u≲δ^k(1-ε_0)+1/2s^-3/2ln s
     +s^-1ℰ_m+2(s,u),
       δ^lℒ_Z^m+1 g_s,u+s^-1δ^lZ^m+2x_s,u≲δ^1/2s^1/2
     +ℰ_m+2(s,u),
       δ^lℒ_Z^m+1R_s,u+δ^lZ^m+1υ_s,u≲δ^k(1-ε_0)+1/2s^1/2ln s
       +sℰ_m+2(s,u),
       δ^lℒ_Z^m^(R)π_T_s,u≲(1+^k(1-_0)-1ln s)(δ^k(1-ε_0)+1/2s^-1/2ln s
       +ℰ_m+2(s,u)),
       δ^lZ^m+1Ľ^j_s,u+δ^lZ^m+1ϱ̌_s,u+δ^lℒ_Z^m^(R)π_s,u+δ^lℒ_Z^m^(R)π_ L_s,u
                               ≲δ^k(1-ε_0)+1/2s^-1/2ln s+ℰ_m+2(s,u),   j=1,2,
       δ^lℒ_Z^m^(T)π_ L_s,u≲δ^k(1-ε_0)-1/2s^-1/2ln s
       +δ^(k-1)(1-ε_0)s^-1/2(√(Ẽ_1,≤ m+2(s,u))
                       +ln s√(Ẽ_2,≤ m+2(s,u))+^k(1-_0)-1√(∫_0^u F̃_1(s,u')du')),
       δ^lℒ_Z^m^(T)π_s,u≲(δ^1/2+^k(1-_0)-1/2ln s)s^-1/2
       +δ^(k-1)(1-ε_0)s^-1/2(√(Ẽ_1,≤ m+2(s,u))
                       +ln s√(Ẽ_2,≤ m+2(s,u))+(s^-1/2+^k(1-_0)-1)√(∫_0^u F̃_1(s,u')du')),

 where ℰ_m+2(s,u)=^k(1-_0)s^-ι√(Ẽ_1,≤ m+2(s,u))+^k(1-_0)+_0(1+^k(1-_0)-1)s^-1/2ln^2 s√(Ẽ_2,≤ m+2(s,u))+^(k-1)(1-_0)√(∫_0^u F̃_1(s,u')du') with Ẽ_i,≤ m+2(s,u)=sup_t_0≤τ≤ sE_i,≤ m+2(τ,u) (i=1,2) and F̃_1,≤ m+2(s,u) =sup_t_0≤τ≤ sF_1,≤ m+2(τ,u).



	By Lemma <ref>, the explicit expressions of ϱ LĽ^i, TĽ^i and RĽ^i
	can be given since ϱ LĽ^i=ϱ L L^i-Ľ^i, TĽ^i=T L^i+μϱg^0i+μϱĽ^i+μ-1ϱ^2x^i and RĽ^i=R^X d_XĽ^i
	hold.
	One then gets
	
    δ^lZ^m+1Ľ^i_s,u≲   δ^l_1Z^m_1Ľ^i_s,u+sδ^l_1ℒ_Z^m_1χ̌_s,u+δ s^-1δ^l_0Z^m_0μ_s,u
       +δ^k(1-ε_0)s^-2ln sδ^l_0Z^m_0x_s,u+δ^k(1-ε_0)s^-1ln sδ^l_1ℒ_Z^m_1 g_s,u
       +δ^(k-1)(1-ε_0)s^-(k-1)/2δ^l_0Z^m_0φ_γ_s,u,

	here l and l_p (p=0,1) are the numbers of T in Z^m+1 and Z^m_p respectively with 1≤ m_p≤ m+1-p.
	Note that |Ľ^i|≲δ^k(1-ε_0)s^-1ln s by (<ref>).
	Together with 1_s,u≲δ^1/2ϱ(s,u)^1/2, this yields
	Ľ^i_s,u≲δ^k(1-ε_0)+1/2s^-1/2ln s. Thus, with the help of (<ref>),
	
    δ^lZ^m+1Ľ^i_s,u≲   δ^k(1-ε_0)+1/2s^-1/2ln s+sδ^l_1ℒ_Z^m_1χ̌_s,u+δ s^-1δ^l_0Z^m_0μ_s,u
       +δ^k(1-ε_0)s^-2ln sδ^l_0Z^m_0x_s,u+δ^k(1-ε_0)s^-1ln sδ^l_1ℒ_Z^m_1 g_s,u
       +δ^k(1-ε_0)+ε_0s^-(k-1)/2{s^-ι√(E_1,≤ m+2(s,u))+√(E_2,≤ m+2(s,u))}.

	Similarly, due to ϱ Lx^i=ϱĽ^i+x^i, Tx^i=μ(-g^0i-Ľ^i-x^iϱ) and Rx^a=Ø^a+υ(g^0a+Ľ^a+x^aϱ), then
	
    δ^lZ^m+2x   _s,u≲δ^12s^32+δ^l_0Z^m_0υ_s,u
    	+s∑_j=1^2δ^l_0Z^m_0Ľ^j_s,u+δδ^l_0Z^m_0μ_s,u
       +δ^k(1-ε_0)+ε_0+1s^-(k-1)/2ln s{s^-ι√(E_1,≤ m+2(s,u))+√(E_2,≤ m+2(s,u))}.

	And it follows from ℒ_ϱ LR=ϱ^(R)π_ L, ℒ_TR=^(R)π_T and (<ref>)  in Appendix A that
	
    δ^lℒ_Z^m+1R   _s,u≲δ^k(1-ε_0)+1/2s^1/2ln s+δ^k(1-ε_0)s^-1ln sδ^l_0Z^m_0x_s,u
       +δ^k(1-ε_0)s^-1δ^l_1Z^m_1υ_s,u+s^2δ^l_1ℒ_Z^m_1χ̌_s,u+∑_j=1^2sδ^l_0Z^m_1Ľ^j_s,u
       +δ^k(1-ε_0)ln sδ^l_1ℒ_Z^m_1 g_s,u+δ^k(1-ε_0)+1s^-1ln sδ^l_0Z^m_0μ_s,u
       +δ^k(1-ε_0)+ε_0s^-(k-3)/2{s^-ι√(E_1,≤ m+2(s,u))+√(E_2,≤ m+2(s,u))}.

	Additionally, by (<ref>), (<ref>) and the facts that ℒ_ϱ L g=ϱχ̌+ g, ℒ_T g=^(T)π and ℒ_R g=^(R)π, it holds that
	
    δ^lZ^m+1υ_s,u≲   δ^k(1-ε_0)s^-1ln sδ^l_0Z^k_0x_s,u+sδ^l_0Z^m_0Ľ^1_s,u+sδ^l_0Z^m_0Ľ^2_s,u
       +δ^k(1-ε_0)+ε_0s^-(k-3)/2{ s^-ι√(E_1,≤ m+2(s,u))+√(E_2,≤ m+2(s,u))},

	
    δ^lZ^m+1ϱ̌_s,u≲   δ^2k(1-ε_0)+1/2s^-3/2ln^2s+δ^k(1-ε_0)s^-2ln sδ^l_0Z^k_0x_s,u
    	+δ^l_0Z^m_0Ľ^1_s,u
    
    	+δ^l_0Z^m_0   Ľ^2_s,u+δ^k(1-ε_0)+ε_0s^-(k-1)/2{ s^-ι√(E_1,≤ m+2(s,u))+√(E_2,≤ m+2(s,u))},

	
    δ^lℒ_Z^m+1 g_s,u≲   δ^1/2s^1/2+s^-1δ^l_1Z^m_1υ_s,u+δ s^-1δ^l_0Z^m_0μ_s,u
       +δ^k(1-ε_0)s^-(k+2)/2δ^l_0Z^k_0x_s,u+sδ^l_1ℒ_Z^m_1χ̌_s,u
       +(δ^k(1-ε_0)+1s^-2+δ^2k(1-_0)s^-2ln s)δ^l_0{Z^m_1Ľ^1_s,u+Z^m_1Ľ^2_s,u}
       +δ^k(1-ε_0)+ε_0s^-(k-1)/2{s^-ι√(E_1,≤ m+2(s,u))+√(E_2,≤ m+2(s,u))}.

	
	Collecting (<ref>)-(<ref>) yields
	
    δ^lZ^m+1Ľ^i_s,u≲   δ^k(1-ε_0)+1/2s^-1/2ln s+sδ^l_1ℒ_Z^m_1χ̌_s,u+δ s^-1δ^l_0Z^m_0μ_s,u
       +δ^k(1-ε_0)+ε_0s^-(k-1)/2{s^-ι√(E_1,≤ m+2(s,u))+√(E_2,≤ m+2(s,u))},
    δ^lZ^m+2x^i_s,u≲   δ^1/2s^3/2+s^2δ^l_1ℒ_Z^m_1χ̌_s,u+δδ^l_0Z^m_0μ_s,u
       +δ^k(1-ε_0)+ε_0s^-(k-3)/2{s^-ι√(E_1,≤ m+2(s,u))+√(E_2,≤ m+2(s,u))},
    δ^lℒ_Z^m+1 g_s,u≲   δ^1/2s^1/2+sδ^l_1ℒ_Z^m_1χ̌_s,u+δ s^-1δ^l_0Z^m_0μ_s,u
       +δ^k(1-ε_0)+ε_0s^-(k-1)/2{ s^-ι√(E_1,≤ m+2(s,u))+√(E_2,≤ m+2(s,u))},
    δ^lZ^m+1υ_s,u≲   δ^k(1-ε_0)+1/2s^1/2ln s+s^2δ^l_1ℒ_Z^m_1χ̌_s,u+δδ^l_0Z^m_0μ_s,u
       +δ^k(1-ε_0)+ε_0s^-(k-3)/2{ s^-ι√(E_1,≤ m+2(s,u))+√(E_2,≤ m+2(s,u))},
    δ^lZ^m+1ϱ̌_s,u≲   δ^k(1-ε_0)+1/2s^-1/2ln s+sδ^l_1ℒ_Z^m_1χ̌_s,u+δ s^-1δ^l_0Z^m_0μ_s,u
       +δ^k(1-ε_0)+ε_0s^-(k-1)/2{s^-ι√(E_1,≤ m+2(s,u))+√(E_2,≤ m+2(s,u))},
    δ^lℒ_Z^m+1R_s,u≲   δ^k(1-ε_0)+1/2s^1/2ln s+s^2δ^l_1ℒ_Z^m_1χ̌_s,u+δδ^l_0Z^m_0μ_s,u
       +δ^k(1-ε_0)+ε_0s^-(k-3)/2{s^-ι√(E_1,≤ m+2(s,u))+√(E_2,≤ m+2(s,u))}.

	Thanks to (<ref>)-(<ref>) in Appendix A, one can deduce
	
    δ^lℒ_Z^m^(T)π_s,u≲   δ^1/2s^-1/2
    	+δ^l_1ℒ_Z^m_1χ̌_s,u+s^-1δ^l_0Z^m_0μ_s,u
       +δ^(k-1)(1-ε_0)s^-(k-1)/2{s^-ι√(E_1,≤ m+2(s,u))+√(E_2,≤ m+2(s,u))},
    δ^lℒ_Z^m^(T)π_ L_s,u≲   δ^k(1-ε_0)-1/2s^-1/2+δ^k(1-ε_0)-1s^-(k-2)/2δ^l_1ℒ_Z^m_1χ̌_s,u+ s^-1δ^l_0Z^m_0μ_s,u
       +δ^(k-1)(1-ε_0)s^-(k-1)/2{s^-ι√(E_1,≤ m+2(s,u))+√(E_2,≤ m+2(s,u))},
    δ^lℒ_Z^m^(R)π_T_s,u≲   δ^k(1-ε_0)+1/2s^-1/2ln s
    	+sδ^l_1ℒ_Z^m_1χ̌_s,u+δ s^-1ln sδ^l_0Z^m_0μ_s,u
    
    	+(δ^k(1-ε_0)+ε_0   +δ^(2k-1)(1-_0))s^-(k-1)/2{s^-ι√(E_1,≤ m+2(s,u))+√(E_2,≤ m+2(s,u))},
    δ^lℒ_Z^m^(R)π_ L_s,u≲   δ^k(1-ε_0)+1/2s^-1/2ln s+sδ^l_1ℒ_Z^m_1χ̌_s,u
    	+δ s^-1δ^l_0Z^m_0μ_s,u
       +δ^k(1-ε_0)+ε_0s^-(k-1)/2{s^-ι√(E_1,≤ m+2(s,u))+√(E_2,≤ m+2(s,u))},
    δ^lℒ_Z^m^(R)π_s,u≲   δ^k(1-ε_0)+1/2s^-1/2ln s+sδ^l_1ℒ_Z^m_1χ̌_s,u+δ s^-1δ^l_0Z^m_0μ_s,u
       +δ^k(1-ε_0)+ε_0s^-(k-1)/2{s^-ι√(E_1,≤ m+2(s,u))+√(E_2,≤ m+2(s,u))}.

	On the other hand, for the solution ϕ to (<ref>),
	
    δ^lZ^m+1ϕ_s,u≲δ^1+l[T,Z^m+1]ϕ_s,u+δ^1+lZ^m+1(μT̃^iφ_i)_s,u
    ≲   δ^3-_0s^-3/2δ^l_1ℒ_Z^m_1^(R)π_T_s,u+δ^3-_0s^-1/2δ^l_1ℒ_Z^m_1^(T)π_ L_s,u+δ^2-_0s^-1/2δ^l'Z^m'+1μ_s,u
       +δ^(k+1)(1-_0)+1s^-3/2δ^l_1ℒ_Z^m_1 g_s,u+(δ+δ^k(1-_0))s^-1δ^l_0Z^m_0ϕ_s,u+δδ^l_0Z^m_0φ_s,u
       +δ^2-_0s^-1/2δ^l_0(Z̃^m_0Ľ^1_s,u+Z̃^m_0Ľ^2_s,u)+δ^2-_0s^-3/2δ^l_0Z^m_0x_s,u,

	which means
	
    δ^lZ^m+1ϕ_s,u≲   δ^5/2-_0+^2-_0s^1/2δ^l_1ℒ_Z^m_1χ̌_s,u+δ^2-_0 s^-1/2δ^l_0Z^m_0μ_s,u
       +δ^2{s^-ι√(E_1,≤ m+2(s,u))+√(E_2,≤ m+2(s,u))}.

	
	
	Note that all the terms in the left hand side of (<ref>)-(<ref>) are controlled by the L^2-norms of
	the derivatives of χ̌ and μ.
	Next we deal with ℒ_Z^m_1χ̌_s,u and Z^m_0μ_s,u.

	First, we treat the cases that all the vectorfields Z's in the left hand side of (<ref>)-(<ref>)
	are R's. Following the processes to obtain (<ref>)-(<ref>), one can show that the following inequalities
	hold when all Z's are replaced by R's,
	
    R^m+1Ľ^i_s,u+   R^m+1ϱ̌_s,u+ℒ_R^m^(R)π_s,u+ℒ_R^m^(R)π_ L_s,u
    ≲   δ^k(1-ε_0)+1/2s^-1/2ln s+sℒ_R^m_1χ̌_s,u+δ s^-1R^m_0μ_s,u
       +δ^k(1-ε_0)+ε_0s^-(k-1)/2{s^-ι√(E_1,≤ m+2(s,u))+√(E_2,≤ m+2(s,u))},
    R^m+2x_s,u≲   δ^1/2s^3/2+s^2ℒ_R^m_1χ̌_s,u+δR^m_0μ_s,u
       +δ^k(1-ε_0)+ε_0s^-(k-3)/2{s^-ι√(E_1,≤ m+2(s,u))+√(E_2,≤ m+2(s,u))},
    ℒ_R^m+1 g_s,u≲   δ^1/2s^1/2+sℒ_R^m_1χ̌_s,u+δ s^-1R^m_0μ_s,u
       +δ^k(1-ε_0)+ε_0s^-(k-1)/2{ s^-ι√(E_1,≤ m+2(s,u))+√(E_2,≤ m+2(s,u))},
    R^m+1υ_s,u≲   δ^k(1-ε_0)+1/2s^1/2ln s+s^2ℒ_R^m_1χ̌_s,u+δR^m_0μ_s,u
       +δ^k(1-ε_0)+ε_0s^-(k-3)/2{ s^-ι√(E_1,≤ m+2(s,u))+√(E_2,≤ m+2(s,u))},
    ℒ_R^m^(T)π_s,u≲   δ^1/2s^-1/2
    	+ℒ_R^m_1χ̌_s,u+s^-1R^m_0μ_s,u
       +δ^(k-1)(1-ε_0)s^-(k-1)/2{s^-ι√(E_1,≤ m+2(s,u))+√(E_2,≤ m+2(s,u))},
    ℒ_R^m^(T)π_ L_s,u≲   δ^k(1-ε_0)-1/2s^-1/2+δ^k(1-ε_0)-1s^-(k-2)/2ℒ_R^m_1χ̌_s,u+ s^-1R^m_0μ_s,u
       +δ^(k-1)(1-ε_0)s^-(k-1)/2{s^-ι√(E_1,≤ m+2(s,u))+√(E_2,≤ m+2(s,u))},
    ℒ_R^m^(R)π_T_s,u≲   δ^k(1-ε_0)+12s^-12ln s
    	+sℒ_R^m_1χ̌_s,u+δ s^-1ln sR^m_0μ_s,u
    
    	+(δ^k(1-ε_0)+ε_0   +δ^(2k-1)(1-_0))s^-(k-1)/2{s^-ι√(E_1,≤ m+2(s,u))+√(E_2,≤ m+2(s,u))},
    R^m+1ϕ_s,u≲   δ^5/2-_0+^2-_0s^1/2ℒ_R^m_1χ̌_s,u+δ^2-_0 s^-1/2R^m_0μ_s,u
       +δ^2{s^-ι√(E_1,≤ m+2(s,u))+√(E_2,≤ m+2(s,u))},
    R^m Lϕ_s,u≲   δ^5/2-_0s^-1+(^2-_0+^(k+1)(1-_0))s^-1/2ℒ_R^m_1χ̌_s,u+δ^2-_0 s^-3/2R^m_0μ_s,u
       +δ^2{s^-1-ι√(E_1,≤ m+2(s,u))+√(E_2,≤ m+2(s,u))}.


We now estimate the L^2-norm of LR^m+1μ.
Note that it follows from (<ref>), (<ref>) and (<ref>) that
	
    L(ϱ^2trχ̌)=   -ϱ^2G_X L^γ( d^X L_)+12ϱ^2G_ L L^_+12ϱ^2 g^XXG_XX^ L^2_-ϱ^2(trχ̌)^2
       +(-12G_ L L^ L_-G_ T L^ L_+G_X L^ d^X_)(ϱ^2trχ̌+ϱ)+⋯,

	where
	
    G_ L L^_=μ^-1G_ L L^(L(μ L_)+12ϱμ L_+2LT_+1ϱ T_-H_)
    
    	=   μ^-1G_ L L^(L(μ L_)+12ϱμ L_-H_)+2μ^-1{G_ L L^0L(μT̃^i L_i)+G_ L L^ L(μT̃^ig_ j d^Xx^j d_X_i)
       -G_ L L^( L L_)T̃^iT_i-(G_ L L^ L_)(LT̃^i)T_i-(G_ L L^ L_)T̃^i( LT_i)+12ϱ(μ G_ L L^0T̃ L_i
       +μ G_ L L^T̃^ig_ j d^Xx^j d_X_i-(G_ L L^ L_)T̃^iT_i)
    	}.

	Applying (<ref>) again for the above two underline terms yields
	
    G_ L L^_
    	=   -(G_ L L^ L_)T̃^i_i+G_ L L^ L^2_+2G_ L L^0T̃^i( L^2_i)
       +2G_ L L^T̃^ig_ j d^Xx^j( d_X L_i)-(G_ L L^ L_)T̃^i( L^2_i)
       +μ^-1f_1(φ,  dx,  L^1,  L^2)(
    	[ G_ L L^ L_;       ^k-1;            ])
    	(
    	[ L; d;   ])
    	(
    	[ ^k-1T;   μ L;   μ d; μϱ^-1 ])
       +μ^-1(trχ̌) f_2(φ,  dx,  L^1,  L^2)(
    	[ (G_ L L^ L_)T;       μ^k-1 L;       μ^k-1 d ]).

Together with (<ref>) and Proposition <ref>, by the analogous proof of (8.43) in <cit.>, this deduces
	
    R^m(ϱ^2μ G_ L L^_)_s,u+R^m(ϱ^2μ G_X L^γ d^X L_)_s,u+R^m(ϱ^2μ g^XXG_XX^ L^2_)_s,u
    ≲   ^k(1-_0) dR^≤ m+1_s,u+^(k-1)(1-_0)s^1/2(s LR^≤ m L_s,u+ LR^≤ m+1_s,u)
       +^k(1-_0)s^-1(R^≤ mμ_s,u+ℒ_R^≤ m^(R)π_ L_s,u)+^k(1-_0)-_0s^1/2R^≤ mtrχ̌_s,u
       +(1+^k(1-_0)-1ln s){^k(1-_0)s^-1(∑_i=1^2R^≤ mĽ^i_s,u+R^≤ mϱ̌_s,u+ℒ_R^≤ m-1^(R)π_s,u
       +s^-1R^≤ mυ_s,u)+^(k-1)(1-_0)s^-3/2(sR^≤ m Lϕ_s,u+R^≤ m+1ϕ_s,u)	}
       +^k(1-_0)+_0(1+^k(1-_0)-1)s^-1/2(s^-ι√(E_1,≤ m+2(s,u))+√(E_2,≤ m+2(s,u))).

Collecting (<ref>), (<ref>) and (<ref>), we have
	
    LR^m(μϱ^2trχ̌)_s,u
    ≲   ^k(1-_0)+1/2s^-1/2+^(k-1)(1-_0)s^1/2(s LR^≤ m L_s,u+ LR^≤ m+1_s,u)
       +^k(1-_0)s^-ι√(E_1,≤ m+2(s,u))+^k(1-_0)+_0(1+^k(1-_0)-1)s^-1/2√(E_2,≤ m+2(s,u))
       +^k(1-_0)-_0s^1/2R^≤ m(μtrχ̌)_s,u+^k(1-_0)s^-1R^≤ mμ_s,u
       +^k(1-_0)+1(1+^k(1-_0)-1)s^-2ln sR^≤ m+1μ_s,u.

On the other hand, by virtue of (<ref>) and the commutation of vector fields, one has
	
    δ^l LR^m+1μ_s,u
    ≲   ^k(1-_0)-1s^-1(ℒ_R^≤ m^(R)π_ L_s,u+R^m+1Ľ^1_s,u+R^m+1Ľ^2_s,u+s^-1R^≤ m+1υ_s,u)
       +^2k(1-_0)-_0s^-2ln sℒ_R^≤ m-1^(R)π_s,u+^k(1-_0)s^-2ln sR^≤ m+1μ_s,u
       +^(k-1)(1-_0)-1s^-1/2(R^≤ m+1_s,u+TR^≤ m+1_s,u+ dR^≤ m L_s,u)
       +^k(1-_0)s^-2ℒ_R^≤ m^(R)π_T_s,u
    ≲   δ^2k(1-ε_0)-1/2s^-3/2ln s+δ^k(1-ε_0)-1R^≤ m(μtrχ̌)_s,u
       +δ^(k-1)(1-ε_0)s^-1/2{s^-ι√(E_1,≤ m+2(s,u))+√(E_2,≤ m+2(s,u))}
       +δ^k(1-ε_0)s^-2ln sR^≤ m+1μ_s,u+δ^2k(1-ε_0)-1s^-2ln sR^≤ mμ_s,u.


	
	Set F(s,u,ϑ)=ϱ(s,u)^2(R^m(μtrχ̌))(s,u,ϑ)-ϱ(t_0,u)^2(R^m(μtrχ̌))(t_0,u,ϑ) in (<ref>).
Since for any 2-form ξ on S_s,u, | L(ϱ^2trξ)|≲ϱ^2|ℒ_ Lξ|+δ^k(1-ε_0)s^-2ln s(ϱ^2|trξ|) holds. Then
	applying (<ref>) and Grownwall's equality simultaneously yields
	
    s^3/2R^m(μtrχ̌)_s,u
    ≲   δ^k(1-ε_0)+1/2ln s
    	+δ^k(1-ε_0)+1(1+^k(1-_0)-1)∫_t_0^sτ^-5/2lnτR^≤ m+1μ_τ,udτ
       +δ^k(1-ε_0)s^1/2-ι√(Ẽ_1,≤ m+2(s,u))+^k(1-_0)+_0(1+^k(1-_0)-1)ln s√(Ẽ_2,≤ m+2(s,u))
       +^k(1-_0)∫_t_0^sτ^-3/2R^≤ mμ_τ,udτ+^(k-1)(1-_0)s^1/2√(∫_0^uF_1,≤ m+2(s,u')du').

	
	
	Analogously, (<ref>) shows that
	
    s^-1/2R^m+1μ_s,u≲   δ^k(1-ε_0)-_0+1/2
    	+δ^k(1-ε_0)-1∫_t_0^sτ^-1/2R^≤ m(μtrχ̌)_τ,udτ
       +δ^2k(1-ε_0)-1∫_t_0^sτ^-5/2lnτR^≤ mμ_τ,udτ
       +δ^(k-1)(1-ε_0)√(Ẽ_1,≤ m+2(s,u))+^(k-1)(1-_0)ln s√(Ẽ_2,≤ m+2(s,u)).

	When m≤ N-1, it follows from Proposition <ref> that |R^m+1μ|≤^k(1-_0)-_0, which implies R^m+1μ_s,u≤^k(1-_0)-_0+1/2s^1/2. For N≤ m≤ 2N-5, one can make the following induction:
	
    s^-1/2R^mμ_s,u≲   δ^k(1-ε_0)-_0+1/2+δ^(k-1)(1-ε_0)√(Ẽ_1,≤ m+2(s,u))
    
    	+^(k-1)(1-_0)   ln s√(Ẽ_2,≤ m+2(s,u))+^(2k-1)(1-_0)-1√(∫_0^uF_1,≤ m+2(s,u')du').

	Substitute (<ref>) into the right hands of (<ref>) and (<ref>) to obtain
	
    s^3/2R^m(μtrχ̌)_s,u
    ≲   δ^k(1-ε_0)+1/2ln s
    	+δ^k(1-ε_0)+1(1+^k(1-_0)-1)∫_t_0^sτ^-5/2lnτR^≤ m+1μ_τ,udτ
       +δ^k(1-ε_0)s^1/2-ι√(Ẽ_1,≤ m+2(s,u))+^k(1-_0)+_0(1+^k(1-_0)-1)ln^2 s√(Ẽ_2,≤ m+2(s,u))
       +^(k-1)(1-_0)s^1/2√(∫_0^uF̃_1,≤ m+2(s,u')du')

	and
	
    s^-1/2R^m+1μ_s,u
    ≲   δ^k(1-ε_0)-_0+1/2
    	+δ^k(1-ε_0)-1∫_t_0^sτ^-1/2R^≤ m(μtrχ̌)_τ,udτ
       +δ^(k-1)(1-ε_0)(√(Ẽ_1,≤ m+2)+ln s√(Ẽ_2,≤ m+2))+^(4k-1)(1-_0)-2√(∫_0^uF̃_1,≤ m+2(s,u')du').

	And hence,
	
    R^m(μtrχ̌)_s,u   ≲δ^k(1-ε_0)+1/2s^-3/2ln s+δ^k(1-ε_0)s^-1-ι√(Ẽ_1,≤ m+2(s,u))
       +^k(1-_0)+_0(1+^k(1-_0)-1)s^-3/2ln^2 s√(Ẽ_2,≤ m+2(s,u))
       +^(k-1)(1-_0)s^-1√(∫_0^uF̃_1,≤ m+2(s,u')du'),
    R^m+1μ_s,u   ≲δ^k(1-ε_0)-_0+1/2s^1/2+δ^(k-1)(1-ε_0)s^1/2{√(Ẽ_1,≤ m+2)+ln s√(Ẽ_2,≤ m+2)}
       +^(2k-1)(1-_0)-1s^1/2√(∫_0^uF̃_1,≤ m+2(s,u')du').

	Thus,
	
    ℒ_R^mχ̌_s,u≲δ^k(1-ε_0)+1/2s^-3/2ln s
    	+s^-1ℰ_m+2(s,u)

	and then (<ref>) hold.
	
	If ℒ_Z^mχ̌=ℒ_Z^m_1ℒ_Tℒ_R^m_2χ̌ with m_1+m_2=m-1, then ℒ_Z^mχ̌=ℒ_Z^m_1[ℒ_T,ℒ_R^m_2]χ̌+ℒ_Z^m_1ℒ_R^m_2ℒ_Tχ̌ holds. Hence
	it follows from Lemma <ref>, (<ref>) and  (<ref>)-(<ref>) that
	
    δ^lℒ_Z^mχ̌_s,u
    ≲   δ^k(1-ε_0)+1/2s^-3/2+δ^k(1-ε_0)s^-1δ^l_1ℒ_Z^m_1χ̌_s,u+δ s^-2δ^l_0Z^m_0μ_s,u
       +δ^k(1-ε_0)+ε_0s^-(k+1)/2{s^-ι√(E_1,≤ m+2)+√(E_2,≤ m+2)}.

	
	
	If ℒ_Z^mχ̌=ℒ_Z^m_1ℒ_ϱ Lℒ_R^m_2χ̌ with m_1+m_2=m-1, then ℒ_Z^mχ̌=ℒ_Z^m_1[ℒ_ϱ L,ℒ_R^m_2]χ̌+ℒ_Z^m_1ℒ_R^m_2(ϱℒ_ Lχ̌), and it follows from (<ref>) that
	
    δ^lℒ_Z^mχ̌_s,u
    ≲   δ^k(1-ε_0)+1/2s^-3/2+δ^k(1-ε_0)s^-1ln sδ^l_1ℒ_Z^m_1χ̌_s,u+δ^k(1-ε_0)+1 s^-3ln sδ^l_0Z^m_0μ_s,u
       +δ^k(1-ε_0)+ε_0s^-(k+1)/2{s^-ι√(E_1,≤ m+2)+√(E_2,≤ m+2)}.

	
	
	Therefore, for any Z∈{ϱ L,T,R}, by (<ref>), (<ref>) and (<ref>), we obtain
	
    δ^lℒ_Z^mχ̌_s,u≲   δ^k(1-ε_0)+1/2s^-3/2ln s+δ s^-2δ^l_0Z^m_0μ_s,u+s^-1ℰ_m+2(s,u).

	
	

  * For Z^m+1μ=Z̅^m+1μ with Z̅∈{T, R},
taking F(s,u,ϑ)=δ^lZ̅^m+1μ(s,u,ϑ)-δ^lZ̅^m+1μ(t_0,u,ϑ) in (<ref>),
meanwhile using

    δ^l LZ̅^m+1μ_s,u
    ≲   δ^k(1-ε_0)-ε_0s^-1δ^l_1ℒ_Z̅^m_1^(R)π_ L_s,u+δ^k(1-ε_0)s^-2ln sδ^l_0Z̅^m_0μ_s,u
       +δ^(k-2)(1-ε_0)-_0s^-1/2δ^l_0Z̅^ m_0φ_s,u+δ^(k+1)(1-ε_0)s^-1δ^l_1ℒ_Z̅^k_1^(T)π_ L_s,u
       +δ^k(1-ε_0)-1s^-k/2δ^l_0Z̅^m_0Ľ^i_s,u+δ^k(1-ε_0)s^-2δ^l_1ℒ_Z̅^m_1^(R)π_T_s,u
       +δ^k(1-ε_0)-1s^-k/2δ^l_0ℒ_Z̅^m_0 dx^i_s,u+δ^(k-1)(1-ε_0)s^-(k-1)/2δ^l_0 LZ̅^m_0φ_s,u
       +δ^(k-1)(1-ε_0)s^-1/2δ^l_0TZ̅^m_0φ_s,u+δ^2k(1-ε_0)-ε_0s^-2ln sδ^l_1ℒ_Z̅^m_1 g_s,u
    ≲   δ^k(1-ε_0)-1/2s^-1/2+δ^k(1-ε_0)-1δ^l_1ℒ_Z̅^m_1χ̌_s,u+δ^k(1-ε_0)s^-2ln sδ^l_0Z̅^m_0μ_s,u
       +δ^(k-1)(1-ε_0)s^-1/2{s^-ι√(E_1,≤ m+2(s,u))+√(E_2,≤ m+2(s,u))}


and (<ref>), one has
	
    δ^lZ̅^m+1μ_s,u≲   δ^k(1-ε_0)-1/2s^1/2ln s+δ^(k-1)(1-ε_0)s^1/2{√(Ẽ_1,≤ m+2)+ln s√(Ẽ_2,≤ m+2)}
       +^(2k-1)(1-_0)-1s^1/2√(∫_0^uF̃_1,≤ m+2(s,u')du').

	
	
	If Z^m+1μ=Z^m_1(ϱ L)Z̅^m_2μ with m_1+m_2=m, with the help of (<ref>),
	then it holds that
	
    δ^lZ^m+1μ_s,u=δ^lZ^m_1[ϱ L,Z̅^m_2]μ+Z^m_1Z̅^m_2(ϱ L)μ_s,u
    ≲   δ^k(1-ε_0)-1/2s^1/2+δ^k(1-ε_0)-1sδ^l_1ℒ_Z^m_1χ̌_s,u+δ^k(1-ε_0) s^-1ln sδ^l_0Z^m_0μ_s,u
       +δ^(k-1)(1-ε_0){s^-ι√(E_1,≤ m+2(s,u))+√(E_2,≤ m+2(s,u))}
    ≲   δ^k(1-ε_0)-1/2s^1/2+δ^k(1-ε_0) s^-1ln sδ^l_0Z^m_0μ_s,u
       +δ^(k-1)(1-ε_0){s^-ι√(E_1,≤ m+2(s,u))+√(E_2,≤ m+2(s,u))},

	where (<ref>) is  used in the last inequality.
	
	Thus, for any Z∈{ϱ L,T,R}, it follows from (<ref>) and (<ref>) that
	
    δ^lZ^m+1μ_s,u≲   δ^k(1-ε_0)-1/2s^1/2ln s+δ^(k-1)(1-ε_0)s^1/2{√(Ẽ_1,≤ m+2)+ln s√(Ẽ_2,≤ m+2)}
       +^(2k-1)(1-_0)-1s^1/2√(∫_0^uF̃_1,≤ m+2(s,u')du'),

	which, together with (<ref>), implies that
	
    δ^lℒ_Z^mχ̌_s,u≲   δ^k(1-ε_0)+1/2s^-3/2ln s+s^-1ℰ_m+2(s,u).

	
	Substituting (<ref>)-(<ref>) into (<ref>)-(<ref>)
	yields the proof of Proposition <ref>.



It follows from the L^∞ estimates in Subsection <ref> that |Z^m+1μ|≲δ^k(1-_0)-_0-l holds
when m≤ N-3, which implies δ^lZ^m+1μ_s,u≲δ^k(1-_0)-_0+1/2s^1/2 (m≤ N-3) due to 1_s,u≲^1/2s^1/2.
Note that the upper bound δ^k(1-_0)-1/2-ls^1/2ln s of the L^2 norm of Z^m+1μ for m≤ 2N-6
(see Proposition <ref>)
is much worse than δ^k(1-_0)-_0+1/2-ls^1/2.
This leads to some essential difficulties for obtaining the global uniform estimates of ϕ in A_2 when _0 is near _k^*.
Therefore, one needs to improve the L^2 norm of Z^m+1μ for N-2≤ m≤ 2N-6,
which will be achieved by studying the transport equation (<ref>) carefully with special attention to the term 12G_ L L^γ Tφ_γ in  (<ref>). According to (<ref>) and (<ref>), one may estimate  Z^m+1 Lϕ_s,u and ℒ_Z^m+1 dϕ_s,u, and even further obtain better estimates on the L^2 norm
of Z^m+1 Lμ. It is emphasized here that one cannot treat Lϕ simply as L^φ_,
otherwise, the better smallness and time-decay rate of Lϕ will be lost. The same caution applies to dϕ.

We now apply (<ref>) to establish the estimates of Z^m+1 Lϕ_s,u and ℒ_Z^m+1 dϕ_s,u.


	Under the assumption (⋆), when δ>0 is small, it holds that for m≤ 2N-6,
	
    δ^lZ̅^m+1 Lϕ_s,u   ≲δ^5/2-_0s^-1+δ s^-ι√(Ẽ_1,≤ m+2(s,u))+δ^2s^-1√(Ẽ_2,≤ m+2(s,u))
       +(δ^3-_0+δ^(k+1)(1-_0)+1)s^-1/2R^mμ_s,u+δ^2-_0s^-3/2δ^l'Z̅^m'+1μ_s,u
       +^k(1-_0)(+^k(1-_0))s^-3/2√(∫_0^u F̃_1(s,u')du'),

	
    δ^lℒ_Z̅^m+1 dϕ_s,u≲   δ^5/2-_0s^-1+δ s^-ι√(Ẽ_1,≤ m+2(s,u))+δ^2s^-1√(Ẽ_2,≤ m+2(s,u))
       +δ^2-_0s^1/2R^mμ_s,u+δ^2-_0s^-3/2δ^l'Z̅^m'+1μ_s,u,                
       +^k(1-_0)+1s^-3/2√(∫_0^u F̃_1(s,u')du')

		
    δ^lZ̅^m L^2ϕ_s,u   ≲δ^5/2-_0s^-2+δ s^-1-ι√(Ẽ_1,≤ m+2(s,u))+δ^2s^-2√(Ẽ_2,≤ m+2(s,u))
    
    	+   δ^(k+1)(1-_0)+1(δ+δ^k(1-_0))s^-5/2R^mμ_s,u+δ^2-_0s^-5/2δ^l'Z̅^m'+1μ_s,u
       +^k(1-_0)(+^k(1-_0))s^-5/2√(∫_0^u F̃_1(s,u')du'),

	where Z̅∈{T,R}, the numbers of T in Z̅^m+1 and Z̅^m'+1 are l and l' respectively, m'≤ m.



	For Z̅^m+1=R^m+1, due to (<ref>), one has
	
    R^m+1 Lϕ_s,u≲δTR^m+1 Lϕ_s,u
    ≲   δR^m+1T( L^φ_)_s,u+δ^3-_0s^-5/2ℒ_R^m_1^(R)π_T_s,u+δ s^-1R^m_0 Lϕ_s,u,

	where T( L^φ_)=(T L^i)φ_i+μT̃^i( Lφ_i).
Recall that T L^i satisfies (<ref>). Then
	
    T( L^φ_)=μT̃^i( Lφ_i)+( d_Xμ) d^Xϕ-(G_X L^γ Tφ_γ) d^Xϕ+12(G_ L L^γ Tφ_γ)T̃^iφ_i+….

	With the help of (<ref>) and (<ref>), after substituting (<ref>) into (<ref>) and absorbing all R^m_0 Lϕ_s,u in the right hand side by ∑_m≤ 2N-6R^m+1 Lϕ_s,u in (<ref>), one obtains
	
    R^m+1 Lϕ_s,u
    ≲   (δ^2-_0+δ^(k+1)(1-_0))s^-3/2(R^m_0Ľ^1_s,u+R^m_0Ľ^2_s,u)+δ^(k+1)(1-_0)s^-3/2R^m_0ϱ̌_s,u
       +δ^2-_0s^-5/2R^m_-1x_s,u+δ^(k+1)(1-_0)+1s^-3/2TR^m_0φ_s,u
       +δ^k(1-_0)+2s^-2 dR^m_0φ_s,u+δ^2-_0s^-3/2ℒ_R^m_1^(R)π_ L_s,u+δ s^-1R^m_0φ_s,u
       +δ LR^m_0φ_s,u+δ^3-_0s^-5/2ℒ_R^m_1^(R)π_T_s,u+δ^k(1-_0)s^-1 dR^m_0ϕ_s,u
       +δ^3-_0s^-3/2 dR^m+1μ_s,u+δ^2-_0s^-3/2R^m'+1μ_s,u.

	where 1≤ m_p≤ m+1-p with p=-1,0,1. Using Proposition <ref> to estimate the second to fourth lines in (<ref>)
yields
	
    R^m+1 L   ϕ_s,u≲δ^5/2-_0s^-1+δ s^-ι√(Ẽ_1,≤ m+2(s,u))
    +δ^2s^-1√(Ẽ_2,≤ m+2(s,u))
       +δ^k(1-_0)s^-1 dR^m_0ϕ_s,u+δ^3-_0s^-3/2 dR^m+1μ_s,u
       +δ^2-_0s^-3/2R^m'+1μ_s,u+^k(1-_0)(+^k(1-_0))s^-3/2√(∫_0^u F̃_1(s,u')du').

	
	If Z̅^m+1=Z̅^p_1TR^p_2 with p_1+p_2=m, then Z̅^m+1 Lϕ=Z̅^p_1R^p_2T( L^φ_)+Z̅^p_1[T,R^p_2] Lϕ. Using (<ref>) again, one can get similarly that
	
    δ^lZ̅^m+1 Lϕ_s,u≲   δ^5/2-_0s^-1+δ s^-ι√(Ẽ_1,≤ m+2(s,u))+δ^2s^-1√(Ẽ_2,≤ m+2(s,u))
    
    	+δ^k(1-_0)   s^-1δ^l_0 dZ̅^m_0ϕ_s,u+δ^2-_0s^-3/2δ^l'Z̅^m'+1μ_s,u
       +^k(1-_0)(+^k(1-_0))s^-3/2√(∫_0^u F̃_1(s,u')du').

	Thus, for any Z̅^m+1 which contains l vector fields T, it follows from (<ref>) and (<ref>) that
		
    δ^lZ̅   ^m+1 Lϕ_s,u≲δ^5/2-_0s^-1+δ s^-ι√(Ẽ_1,≤ m+2(s,u))+δ^2s^-1√(Ẽ_2,≤ m+2(s,u))
       +δ^k(1-_0)s^-1δ^l_0 dZ̅^m_0ϕ_s,u+δ^3-_0s^-1/2 R^mμ_s,u+δ^2-_0s^-3/2δ^l'Z̅^m'+1μ_s,u
       +^k(1-_0)(+^k(1-_0))s^-3/2√(∫_0^u F̃_1(s,u')du').

	
Due to |ℒ_Z̅^m+1 dϕ|≲ s^-1|RZ̅^m+1ϕ|,
one needs to estimate the L^2 norm of RZ̅^m+1ϕ. As before, for Z̅^m+1=R^m+1, there holds
	
    R^m+2ϕ_s,u≲δR^m+1TRϕ_s,u+δ^3-_0s^-3/2ℒ_R^m_1^(R)π_T_s,u+δ s^-1R^m_0Rϕ_s,u.

	Note that TRϕ=^(R)π_TX d^Xϕ+(Rμ)T̃^iφ_i-μ(Rg^0i)φ_i-μ(R L^i)φ_i+μT̃^iRφ_i. In addition, it follows from (<ref>) in Appendix A and (<ref>) that
	
    ^(R)π_TX d^Xϕ-μ(R L^i)φ_i
    
    	=   υ d^Xμ d_Xϕ+μ(G_X L^ Rφ_) d^Xϕ+μ(G_XT̃^ Rφ_) d^Xϕ+μϵ_i^aŤ^ig_ab( d_Xx^b d^Xϕ)-μϱRϕ
       +υ(G_XT̃^ Tφ_) d^Xϕ-12υμ(G_T̃T̃^ d^Xφ_) d_Xϕ+μ(G_ LT̃^ Rφ_)T̃^iφ_i+12μ(G_T̃T̃^ Rφ_)T̃^iφ_i.

	Thus,
	
    R^m+2ϕ_s,u≲   δ^2-_0s^-1/2{s dR^m+1μ_s,u+R^m'+1μ_s,u}+δ^3-_0s^-3/2ℒ_R^m_1^(R)π_T_s,u
       +δZ^m_0φ_s,u+δ s dR^m+1φ_s,u+δ^2k(1-_0)+2s^-2ln sTR^m+1φ_s,u
       +δ^(k+1)(1-_0)+1s^-5/2R^m_0υ_s,u+δ^(k+1)(1-_0)+1s^-3/2ln sℒ_R^m_0 g_s,u
       +δ^2-_0s^-1/2(R^m_0Ľ^1_s,u+R^m_0Ľ^2_s,u)+δ^2-_0s^-3/2R^m_-1x_s,u
    ≲δ^5/2-_0   +δ s^1-ι√(Ẽ_1,≤ m+2(s,u))+δ^2√(Ẽ_2,≤ m+2(s,u))+δ^2-_0s^1/2 dR^m+1μ_s,u
    
    	+δ^2-_0   s^-1/2R^m'+1μ_s,u+^k(1-_0)+1s^-1/2√(∫_0^u F̃_1(s,u')du').

	For Z̅^m+1=Z̅^p_1TR^p_2 with p_1+p_2=m and the number of T in Z̅^p_1 being q_1,
due to RZ̅^m+1ϕ=RZ̅^p_1R^p_2(μT̃^iφ_i)+RZ̅^p_1[T,R^p_2]ϕ, then it holds that
	
    δ^lRZ̅^m+1ϕ_s,u≲   δ^5/2-_0+δ^2 s^-ι√(Ẽ_1,≤ m+2(s,u))+δ^2√(Ẽ_2,≤ m+2(s,u))+δ^2-_0s^-1/2δ^l'Z̅^m'+1μ_s,u
       +δ s^-2ln sδ^l_1RZ̅^m_1ϕ_s,u+^k(1-_0)+1s^-1/2√(∫_0^u F̃_1(s,u')du'),

	where 1≤ m_1≤ m and the number of T in Z̅^m_1 is at most q_1. Then it follows from induction that
	
    δ^lRZ̅^m+1ϕ_s,u≲   δ^5/2-_0+δ^2 s^-ι√(Ẽ_1,≤ m+2(s,u))
    +δ^2√(Ẽ_2,≤ m+2(s,u))
       +δ^2-_0s^-1/2δ^l'Z̅^m'+1μ_s,u+δ s^-2ln sR^m_1+1ϕ_s,u
       +^k(1-_0)+1s^-1/2√(∫_0^u F̃_1(s,u')du')
    ≲   δ^5/2-_0+δ^2 s^-ι√(Ẽ_1,≤ m+2(s,u))+δ^2√(Ẽ_2,≤ m+2(s,u))
       +δ^2-_0s^-1/2δ^l'Z̅^m'+1μ_s,u+^k(1-_0)+1s^-1/2√(∫_0^u F̃_1(s,u')du').

Note that the last inequality in (<ref>) holds when the number m+2 is replaced by m_1+1 in (<ref>).
Therefore, (<ref>) follows from (<ref>) and (<ref>), and furthermore, (<ref>) is obtained
by inserting (<ref>) into (<ref>).
	
Analogously, due to
	
    R^m L^2ϕ_s,u≲δ[T,R^m L] Lϕ_s,u
    +R^m LT( L^φ_)_s,u,

	 then it follows from (<ref>) that for  Z̃∈{ϱ L, R},
	
    R^m L^2ϕ_s,u
    ≲   (δ^2-_0+δ^(k+1)(1-_0))s^-5/2(Z̃^m_0Ľ^1_s,u+Z̃^m_0Ľ^2_s,u)+δ^(k+1)(1-_0)s^-5/2Z̃^m_0ϱ̌_s,u
       +δ^2-_0s^-7/2Z̃^m_-1x_s,u+δ^3-_0s^-5/2ℒ_R^m_1^(T)π_ L_s,u+δ^3-_0s^-7/2ℒ_R^m_1^(R)π_T_s,u
       +δ^k(1-_0)s^-2R^m_0 Lϕ_s,u+δ^k(1-_0)s^-3R^m_0ϕ_s,u+δ^3-_0s^-3/2 dR^m Lμ_s,u
       +δ^2-_0s^-3/2R^m_0 Lμ_s,u+δ^2-_0s^-5/2R^m_0μ_s,u+ s^-1-ι√(E_1,≤ m+2)+^2 s^-2√(E_2,≤ m+2).

	Utilizing (<ref>), (<ref>) and (<ref>) leads to
	
    R^m L^2ϕ_s,u
    ≲   (δ^2-_0+δ^(k+1)(1-_0))s^-5/2(Z̃^m_0Ľ^1_s,u+Z̃^m_0Ľ^2_s,u)+δ^(k+1)(1-_0)s^-5/2Z̃^m_0ϱ̌_s,u
       +δ^2-_0s^-7/2Z̃^m_-1x_s,u+δ^3-_0s^-5/2ℒ_R^m_1^(T)π_ L_s,u+δ^3-_0s^-7/2ℒ_R^m_1^(R)π_T_s,u
       +δ^k(1-_0)s^-2R^m_0 Lϕ_s,u+δ^k(1-_0)+1s^-3 dR^m+1ϕ_s,u+δ^k(1-_0)s^-3R^m_0ϕ_s,u
       +δ^2-_0s^-5/2R^m_0μ_s,u+ s^-1-ι√(E_1,≤ m+2)+^2 s^-2√(E_2,≤ m+2)
    ≲   δ^5/2-_0s^-2+δ s^-1-ι√(Ẽ_1,≤ m+2(s,u))+δ^2s^-2√(Ẽ_2,≤ m+2(s,u))
       +δ^(k+1)(1-_0)+1(δ+δ^k(1-_0))s^-5/2R^mμ_s,u
    +δ^2-_0s^-5/2δ^l'R^m'+1μ_s,u
       +^k(1-_0)(+^k(1-_0))s^-5/2√(∫_0^u F̃_1(s,u')du'),

	where (<ref>) and (<ref>) have been used. For the case of Z̅^p_1TR^p_2 L^2ϕ with p_1+p_2=m-1, the desired result can be obtained by using Z̅^p_1TR^p_2 L^2ϕ=Z̅^p_1R^p_2[T, L] Lϕ+Z̅^p_1[T,R^p_2] L^2ϕ+Z̅^p_1R^p_2 LT( L^φ_).


(<ref>) implies that when there is at least one T in Z̅^m+1, it then holds that

    δ^lℒ_Z̅^m+1 dϕ_s,u≲   δ^5/2-_0s^-1+δ^2 s^-1-ι√(Ẽ_1,≤ m+2(s,u))+δ^2s^-1√(Ẽ_2,≤ m+2(s,u))
       +δ^2-_0s^-3/2δ^l'Z̅^m'+1μ_s,u+^k(1-_0)+1s^-3/2√(∫_0^u F̃_1(s,u')du').


In addition, it follows from (<ref>) that

    δ^lZ^m+1ϕ_s,u≲   δ^5/2-_0+δ^2s^-ι√(Ẽ_1,≤ m+2(s,u))+δ^2√(Ẽ_2,≤ m+2(s,u))
       +δ^2-_0s^-1/2δ^l'Z̅^m'+1μ_s,u+^k(1-_0)+1s^-1/2√(∫_0^u F̃_1(s,u')du').


One can check easily from (<ref>) and (<ref>) that Z̅^m+1 Lϕ_s,u
has admitted the better smallness order and time decay rate than ℒ
_Z̅^m+1 dϕ_s,u (see the coefficients of R^mμ_s,u).
In addition, it should be pointed out that δ^2-_0s^1/2R^mμ_s,u in (<ref>) is not enough to
close the energy estimates derived in Section <ref> below  when _0 is near _k^*.
To improve L^2 norm estimate of Z̅^m+1μ by the transport equation for LZ̅^m+1μ
 derived from (<ref>), our main strategy is to explore the possibility that the related term 12G_ L L^ Tφ_ in (<ref>) may admit
more precise  smallness and better time decay rate information by some suitable combinations of dϕ.
This will be the next main task.

It follows from (<ref>) that φ__j=δ__j^0 Lϕ- L__jT̃^l_jφ_l_j
+g__ja d^Xx^a d_Xϕ. Then

    g^β,_2⋯_kφ__2⋯φ__k
    
    =   (-1)^k-1g^β,_2⋯_k L__2⋯ L__k(T̃^l_2⋯T̃^l_k)(φ_l_2⋯φ_l_k)
       +(-1)^k(k-1)g^β,_2⋯_k(g__2a d^Xx^a d_Xϕ) L__3⋯ L__k(T̃^l_3⋯T̃^l_k)(φ_l_3⋯φ_l_k)_vanish when k=2
       +f(φ, Ľ^1, Ľ^2,xϱ)(
    [                        ( Lϕ)φ^k-2; ( d^Xx)( d_Xϕ)( d^Xx)( d_Xϕ)φ^k-3;                                   ])
    
    =   (-1)^k-1g^β,_2⋯_kø__2⋯ø__k(T̃^l_2⋯T̃^l_k)(φ_l_2⋯φ_l_k)
       +(-1)^k(k-1)g^β, a_3⋯_k(m_ab d^Xx^b d_Xϕ)ø__3⋯ø__k(T̃^l_3⋯T̃^l_k)(φ_l_3⋯φ_l_k)_vanish when k=2
       +f(φ, Ľ^1, Ľ^2,xϱ,ϱ̌)(
    [                          Ľ^1φ^k-1;                          Ľ^2φ^k-1;                           ϱ̌φ^k-1;                        ( Lϕ)φ^k-2;              ( d^Xx)( d_Xϕ)φ^2k-2; ( d^Xx)( d_Xϕ)( d^Xx)( d_Xϕ)φ^k-3;                                   ]).

Substituting (<ref>) into the third line of (<ref>) and using the null condition (<ref>) yield

    (_φ_γg^β) L_ L_β L_γ
    
    =   (-1)^kk(k-1)g^β, a_3⋯_kø_ø_βø_(m_ab d^Xx^b d_Xϕ)ø__3⋯ø__k(T̃^l_3⋯T̃^l_k)(φ_l_3⋯φ_l_k)_vanish when k=2
       +f(φ, Ľ^1, Ľ^2,xϱ,ϱ̌)(
    [                               φ^k;                          Ľ^1φ^k-1;                          Ľ^2φ^k-1;                           ϱ̌φ^k-1;                        ( Lϕ)φ^k-2;              ( d^Xx)( d_Xϕ)φ^2k-2; ( d^Xx)( d_Xϕ)( d^Xx)( d_Xϕ)φ^k-3;                                   ]).

Based on (<ref>), we can introduce a good unknown as

    𝒜
    =g^β, a_3⋯_kø_ø_βø_(m_ab d^Xx^b d_Xϕ)ø__3⋯ø__k,

which is a combination of  dϕ.


	Under the assumption (⋆), when δ>0 is small, it holds that for m≤ 2N-6,
	
    δ^l   Z̅^m+1𝒜_s,u≲δ^3/2-_0(δ+δ^k(1-_0)ln s)s^-1+δ s^-ι√(Ẽ_1,≤ m+2(s,u))
       +(δ^2+δ^k(1-_0)(+^k(1-_0))ln^2 s) s^-1√(Ẽ_2,≤ m+2(s,u))+δ^2-_0s^-3/2δ^l'Z̅^m'+1μ_s,u
       +(δ^3-_0+δ^(k+1)(1-_0)+1)s^-1/2R^mμ_s,u+^k(1-_0)s^-1/2√(∫_0^u F̃_1(s,u')du'),

	where Z̅∈{T,R}, the numbers of T in Z̅^m+1 and Z̅^m'+1 are l and l' respectively, m'≤ m.



	Note that the coefficients of R^mμ_s,u
in (<ref>) and (<ref>) are the same.
We will use (<ref>) instead of (<ref>) to estimate LZ̅^m+1μ_s,u later.



	Due to d^Xx^b d_Xϕ=(g^bj- L^bg^0j- L^jg^0b- L^b L^j)φ_j by (<ref>) and m_ab L^b=ø_a+m_abĽ^b+ϱ̌ø_a, then it holds that
	
    𝒜=   g^β, a_3⋯_kø_ø_βø_ m_ab(g^bj- L^bg^0j- L^jg^0b)φ_jø__3⋯ø__k
       -g^β, a_3⋯_kø_ø_βø_ø_aø__3⋯ø__k( L^jφ_j)+f(ø)(
    	[ Ľ^1; Ľ^2;  ϱ̌;     ])( L^jφ_j)
    
    	=   g^β, a_3⋯_kø_ø_βø_ m_ab(g^bj- L^bg^0j- L^jg^0b)φ_jø__3⋯ø__k
       -g^β, 0_3⋯_kø_ø_βø_ø__3⋯ø__k( L^jφ_j)+f(ø)(
    	[ Ľ^1; Ľ^2;  ϱ̌;     ])( L^jφ_j),

	where the last identity comes from (<ref>). This, together with (<ref>), yields
	
    R^m+1𝒜_s,u
    ≲   R^m+1{f(ø)(
    	[ Ľ^1; Ľ^2;  ϱ̌;     ])( L^jφ_j)}_s,u
       +δ[R^m+1,T]{g^β, a_3⋯_kø_ø_βø_ m_ab(g^bj- L^bg^0j- L^jg^0b)φ_jø__3⋯ø__k
                     -g^β, 0_3⋯_kø_ø_βø_ø__3⋯ø__k( L^jφ_j)}_s,u
       +δR^m+1{T(g^β, a_3⋯_kø_ø_βø_ø__3⋯ø__k m_ab(g^bj- L^bg^0j- L^jg^0b))φ_j}_s,u
       +δR^m+1{T(g^β, 0_3⋯_kø_ø_βø_ø__3⋯ø__k) L^jφ_j}_s,u
       +δR^m+1{g^β, a_3⋯_kø_ø_βø_ø__3⋯ø__k m_ab(g^bj- L^bg^0j- L^jg^0b)Tφ_j
                 -g^β, 0_3⋯_kø_ø_βø_ø__3⋯ø__k( L^jTφ_j)}_s,u
       +δR^m+1{g^β, 0_3⋯_kø_ø_βø_ø__3⋯ø__k(T L^j)φ_j}_s,u.

	Denote the first five lines in the right hand side of (<ref>) as ℐ, the sixth-seventh
lines as ℐℐ, and the last line as ℐℐℐ, which will be estimated separately as follows.
	
		
  * Since ø^i=1ϱ̌+1·x^iϱ, Tø^i=μT̃^iϱ(ϱ̌+1)-∑_j=1^3μ x^ix^jT̃^jϱ^3(ϱ̌+1)^3 and T L^i satisfies (<ref>), then
		
    ℐ≲   δ^1-_0s^-1/2(∑_i=1^2R^m_0Ľ^i_s,u+R^m_0ϱ̌_s,u)+δ^k(1-_0)s^-1ln sR^m_0φ_s,u
       +(δ^(k+1)(1-_0)ln s+δ^2-_0)s^-5/2R^m_-1x_s,u+δ^2-_0s^-3/2ℒ_R^m_1^(R)π_T_s,u
       +δ^2-_0s^-5/2ln s(δ^k(1-_0)+δ^2k(1-_0)-1)ℒ_R^m_1 g_s,u+δ^2-_0s^-3/2R^m'+1μ_s,u
       +δ^(k+1)(1-_0)+1s^-3/2 dR^m+1μ_s,u+δ^k(1-_0)+1s^-1TR^m'+1φ_s,u
       +δ^2k(1-_0)+1s^-2 dR^m+1φ_s,u+δ^2k(1-_0)+1s^-2R^m'+1 Lφ_s,u,

		where the second boxed term above arises from δR^m+1(g^0jT L^b+g^0bT L^j)φ_j_s,u.
		
		
  * Making use of (<ref>) again, that is, g^β, 0_3⋯_kø_ø_βø_ø__3⋯ø__k
=g^β, a_3⋯_kø_ø_βø_ø_aø__3⋯ø__k, then one can  rewrite ø_a=m_ab L^b+(1ϱ̌+1-1)m_ab L^b-1ϱ̌+1m_abĽ^b to get
		
    ℐℐ=   δR^m+1{g^β, a_3⋯_kø_ø_βø_ø__3⋯ø__k(m_ab d^Xx^b d_Xφ_j)μT̃^j}_s,u
       +δR^m+1{f(ø, L^1, L^2,ϱ̌)(
    		[ Ľ^1; Ľ^2;  ϱ̌;     ])( L^jTφ_j)}_s,u,

		which implies
		
    ℐℐ≲   δ dR^m+1φ_s,u+δ^2-_0s^-5/2R^m_-1x_s,u+δ^1-_0s^-1/2R^m_0ϱ̌_s,u
       +δ^1-_0s^-1/2(R^m_0Ľ^1_s,u+R^m_0Ľ^2_s,u)+δ^2-_0s^-3/2R^m'+1μ_s,u
       +δ s^-1R^m_0φ_s,u+δ^k(1-_0)+1s^-1ln sTR^m'+1φ_s,u
       +δ^(k+1)(1-_0)+1s^-5/2ln sℒ_R^m_1^(R)π_T_s,u+δ^2-_0s^-3/2ℒ_R^m_1 g_s,u.

		
		
  * It follows from (<ref>) and (<ref>) that
		
    ℐℐℐ≲   δ^3-_0s^-3/2 dR^m+1μ_s,u+(δ^(k+1)(1-_0)+1+δ^3-_0)s^-5/2R^m'+1μ_s,u
       +δ^(k+1)(1-_0)+1s^-5/2ℒ_R^m_1 g_s,u+δ^k(1-_0)s^-1 dR^m'+1ϕ_s,u
       +δ^k(1-_0)s^-1R^m'+1 Lϕ_s,u+δ^(k+2)(1-_0)s^-5/2R^m_0+1x_s,u
       +δ^(k+1)(1-_0)s^-3/2(R^m_0Ľ^1_s,u+R^m_0Ľ^2_s,u+R^m_0ϱ̌_s,u)
       +δ^(k+1)(1-_0)+1s^-3/2TR^m'+1φ_s,u+δ^(k+2)(1-_0)+1s^-3ℒ_R^m_1^(R)π_T_s,u
       +δ^k(1-_0)+1s^-1R^m'+1 Lφ_s,u+δ^k(1-_0)+1s^-1 dR^m+1φ_s,u
       +δ^(k+1)(1-_0)s^-3/2R^m_0φ_s,u.

	

Substituting (<ref>)-(<ref>) to (<ref>), using Proposition <ref> and Lemma <ref> to estimate all terms except the boxed ones, and adopting (<ref>) and (<ref>) in Appendix A to find the relationship between dR^m+1μ and R^mμ,
then one can get eventually that
	
    R^m+1   𝒜_s,u≲δ^3/2-_0(δ+δ^k(1-_0)ln s)s^-1+δ s^-ι√(Ẽ_1,≤ m+2(s,u))
       +(δ^2+δ^k(1-_0)(+^k(1-_0))ln^2 s) s^-1√(Ẽ_2,≤ m+2(s,u))+δ^2-_0s^-3/2R^m'+1μ_s,u
       +(δ^3-_0+δ^(k+1)(1-_0)+1)s^-1/2R^mμ_s,u+^k(1-_0)s^-1/2√(∫_0^u F̃_1(s,u')du').


For Z̅^m+1=Z̅^p_1TR^p_2 with p_1+p_2=m, it follows from the expression of 𝒜, Proposition <ref>, (<ref>) and (<ref>) that

    δ^lZ̅^m+1𝒜_s,u
    ≲   δ^2-_0s^-3/2{δ^l_0Z̅^m_0ϱ̌_s,u+s^-1δ^l_-1Z̅^m_-1x_s,u}+δ^lℒ_Z̅^m+1 dϕ_s,u+s^-1δ^l_0Z̅^m_0ϕ_s,u
    ≲   δ^5/2-_0s^-1+δ^2 s^-1-ι√(Ẽ_1,≤ m+2)+δ^2s^-1√(Ẽ_2,≤ m+2)+δ^2-_0s^-3/2δ^l'Z̅^m'+1μ_s,u
       +^k(1-_0)+1s^-3/2√(∫_0^u F̃_1(s,u')du').

Therefore, (<ref>) follows from (<ref>) and (<ref>).


Next we can refine the L^2 norm of Z^m+1μ despite the appearance of R^mμ which will be dealt with later.

	Under the assumption (⋆), when δ>0 is small, it holds that for m≤ 2N-6,
	
    δ^lZ^m+1μ_s,u≲δ^k(1-_0)-_0+1/2s^1/2+δ^(k-1)(1-_0) s^1/2√(Ẽ_1,≤ m+2(s,u))
           +δ^k(1-_0)s^1/2√(Ẽ_2,≤ m+2(s,u))+δ^k(1-_0)(δ+δ^k(1-_0))s^1/2∫_t_0^sτ^-3/2R^mμ_τ,udτ
           +^(2k-1)(1-_0)-1s^1/2√(∫_0^u F̃_1(s,u')du').



	For any Z̅∈{T, R}, (<ref>) together with (<ref>) and (<ref>) gives
	
    δ^l LZ̅^m+1μ_s,u
    ≲   δ^k(1-_0)s^-1δ^lTZ̅^m+1φ_s,u+δ^(k+1)(1-_0)s^-5/2δ^l_1ℒ_Z̅^m_1^(R)π_T_s,u
       +δ^k(1-_0)-1s^-1δ^l_0(Z̅^m_0ϱ̌_s,u+Z̅^m_0Ľ^1_s,u+Z̅^m_0Ľ^2_s,u)
       +δ^(k-1)(1-_0)-1s^-1/2δ^l_0Z̅^m_0 Lϕ_s,u+δ^k(1-_0)-1s^-1δ^l_0Z̅^m_0φ_s,u
       +δ^(k-1)(1-_0)-1(δ+δ^k(1-_0))s^-3/2δ^l_0ℒ_Z̅^m_0 dϕ_s,u+δ^k(1-_0)-_0s^-1δ^l_1ℒ_Z̅^m_1^(R)π_ L_s,u
       +δ^k(1-_0)-_0s^-5/2δ^l_-1Z̅^m_-1x_s,u+δ^(k-1)(1-_0)s^-1/2δ^l dZ̅^m+1φ_s,u
       +δ^(k-1)(1-_0)-1s^-1/2δ^l_0Z̅^m_0𝒜_s,u+δ^(k-1)(1-_0)s^-1/2δ^l_0 LZ̅^m_0φ_s,u        
       +δ^(k+1)(1-_0)s^-1δ^l_1ℒ_Z̅^m_1^(T)π_ L_s,u+δ^2k(1-_0)-_0s^-2ln sδ^l_1ℒ_Z̅^m_1 g_s,u
       +δ^k(1-_0)s^-2δ^l_0Z̅^m_0μ_s,u,

where l_p (p=-1,0,1) are the numbers of T in Z^m_p and 1≤ m_p≤ m+1-p.
It follows from Proposition <ref>, Lemma <ref> and (<ref>) that
	
    δ^l LZ̅^m+1μ_s,u≲   δ^k(1-_0)(δ+δ^k(1-_0))s^-1R^mμ_s,u+δ^k(1-_0)s^-2δ^l_0Z̅^m_0μ_s,u
       +δ^k(1-_0)-_0+1/2s^-1+δ^(k-1)(1-_0)s^-1/2-ι√(Ẽ_1,≤ m+2(s,u))
       +δ^k(1-_0)s^-1√(Ẽ_2,≤ m+2(s,u))+^(2k-1)(1-_0)-1s^-1√(∫_0^u F̃_1(s,u')du').

	As in (<ref>), by taking F(s,u,ϑ)=δ^l(Z̅^m+1μ(s,u,ϑ)-Z̅^m+1μ(t_0, u, ϑ)) in (<ref>), one gets
	
    δ^lZ̅^m+1μ_s,u≲δ^k(1-_0)-_0+1/2s^1/2+δ^(k-1)(1-_0) s^1/2√(Ẽ_1,≤ m+2(s,u))
           +δ^k(1-_0)s^1/2√(Ẽ_2,≤ m+2(s,u))+δ^k(1-_0)(δ+δ^k(1-_0))s^1/2∫_t_0^sτ^-3/2R^mμ_τ,udτ
           +^(2k-1)(1-_0)-1s^1/2√(∫_0^u F̃_1(s,u')du').

	
	For Z^m+1=Z^p_1(ϱ L)Z̅^p_2 with p_1+p_2=m, due to (<ref>), (<ref>) and (<ref>), one can get by Proposition <ref> and (<ref>) that
	
    δ^lZ^p_1(ϱ L)Z̅^p_2μ_s,u≲δ^lZ^p_1(ϱZ̅^p_2 Lμ)_s,u+δ^lZ^p_1(ϱ[ L,Z̅^p_2]μ)_s,u
    ≲   δ^k(1-_0)-_0δ^l_1ℒ_Z^m_1^(R)π_ L_s,u+δ^(k+1)(1-_0)δ^l_1ℒ_Z^m_1^(T)π_ L_s,u+δ^k(1-_0)s^-1ln sδ^l_1Z^m_1μ_s,u
       +δ^2k(1-_0)-_0s^-1ln sδ^l_1ℒ_Z^m_1 g_s,u+δ^k(1-_0)-1δ^l_1(Z^m_1Ľ^1_s,u+Z^m_1Ľ^2_s,u+Z^m_1ϱ̌_s,u)
    	
       +δ^k(1-_0)-_0s^-3/2δ^l_0Z^m_0x_s,u+δ^k(1-_0)-1δ^l_0Z^m_0φ_s,u+δ^(k-1)(1-_0)-1s^-1/2δ^l_0Z^m_0ϕ_s,u
    ≲   δ^k(1-_0)-_0+1/2+δ^k(1-_0)(1+^k(1-_0)-1) s^-ι√(Ẽ_1,≤ m+2(s,u))+δ^k(1-_0)√(Ẽ_2,≤ m+2(s,u))
       +δ^k(1-_0)s^-1ln sδ^l_1Z^m_1μ_s,u+^(2k-1)(1-_0)-1√(∫_0^u F̃_1(s,u')du').

	
	Thus, (<ref>) follows from (<ref>) and (<ref>).


It follows from (<ref>), Lemma <ref>, (<ref>) and (<ref>) that

	Under the assumption (⋆), for small δ>0, it holds that for m≤ 2N-6,
	
    δ^lZ̅^m+1 Lϕ   _s,u≲δ^5/2-_0s^-1+δ s^-ι√(Ẽ_1,≤ m+2(s,u))+δ^2s^-1√(Ẽ_2,≤ m+2(s,u))
    
    +   δ^2-_0(δ+δ^k(1-_0)){s^-1/2R^mμ_s,u
    +δ^k(1-_0)s^-1∫_t_0^sτ^-3/2R^mμ_τ,udτ}
    
    +   ^k(1-_0)( s^-3/2+^k(1-_0)s^-1)√(∫_0^u F̃_1(s,u')du'),
    δ^lℒ_Z̅^m+1 dϕ   _s,u≲δ^5/2-_0s^-1
    +δ s^-ι√(Ẽ_1,≤ m+2(s,u))+δ^2s^-1√(Ẽ_2,≤ m+2(s,u))
    
    	+δ^2-_0   s^1/2R^mμ_s,u+δ^(k+1)(1-_0)+1(δ
    +δ^k(1-_0))s^-1∫_t_0^sτ^-3/2R^mμ_τ,udτ
    
    +^k(1-_0)   ( s^-3/2+^k(1-_0)s^-1)√(∫_0^u F̃_1(s,u')du'),
    δ^lZ̅^m L^2ϕ   _s,u≲δ^5/2-_0s^-2+δ s^-1-ι√(Ẽ_1,≤ m+2(s,u))+δ^2s^-2√(Ẽ_2,≤ m+2(s,u))
    
    	+   δ^(k+1)(1-_0)+1(δ+δ^k(1-_0))s^-2{s^-1/2R^mμ_s,u
    +∫_t_0^sτ^-32R^mμ_τ,udτ}
    
    +   ^k(1-_0)( s^-5/2+^k(1-_0)s^-2)√(∫_0^u F̃_1(s,u')du'),
    δ^lZ^m+1ϕ   _s,u≲δ^5/2-_0+δ(δ s^-ι+δ^k(1-_0))√(Ẽ_1,≤ m+2(s,u))+δ^2√(Ẽ_2,≤ m+2(s,u))
             +δ^(k+1)(1-_0)+1(δ+δ^k(1-_0))∫_t_0^sτ^-3/2R^mμ_τ,udτ
             +^k(1-_0)( s^-1/2+^k(1-_0))√(∫_0^u F̃_1(s,u')du'),

	and
	
    δ^lZ̅^m+1   𝒜_s,u≲δ^3/2-_0(δ+δ^k(1-_0)ln s)s^-1+δ s^-ι√(Ẽ_1,≤ m+2(s,u))
       +(δ^2+δ^k(1-_0)(+^k(1-_0))ln^2 s) s^-1√(Ẽ_2,≤ m+2(s,u))
       +(δ+δ^k(1-_0))δ^2-_0{s^-1/2R^mμ_s,u
    +^k(1-_0)s^-1∫_t_0^sτ^-3/2R^mμ_τ,udτ}
       +^k(1-_0)s^-1/2√(∫_0^u F̃_1(s,u')du').






§ TOP ORDER L^2 ESTIMATES ON THE DERIVATIVES OF Χ AND Μ


	It follows from Proposition <ref> that to complete the estimate of Z^m+1μ_s,u, it remains to bound R^mμ_s,u. Furthermore, by the energy estimate (<ref>) and (<ref>), one checks easily that the top order derivatives of φ
in the energy estimate for (<ref>) are of 2N-4. On the other hand, by the expression of ^(Z)D_γ,2^m
 in (<ref>), the top order derivatives of χ and μ in (<ref>) are of 2N-5 and 2N-4 respectively as in <cit.>. However,
 it follows from  Proposition <ref> that the L^2 estimates on the 2N-5 top order derivatives of χ
and the 2N-4 top order derivatives of
μ should be controlled by the energies of  with the orders 2N-3, which
is beyond the range of energies containing up to 2N-4 order derivatives of .
To overcome this difficulty, as in <cit.>, also motivated by <cit.>,
we will
treat the smallness orders and the time decay rates of trχ and μ
with the corresponding top order derivatives. It should be pointed out that although the main ideas here are similar to
<cit.>, yet it seems hard to adopt the analysis in <cit.> to close the energies in
Section <ref> when _0 is near _k^* in this paper. For instance, the quantity e in equation (<ref>) and the factor ( LG_β^γ) L^ L^β Lφ_ in equation (<ref>) below should be handled more carefully by the null structure (<ref>) and a new identity (<ref>). In this case, the resulting new transport
equation (<ref>) is much more suitable for estimating the L^2 norm of the top derivative of trχ in the paper.







 §.§ Estimates on the derivatives of dtrχ and ∇χ



1. If there is at least one vector field ϱ L in Z^m, that is, Z^m=Z^p_1(ϱ L)Z̅^p_2 and p_1+p_2=m-1 with Z̅∈{T,R}, then
it follows from (<ref>) and Proposition <ref> that

    δ^l dZ^m(trχ̌)_s,u=δ^l dZ^p_1Z̅^p_2(ϱ L) (trχ̌)
    + dZ^p_1[ϱ L,Z̅^p_2](trχ̌)_s,u
    ≲   s^-1δ^l_1ℒ_Z^m_1χ̌_s,u+δ^k(1-ε_0)s^-3ln sδ^l_1(ℒ_Z^m_1^(R)π_ L_s,u+δℒ_Z^m_1^(T)π_ L_s,u)
       +δ^k(1-ε_0)s^-3{δ^l_1Z^m_1Ľ^1_s,u+δ^l_1Z^m_1Ľ^2_s,u+s^-1δ^l_0Z^k_0x_s,u+ln sδ^l_1ℒ_Z^m_1 g_s,u}
       +δ^(k-1)(1-ε_0)s^-5/2δ^l_0Z^m_0φ_s,u+δ^(k-1)(1-ε_0)s^-3/2δ^l_0 dZ^m_0φ_s,u
    ≲   δ^k(1-ε_0)+1/2s^-5/2ln s+δ^(k-1)(1-ε_0)s^-3/2-ι√(Ẽ_1,≤ m+2)
       +δ^k(1-ε_0)+ε_0(1+^k(1-_0)-1) s^-5/2ln^2 s√(Ẽ_2,≤ m+2)+^(k-1)(1-_0)s^-2√(∫_0^u F̃_1(s,u')du').

Thus,

    δ^l∇ℒ_Z^mχ̌_s,u≲   δ^k(1-ε_0)+1/2s^-5/2ln s+δ^(k-1)(1-ε_0)s^-3/2-ι√(Ẽ_1,≤ m+2)
       +δ^k(1-ε_0)+ε_0(1+^k(1-_0)-1) s^-5/2ln^2 s√(Ẽ_2,≤ m+2)
       +^(k-1)(1-_0)s^-2√(∫_0^u F̃_1(s,u')du').


2. When all vectorfields Z's are in {T, R}, then (<ref>) implies

    L(tr   χ)=-|χ̌|^2-2ϱtrχ+1ϱ^2-G_X L^γ d^X Lφ_γ
    +12G_ L L^γφ_γ+12 g^XXG_XX^γ L^2φ_γ
       -{12G_ L L^γ( Lφ_γ)+G_T̃ L^γ( Lφ_γ)-G_X L^γ( d^Xφ_γ)}trχ+f(φ,  dx, L^1, L^2)φ^k-2(
    [ Lφ; dφ;    ])^2,

where

    φ_=μ^-1( LLφ_+12ϱLφ_-H_)

by (<ref>).
Note that

    G_X L^γ d^X Lφ_γ=    L(G_X L^γ d^Xφ_γ)-( LG_β^) L^β d_Xx^ d^Xφ-G_β( d_X L^) L^β d^Xφ_
       -G_β d_Xx^( L L^β) d^Xφ_+2trχ (G_X L^ d^Xφ_),

and the term L(G_X L^γ d^Xφ_γ) can be moved to the left hand of (<ref>).
Analogous treatments can be carried out for 12 g^XXG_XX^γ L^2φ_γ and 12μ^-1G_ L L^γ LLφ_.
Thus, with the help of (<ref>), (<ref>), (<ref>) and (<ref>), (<ref>) can be written as a new form containing only the first order
or zero-th order derivatives of φ_ on the right hand side of the equality

    L(trχ-E)=(-2ϱ+ℰ)trχ+1ϱ^2-|χ̌|^2-12μ^-1 L(G_β^γ) L^ L^β Lφ_+e,

where

    E=   -G_X L^γ d^Xφ_γ+12μ^-1G_ L L^γLφ_γ
    +12 g^XXG_XX^γ Lφ_γ,
    ℰ=   -12G_ L L^γ Lφ_γ-G_T̃ L^γ Lφ_γ+12μ^-1G_ L L^γ Tφ_γ,

and

    e=   12μ^-2(G_ L L^γ Tφ_γ)^2+f_1(φ,  dx^i, L^i, g)φ^k-2(
    [ Lφ; dφ;    ])^2+f_2(φ,  dx^i, L^i, g)μ^-1φ^2k-2 Lφ Tφ
       +f_3(φ,  dx^i, L^i, g)μ^-1φ^k-1(G_ L L^ d_Xφ_)Tφ
    +f_4(φ,  dx^i, L^i, g)μ^-1φ^k-1(G_ L L^ Tφ_) dφ.

Here note that for the global estimate on the solution ϕ
to (<ref>) with (<ref>) for all _0<_k^* especially for _0 near the critical
exponent _k^*, we will make full use of the null structure (<ref>) to
write explicitly the factors G_ L L^ d_Xφ_ and G_ L L^ Tφ_
in the expression of e. In fact, according to (<ref>),

    G_ L L^ Tφ_=-(G_ L L^ L_)T̃^iTφ_i
    +μ G_ L L^0T̃^i( Lφ_i)
    +μ g_γ iG_ L L^γ ( d^Xx^i)T̃^j d_Xφ_j.

In addition, inserting (<ref>) into (<ref>) gives

    d^Xx^i d_Xφ_i=μ^-1 L^ Lφ_
    =-( L^+2g^0) Lφ_,

which, together with (<ref>), implies

    G_ L L^ d_Xφ_=G_ L L^ d_Xx^i(_φ_i)
    
    =   _^0G_ L L^( d_Xx^i Lφ_i)-(G_ L L^ L_)T̃^j d_Xφ_j+g_ jG_ L L^ d_Xx^j( d^Xx^i d_Xφ_i)
    
    =   _^0G_ L L^( d_Xx^i Lφ_i)-(G_ L L^ L_)T̃^j d_Xφ_j-g_ jG_ L L^ d_Xx^j( L^+2g^0) Lφ_.

Substituting (<ref>) and (<ref>) into (<ref>) yields

    e=   12μ^-2(G_ L L^γ Tφ_γ)^2+f_1(φ,  dx, L^1, L^2, g)φ^k-2(
    [ Lφ; dφ;    ])^2
       +f_2(φ,  dx, L^1, L^2, g)μ^-1φ^2k-2 Lφ Tφ+f_3(φ,  dx, L^1, L^2, g)μ^-1φ^k-1(G_ L L^ L_)Tφ dφ.

Let F^m= dZ̅^mtrχ- dZ̅^m E with Z̅∈{T,R}. Then by an induction
argument on (<ref>), one can get

    ℒ_ LF^m=   (-2ϱ+ℰ)F^m+(-2ϱ+ℰ) dZ̅^m E- dZ̅^m(|χ̌|^2)
       -12 dZ̅^m(μ^-1 L(G_β^γ) L^ L^β Lφ_)+e^m,

where

    e^m=   ℒ_Z̅^m e^0+∑_p_1+p_2=m-1ℒ_Z̅^p_1ℒ_[ L,Z̅]F^p_2+∑_[ p_1+p_2=m;    p_1≥ 1 ]Z̅^p_1(-2ϱ+ℰ) dZ̅^p_2trχ_vanish when m=1

and

    e^0=    de+( dℰ)trχ.


Similarly,
the factor ( LG_β^γ) L^ L^β Lφ_ in (<ref>)
can also be dealt with (<ref>). In fact, due to 
    G_β^=-kg_'g_ββ'g^'β',_2⋯_kφ__2⋯φ__k+f(φ)φ^k,

then by (<ref>) and L L^i=-12(G_ L L^ L_)T̃^j d^Xx^i d_Xφ_j+O(φ^k-1 Lφ) from (<ref>) and (<ref>), one can obtain

    LG_β^=   -k(k-1)g_'g_ββ'g^'β',_2⋯_k( L__2ϕ)φ__3⋯φ__k+f(φ)φ^k-1 Lφ
    
    =   k(k-1)g_'g_ββ'g^'β',_2⋯_k L__2T̃^i_2( Lφ_i_2)φ__3⋯φ__k
       -k(k-1)g_'g_ββ'g^'β',_2⋯_k L(δ__2^0 Lϕ+g__2i d^Xx^i d_Xϕ)φ__3⋯φ__k
       +f(φ, L^1, L^2)φ^k-1(
    [                     Lφ; (G_ L L^ L_) d^Xx d_Xφ ]).

Applying (<ref>) to φ__p=__pϕ (3≤ p≤ k) again leads to

    LG_β^=   (-1)^kk(k-1)g_'g_ββ'g^'β',_2⋯_k L__2⋯ L__kT̃^i_2⋯T̃^i_k( Lφ_i_2)φ_i_3⋯φ_i_k
       -(-1)^kk(k-1)(k-2)g_'g_ββ'g^'β',_2⋯_k L__2T̃^i( Lφ_i)(g__3j d^Xx^j d_Xϕ)
                           · L__4⋯ L__kT̃^i_4⋯T̃^i_kφ_i_4⋯φ_i_k
       -k(k-1)g_'g_ββ'g^'β',_2⋯_k L(δ__2^0 Lϕ+g__2i d^Xx^i d_Xϕ)φ__3⋯φ__k
       +f(φ, L^1,  L^2)(
    [                        φ^k-1 Lφ;     φ^k-1(G_ L L^ L_) d^Xx d_Xφ;                      Lφ Lϕφ^k-3; Lφ( d_Xx d^Xϕ)( d_Xx d^Xϕ)φ^k-4 ]).

Thus,

    ( LG_β^γ) L^ L^β Lφ_=( LG_β^γ) L^ L^β L^ν_φ_ν
    
    =   (-1)^k-1k(k-1)g^β,_2⋯_k L_ L_β L_ L__2⋯ L__kT̃^i_2⋯T̃^i_k( Lφ_i_2)φ_i_3⋯φ_i_k(μ^-1 L^ν Tφ_ν)
       +(-1)^kk(k-1)(k-2)g^β,_2⋯_k L_ L_β L_ L__2(g__3j d^Xx^j d_Xϕ) L__4⋯ L__k
                           ·T̃^i( Lφ_i)T̃^i_4⋯T̃^i_kφ_i_4⋯φ_i_k(μ^-1 L^ν Tφ_ν)
       -k(k-1)g_'g_ββ'g^'β',_2⋯_k L(δ__2^0 Lϕ+g__2i d^Xx^i d_Xϕ)φ__3⋯φ__k L^ L^β Lφ_
       +f(φ, L^1,  L^2,xϱ)(
    [                        φ^k-1 Lφ( Lφ);     φ^k-1(G_ L L^ L_) d^Xx d_Xφ( Lφ);                      Lφ Lϕφ^k-3( Lφ); Lφ( d_Xx d^Xϕ)( d_Xx d^Xϕ)φ^k-4( Lφ);                        μ( Lφ)^2φ^k-2;                μ Lφ( d^Xx d_Xφ)φ^k-2 ]).

As for (<ref>), the null condition (<ref>) implies that

    ( LG_β^γ) L^ L^β Lφ_
    
    =   (-1)^kk(k-1)(k-2)g^β, j'_3⋯_kø_ø_βø_(m_jj' d^Xx^j d_Xϕ)ø__3⋯ø__k^𝒜        
                           ·T̃^i( Lφ_i)T̃^i_4⋯T̃^i_kφ_i_4⋯φ_i_k(μ^-1 L^ν Tφ_ν)
       -k(k-1)g_'g_ββ'g^'β',_2⋯_k L(δ__2^0 Lϕ+g__2i d^Xx^i d_Xϕ)φ__3⋯φ__k L^ L^β Lφ_
       +f(φ, L^1, L^2,xϱ,ϱ̌)(
    [                        φ^k-1 Lφ( Lφ);     φ^k-1(G_ L L^ L_) d^Xx d_Xφ( Lφ);                      Lφ Lϕφ^k-3( Lφ); Lφ( d_Xx d^Xϕ)( d_Xx d^Xϕ)φ^k-4( Lφ);                        μ( Lφ)^2φ^k-2;                μ Lφ( d^Xx d_Xφ)φ^k-2;            ( Lφ)Tφ( d^Xx d_Xϕ)φ^2k-3;                    Ľ^1( Lφ)φ^k-2(Tφ);                    Ľ^2( Lφ)φ^k-2(Tφ);                     ϱ̌( Lφ)φ^k-2(Tφ);        Ľ^1( d^Xx d_Xϕ)( Lφ)φ^k-3(Tφ);        Ľ^2( d^Xx d_Xϕ)( Lφ)φ^k-3(Tφ);         ϱ̌( d^Xx d_Xϕ)( Lφ)φ^k-3(Tφ) ])_ℬ.


Observe that the first term in the right hand side of (<ref>) contains
the good known 𝒜 which has been estimated in (<ref>). Note also that
if one substitutes the second term of (<ref>) into (<ref>) and estimates
the resulting term directly, then the final energy estimates cannot be closed since
the highest m+3  order derivatives of ϕ will  appear on the right hand side
of the related energy inequality (Corollary <ref> only includes the estimates of at most
m+2  order derivatives of ϕ). Fortunately, there exists
a crucial factor L(δ__2^0 Lϕ+g__2i d^Xx^i d_Xϕ) in
the second term of (<ref>), which helps us rewrite the corresponding term in (<ref>) as
 -ℒ_ L𝒞^m+⋯, and -ℒ_ L𝒞^m
 can be moved to the left hand side of (<ref>) (see (<ref>) below and 𝒞^m
 admits some good properties). In addition, the last term ℬ can be regarded
as the higher order error term, which can be treated easily. To deal with the term -12ℒ_Z̅^m d(( LG_β^γ) L^ L^β Lφ_) in (<ref>), as for (<ref>), one can get from (<ref>) that

    dZ̅^m(μ^-1( LG_β^γ) L^ L^β Lφ_)
    
    =   ℒ_ L𝒞^m+𝒟^m+ dZ̅^m(μ^-1ℬ)
       +(-1)^kk(k-1)(k-2) dZ̅^m{𝒜T̃^i( Lφ_i)T̃^i_4⋯T̃^i_kφ_i_4⋯φ_i_k(μ^-2 L^ν Tφ_ν)},

where

    𝒞^m=-k(k-1)g^β,_2⋯_k dZ̅^m(δ__2^0 Lϕ+g__2i d^Xx^i d_Xϕ)μ^-1φ__3⋯φ__k L_ L_β Lφ_

and

    𝒟^m
    =   -k(k-1)g^β,_2⋯_k{([ℒ_Z̅^m,ℒ_ L] d(δ__2^0 Lϕ+g__2i d^Xx^i d_Xϕ))μ^-1φ__3⋯φ__k L_ L_β Lφ_
           - dZ̅^m(δ__2^0 Lϕ+g__2i d^Xx^i d_Xϕ) L(μ^-1φ__3⋯φ__k L_ L_β Lφ_)
           +ℒ_Z̅^m{ L(δ__2^0 Lϕ+g__2i d^Xx^i d_Xϕ) d(μ^-1φ__3⋯φ__k L_ L_β Lφ_)}
           +∑_[ p_1+p_2=m;    p_1≥ 1 ]ℒ_Z̅^p_1 d L(δ__2^0 Lϕ+g__2i d^Xx^i d_Xϕ)Z̅^p_2(μ^-1φ__3⋯φ__k L_ L_β Lφ_)}.

Substituting (<ref>) into (<ref>) and setting

    F̂^m:=F^m+12𝒞^m

and

    ê^m:=   e^m-12𝒟^m-12 dZ̅^m(μ^-1ℬ)
       -12(-1)^kk(k-1)(k-2) dZ̅^m{𝒜T̃^i( Lφ_i)T̃^i_4⋯T̃^i_kφ_i_4⋯φ_i_k(μ^-2 L^ν Tφ_ν)},

then one can rewrite (<ref>) as

    ℒ_ LF̂^m=(-2ϱ+ℰ)F̂^m+(-2ϱ+ℰ)( dZ̅^m E
    -12𝒞^m)- dZ̅^m(|χ̌|^2)+ê^m.

Note that for any one-form ξ on S_s,u, it holds that

    L(ϱ^2|ξ|^2)=-2ϱ^2χ̌^XXξ_Xξ_X
    +2ϱ^2 g^XX(ℒ_ Lξ_X)ξ_X.

By taking ξ=ϱ^2F̂^m in (<ref>) and using (<ref>), one can obtain

    L(ϱ^6|F̂^m|^2)
    =2ϱ^6{   -trχ̌|F̂^m|^2+ℰ|F̂^m|^2+ê^m·F̂^m
       +(-2ϱ+ℰ)( dZ̅^m E-12𝒞^m)·F̂^m- dZ̅^m(|χ̌|^2)·F̂^m}.

Then

    | L(ϱ^3|F̂^m|)|≲   δ^k(1-ε_0)-ε_0s^-3/2ϱ^3|F̂^m|+ϱ^2| dZ̅^m E|+ϱ^2|𝒞^m|+ϱ^3| dZ̅^m(|χ̌|^2)|+ϱ^3|ê^m|.

It follows from (<ref>) and (<ref>) that

    δ^lϱ^3F̂^m   _s,u≲δ^lF̂^m(t_0,·,·)_L^2(Σ_s^ u)+δ^lϱ^1/2∫_t_0^s{δ^k(1-ε_0)
    -ε_0τ^-3/2ϱ^5/2F̂^m_L^2(Σ_τ^ u)
       +τ^3/2 dZ̅^m E_τ,u+τ^3/2𝒞^m_τ,u+τ^5/2 dZ̅^m(|χ̌|^2)_L^2(Σ_τ^u)+τ^5/2ê^m_L^2(Σ_τ^u)}dτ.

This, together with Gronwall's inequality, yields

    ϱ^5/2δ^lF̂
    ^m_s,u≲   δ^k(1-ε_0)-ε_0+1/2+δ^l∫_t_0^s{τ^3/2 dZ̅^m E_τ,u
       +τ^3/2𝒞^m_τ,u+τ^5/2 dZ̅^m(|χ̌|^2)_L^2(Σ_τ^u)+τ^5/2ê^m_L^2(Σ_τ^u)}dτ.

Each term in the integrand of (<ref>) can be estimated as follows.

0.1 true cm

(1) The estimate of dZ̅^m E_L^2(Σ_s^u)
	
Due to E=-G_X L^γ d^Xφ_γ+12μ^-1G_ L L^γLφ_γ+12 g^XXG_XX^γ Lφ_γ, by virtue of (<ref>) and (<ref>),
then it holds that
	
    δ^l dZ̅^m E_s,u
    ≲   δ^k(1-ε_0)-ε_0s^-7/2δ^l_-1Z̅^m_-1x_s,u
    +δ^k(1-ε_0)-1s^-2{∑_i=1^2δ^l_0Z̅^m_0Ľ^i_s,u+δ^l_0Z̅^m_0ϱ̌_s,u}
       +δ^(k-1)(1-ε_0)-1s^-3/2δ^l_0Z̅^m_0 Lϕ_s,u+δ^(k-1)(1-ε_0)(1+^k(1-_0)-1)s^-5/2δ^l_0 dZ̅^m_0ϕ_s,u
       +δ^k(1-ε_0)-1s^-2δ^l_0Z̅^m_0φ_s,u
    +δ^k(1-ε_0)s^-3δ^l_0ℒ_Z̅^m_0 g_s,u
    +^k(1-_0)s^-2δ^l_0TZ̅^m_0φ_s,u
       +δ^k(1-ε_0)-ε_0s^-5/2δ^l_0Z̅^m_0μ_s,u+δ^(k-1)(1-ε_0)s^-3/2δ^l_0 dZ̅^m_0φ_s,u
       +^(k+1)(1-_0)s^-7/2^l_1ℒ_Z̅^m_1^(R)π_T_s,u+^(k-1)(1-_0)-1s^-3/2δ^l_0Z̅^m_0𝒜_s,u
    ≲   δ^k(1-_0)-_0+1/2s^-2+δ^(k-1)(1-_0)s^-3/2-ι√(Ẽ_1,≤ m+2)+δ^k(1-_0)s^-2√(Ẽ_2,≤ m+2)
       +^k(1-_0)(δ+^k(1-_0))s^-2{R^mμ_s,u+δ^k(1-_0)
    -_0∫_t_0^sτ^-3/2R^mμ_τ,udτ}
       +^(2k-1)(1-_0)-1s^-2√(∫_0^u F̃_1(s,u')du'),

where one has used the related L^∞ estimates in Subsection <ref> and the L^2 estimates in
Propositions <ref>, <ref> and Corollary <ref>,  l_p is the number of T
in Z^m_p (p=-1,0,1) and 1≤ m_p≤ m+1-p.

0.1 true cm

(2) The estimate of 𝒞^m_L^2(Σ_s^u)

0.1 true cm

(<ref>) implies that

    ^l𝒞^m_L^2(Σ_s^u)
    ≲   δ^(k-1)(1-_0)-1s^-3/2δ^lRZ̅^m Lϕ_L^2(Σ_s^u)
    +^(2k-1)(1-_0)s^-7/2^l_0Z̅^m_0φ_L^2(Σ_s^u)
       +^2k(1-_0)s^-5^l_0Z̅^m_0x_L^2(Σ_s^u)+^(2m-1)(1-_0)-1s^-7/2^l_0Z̅^m_0ϕ_L^2(Σ_s^u)
       +s^-1^lg^β,_2⋯_kRZ̅^m( d^Xx^i d_Xϕ)g__2iφ__3⋯φ__k L_ L_β Lφ__L^2(Σ_s^u).

Note that the last term in (<ref>) can be estimated by the L^2 norm of 𝒜.
Indeed, by φ__p=- L__pT̃^iφ_i+O(δ^2-_0s^-3/2), Lφ_=- L_( L^+2T̃^)Tφ_+O(^1-_0s^-3/2), L_=ø_+O(^k(1-_0)s^-1ln s) and ø_i=x^iϱ(1+ϱ̌), one has

    ^lg^β,_2⋯_kRZ̅^m( d^Xx^i d_Xϕ)g__2iφ__3⋯φ__k L_ L_β Lφ__L^2(Σ_s^u)
    ≲   ^(k-1)(1-_0)-1s^-1/2^lg^β, i'_3⋯_kRZ̅^m( d^Xx^i d_Xϕ)m_ii'ø__3⋯ø__kø_ø_βø__L^2(Σ_s^u)
       +(^(2k-1)(1-_0)-1ln s+^(k-1)(1-_0))s^-3/2^lRZ̅^m( d^Xx d_Xϕ)_L^2(Σ_s^u)
    ≲   ^(k-1)(1-_0)-1s^-1/2^lRZ̅^m𝒜_L^2(Σ_s^u)+δ^k(1-_0)s^-3^l_-1Z̅^m_-1x_L^2(Σ_s^u)
       +(^(2k-1)(1-_0)-1ln s+^(k-1)(1-_0))s^-3/2^l dRZ̅^mϕ_L^2(Σ_s^u)
       +δ^k(1-_0)s^-2δ^l_0Z̅^m_0ϱ̌_L^2(Σ_s^u)+^(k-1)(1-_0)-1s^-3/2δ^l_0Z̅^m_0ϕ_L^2(Σ_s^u).

Substituting (<ref>) into (<ref>) and using Proposition <ref> and Corollary <ref> lead to

    ^l   𝒞^m_L^2(Σ_s^u)≲δ^k(1-_0)-1/2(+^k(1-_0)ln s)s^-5/2+δ^(k-1)(1-_0)s^-3/2-ι√(Ẽ_1,≤ m+2)
       +{^(2k-1)(1-_0)(1+^k(1-_0)-1)ln^2 s+δ^k(1-_0)+_0}s^-5/2√(Ẽ_2,≤ m+2)
       +^2k(1-_0)(δ+^k(1-_0))s^-5/2∫_t_0^sτ^-3/2R^mμ_τ,udτ
       +^k(1-_0)(+^k(1-_0)ln s)s^-2R^mμ_s,u+^(2k-1)(1-_0)-1s^-2√(∫_0^u F̃_1(s,u')du').


(3) The estimate of dZ̅^m(|χ̌|^2)_L^2(Σ_s^u)

Due to |χ̌|^2=(trχ̌)^2, then dZ̅^m(|χ̌|^2)
=2trχ̌(F̂^m-12𝒞^m+ dZ̅^m E)+2∑_p≤ m-1(Z̅^m-ptrχ̌)( dZ̅^ptrχ̌). This, together with Proposition <ref>, (<ref>) and (<ref>), yields

    δ^l dZ^m(|χ̌|^2)_s,u≲   δ^k(1-ε_0)s^-2ln s(δ^lF̂^m_s,u+^l𝒞^m_s,u
    +δ^l dZ̅^m E_s,u)
       +δ^k(1-ε_0)s^-3ln s{δ^l_1ℒ_Z^m_1χ̌_s,u
    +δ^k(1-ε_0)s^-2ln sδ^l_0ℒ_Z^m_0 g_s,u}
    ≲   δ^k(1-ε_0)s^-2ln sδ^lF̂^m_s,u+δ^2k(1-ε_0)
    +1/2-_0s^-4ln s
       +δ^(2k-1)(1-ε_0)s^-7/2-ιln s√(Ẽ_1,≤ m+2)
    +δ^2k(1-ε_0)s^-4ln s√(Ẽ_2,≤ m+2)
       +^2k(1-_0)(δ+^k(1-_0)ln s)s^-4ln sR^mμ_s,u
       +^3k(1-_0)-_0(δ+^k(1-_0))s^-4ln s∫_t_0^sτ^-3/2R^mμ_τ,udτ
       +^(2k-1)(1-_0)(1+^k(1-_0)-1)s^-4ln s√(∫_0^u F̃_1(s,u')du')


(4) The estimate of ê^m_L^2(Σ_s^u)


	
  * 

One starts with ℒ_Z̅^p_1ℒ_[ L,Z̅]F^p_2
in (<ref>) for p_1+p_2=m-1. Due to
[ L, Z̅]=^(Z̅)π_ L^XX, then
ℒ_[ L,Z̅]F^p_2=^(Z̅)π_ L^X∇_XF^p_2+F^p_2_X∇^(Z̅)π_ L^X.
This implies that

    δ^lℒ_Z̅^m_1ℒ_[ L, Z̅]F^m_2_s,u
    ≲   δ^k(1-ε_0) s^-2ln s{δ^lF̂^m_s,u+^l𝒞^m_s,u
    +s^-1δ^l_1ℒ_ Z^m_1χ̌_s,u+δ^l_2 dZ^m_2E_s,u}
       +δ^k(1-ε_0)-ε_0s^-7/2δ^l_1{ℒ_ Z^m_1^(R)π_ L_s,u+δℒ_ Z^m_1^(T)π_ L_s,u+δ^k(1-ε_0)s^-1ln sℒ_Z^m_1 g_s,u}
    ≲   δ^2k(1-ε_0)-ε_0+1/2s^-4ln s+δ^2k(1-ε_0)(^_0-1ln s+^-_0)s^-7/2-ιln s√(Ẽ_1,≤ m+2)
       +δ^2k(1-ε_0)(1+^k(1-_0)-1)s^-4ln^2 s√(Ẽ_2,≤ m+2)
       +^3k(1-_0)(δ+^k(1-_0))s^-4ln s∫_t_0^sτ^-32R^mμ_τ,udτ+δ^k(1-ε_0) s^-2ln sδ^lF̂^m_s,u
       +^2k(1-_0)(δ+^k(1-_0)ln s)s^-4ln sR^mμ_s,u+^2k(1-_0)-1s^-7/2√(∫_0^u F̃_1(s,u')du').



  * For the term Z̅^p_1(-2ϱ+ℰ) dZ̅^p_2trχ
in (<ref>) with p_1+p_2=m and p_1≥ 1, by making use of (<ref>), (<ref>), Corollary <ref>, Propositions <ref> and <ref>, one gets

    δ^lZ̅^p_1(-2ϱ+ℰ) dZ̅^p_2trχ_s,u
    ≲   s^-5/2δ^l_1ℒ_Z̅^m_1χ̌_s,u+δ^2k(1-ε_0)-_0s^-9/2ln s{δ^l_1Z̅^k_1μ_s,u+s^-1δ^l_0Z̅^m_0x_s,u}
       +δ^2k(1-ε_0)-1s^-4ln s{∑_i=1^2δ^l_1Z̅^m_1Ľ^i_s,u+δ^l_1Z̅^m_1ϱ̌_s,u+δ^l_0Z̅^m_0φ_s,u}
       +δ^(2k-1)(1-ε_0)-1s^-9/2ln s^l_0Z̅^m_0ϕ_s,u+δ^k(1-ε_0)s^-9/2ln sδ^l_1ℒ_Z̅^m_1 g_s,u
    ≲   δ^k(1-ε_0)s^-7/2-ιln s√(Ẽ_1,≤ m+2)+δ^k(1-ε_0)+_0(1+^k(1-_0)-1)s^-4ln^2 s√(Ẽ_2,≤ m+2)
       +δ^k(1-ε_0)+1/2s^-4ln s+^(k-1)(1-_0)s^-7/2√(∫_0^u F̃_1(s,u')du')
       +δ^3k(1-_0)-_0(δ+^k(1-_0))s^-4ln s∫_t_0^sτ^-3/2R^mμ_τ,udτ.



  * Analogously to the treatment on 𝒞^m,
the second term in the right side hand of (<ref>) can be estimated as follows:

    ^lg^β,_2⋯_k dZ̅^m(δ__2^0 Lϕ+g__2i d^Xx^i d_Xϕ) L(μ^-1φ__3⋯φ__k L_ L_β Lφ_)_s,u
    ≲   ^(k-1)(1-_0)-1s^-5/2^lRZ̅^m Lϕ_s,u
       +s^-1^lg^β,_2⋯_kRZ̅^m(g__2i d^Xx^i d_Xϕ)( Lφ__3)φ__4⋯φ__k L_ L_β Lφ__s,u
       +s^-1^lg^β,_2⋯_kRZ̅^m(g__2i d^Xx^i d_Xϕ)φ__3⋯φ__k L_ L_β( L Lφ_)_s,u
       +s^-1^lg^β,_2⋯_kRZ̅^m(g__2i d^Xx^i d_Xϕ)φ__3⋯φ__k L(μ^-1 L_ L_β) Lφ__s,u,

which, together with

    Lφ__3=   - L__3T̃^i_3 Lφ_i_3+O(^1-_0(+^k(1-_0))s^-5/2),
     L Lφ_=   μ^-1( Lμ) Lφ_
    - L_(μ^-1 L^ T Lφ_)+μ^-1 L_(T L^) Lφ_
       +O(^-_0(+^k(1-_0))s^-5/2)

and an analogous analysis for (<ref>), yields

    ^lg^β,_2⋯_k dZ̅^m(δ__2^0 Lϕ+g__2i d^Xx^i d_Xϕ) L(μ^-1φ__3⋯φ__k L_ L_β Lφ_)_s,u
    ≲   ^(k-1)(1-_0)-1s^-5/2^lRZ̅^m Lϕ_s,u+δ^(k-1)(1-_0)-1(+^k(1-_0)-_0)s^-4^l_0Z̅^m_0ϕ_s,u
       +^1-_0s^-5/2^lg^β,_2⋯_kRZ̅^m(g__2i d^Xx^i d_Xϕ) L__3φ__4⋯φ__k L_ L_β Lφ__s,u
       +^k(1-_0)-_0s^-5/2^lg^β,_2⋯_kRZ̅^m(g__2i d^Xx^i d_Xϕ)φ__3⋯φ__k L_ L_β Lφ__s,u
       +^-_0s^-5/2^lg^β,_2⋯_kRZ̅^m(g__2i d^Xx^i d_Xϕ)φ__3⋯φ__k L_ L_β L__s,u
       +δ^(k-1)(1-_0)-1(+^k(1-_0))s^-72^l dZ̅^m+1ϕ_s,u
       +δ^k(1-_0)(+^k(1-_0)-_0)s^-11/2^l_-1Z̅^m_-1x_s,u
    ≲   δ^k(1-_0)-1/2(+^k(1-_0)ln s)s^-72+δ^(k-1)(1-_0)s^-5/2-ι√(Ẽ_1,≤ m+2)
       +δ^(k-1)(1-_0){δ^k(1-_0)(1+^k(1-_0)-1)ln^2 s+} s^-7/2√(Ẽ_2,≤ m+2)
       +^(2k-1)(1-_0)-1s^-3√(∫_0^u F̃_1(s,u')du')+^k(1-_0)(+^k(1-_0)ln s)s^-3R^mμ_s,u
       +^2k(1-_0)(δ+^k(1-_0))s^-7/2∫_t_0^sτ^-3/2R^mμ_τ,udτ.

Similarly, one can derive that

    ^lg^β,_2⋯_k([ℒ_Z̅^m,ℒ_ L] d(δ__2^0 Lϕ+g__2i d^Xx^i d_Xϕ))μ^-1φ__3⋯φ__k L_ L_β Lφ__s,u
    ≲   ^k(1-_0)s^-3ln s^lg^β,_2⋯_kRZ̅^m(g__2i d^Xx^i d_Xϕ)φ__3⋯φ__k L_ L_β Lφ__s,u
       +^k(1-_0)s^-4δ^l_1ℒ_Z̅^m_1^(R)π_ L_s,u+^k(1-_0)+1s^-4δ^l_1ℒ_Z̅^m_1^(T)π_ L_s,u
       +^(2k-1)(1-_0)-1s^-7/2ln s^l_0Z̅^m_0 Lϕ_s,u+^(2k-1)(1-_0)-1s^-9/2ln s^l_0Z̅^m_0ϕ_s,u
       +^2k(1-_0)s^-6ln s^l_0Z̅^m_0x_s,u+^2k(1-_0)s^-5ln s^l_0ℒ_Z̅^m_0 g_s,u
    ≲   ^2k(1-_0)-1/2(+^k(1-_0)ln s)s^-9/2ln s+^(2k-1)(1-_0)s^-7/2-ιln s√(Ẽ_1,≤ m+2)
       +δ^(2k-1)(1-_0){δ^k(1-_0)(1+^k(1-_0)-1)ln^2 s+} s^-9/2ln s√(Ẽ_2,≤ m+2)
       +^2k(1-_0)(+^k(1-_0)ln s)s^-4ln sR^mμ_s,u
       +^3k(1-_0)(+^k(1-_0))s^-9/2ln s∫_t_0^sτ^-3/2R^mμ_τ,udτ
       +^(2k-1)(1-_0)(1+^k(1-_0)-1ln s)s^-4√(∫_0^u F̃_1(s,u')du').

To estimate the last two terms in (<ref>), one can get by direct computations that

    ^lℒ_Z̅^m{ L(δ__2^0 Lϕ+g__2i d^Xx^i d_Xϕ) d(μ^-1φ__3⋯φ__k L_ L_β Lφ_)}
       +∑_[ p_1+p_2=m;    p_1≥ 1 ]ℒ_Z̅^p_1 d L(δ__2^0 Lϕ+g__2i d^Xx^i d_Xϕ)Z̅^p_2(μ^-1φ__3⋯φ__k L_ L_β Lφ_)_s,u
    ≲   ^(k-1)(1-_0)-1s^-3/2(^l_1Z̅^m_1 L^2ϕ_s,u+s^-1^l_0Z̅^m_0 Lϕ_s,u+s^-2^l_0Z̅^m_0ϕ_s,u)
       +^k(1-_0)s^-5(^l_-1Z^m_-1x_s,u+s^l_0Z̅^m_0μ_s,u+s^l_0Z̅^m_0Ľ^1_s,u+s^l_0Z̅^m_0Ľ^2_s,u)
       +^(k-1)(1-_0)s^-7/2^l_0Z̅^m_0φ_s,u+^k(1-_0)+_0s^-7/2^l_0(TZ̅^m_0φ_s,u+ dZ̅^m_0φ_s,u)
       +^k(1-_0)+1s^-5^l_1ℒ_Z̅^m_1^(R)π_T_s,u+^2k(1-_0)(+^k(1-_0))s^-6ln s^l_0ℒ_Z̅^m_0 g_s,u
    ≲   δ^k(1-_0)+1/2s^-7/2+δ^(k-1)(1-_0)s^-5/2-ι√(Ẽ_1,≤ m+2)+δ^k(1-_0)+_0s^-7/2√(Ẽ_2,≤ m+2)
       +^k(1-_0)(+^k(1-_0)ln s)s^-3R^mμ_s,u
       +^2k(1-_0)(δ+^k(1-_0))s^-7/2∫_t_0^sτ^-3/2R^mμ_τ,udτ
       +^(2k-1)(1-_0)-1( s^-1/2+^k(1-_0))s^-7/2√(∫_0^u F̃_1(s,u')du'),

where one has used Proposition <ref> and Corollary <ref> in the last inequality.

It follows from (<ref>)-(<ref>) that

    ^l𝒟^m_s,u
    ≲   δ^k(1-_0)-1/2(+^k(1-_0)ln s)s^-7/2+δ^(k-1)(1-_0)s^-5/2-ι√(Ẽ_1,≤ m+2)
       +δ^(k-1)(1-_0){δ^k(1-_0)(1+^k(1-_0)-1)ln^2 s+} s^-7/2√(Ẽ_2,≤ m+2)
       +^(2k-1)(1-_0)-1s^-3√(∫_0^u F̃_1(s,u')du')+^k(1-_0)(+^k(1-_0)ln s)s^-3R^mμ_s,u
       +^2k(1-_0)(δ+^k(1-_0))s^-7/2∫_t_0^sτ^-3/2R^mμ_τ,udτ.



  * Note that (<ref>) implies |G_ L L^ L_|≲^k(1-_0)s^-1.
Then it follows from the definitions of e and ℬ in (<ref>) and (<ref>) respectively that

    ^lℒ_Z̅^m de_s,u+^l dZ̅^m(μ^-1ℬ)_s,u
    ≲   ^k(1-_0)-_0s^-7/2^l_0Z̅^m_0μ_s,u+^k(1-_0)s^-3^lTZ̅^m+1φ_s,u
       +^k(1-_0)-1s^-2^l LZ̅^m+1φ_s,u+^k(1-_0)-1s^-3^l_0 Z^m_0φ_s,u
       +^(k-1)(1-_0)s^-5/2^l dZ̅^m+1φ_s,u+^(k+1)(1-_0)s^-9/2^l_1ℒ_Z̅^m_1^(R)π_T_s,u
       +^k(1-_0)-_0s^-7/2^l_1ℒ_Z̅^m_1^(R)π_ L_s,u+^(k+1)(1-_0)s^-7/2^l_1ℒ_Z̅^m_1^(T)π_ L_s,u
       +^k(1-_0)-1s^-3^l_0Z̅^m_0ϱ̌_s,u+^k(1-_0)-1s^-3^l_0(Z̅^m_0Ľ^1_s,u+Z̅^m_0Ľ^2_s,u)
       +^k(1-_0)-_0s^-9/2^l_-1Z̅^m_-1x_s,u+^2k(1-_0)-2s^-3^lZ̅^m+1𝒜_s,u
       +^(k-1)(1-_0)-1s^-5/2^lZ̅^m+1 Lϕ_s,u+^(k-1)(1-_0)-1s^-7/2^lZ^m+1ϕ_s,u
       +^(k-1)(1-_0)-1(+^k(1-_0)ln s)s^-7/2^lℒ_Z̅^m+1 dϕ_s,u+^k(1-_0)s^-4^l_0ℒ_Z̅^m_0 g_s,u
    ≲   δ^k(1-_0)+1/2-_0s^-3+δ^(k-1)(1-_0)s^-5/2-ι√(Ẽ_1,≤ m+2)+δ^k(1-_0)s^-3√(Ẽ_2,≤ m+2)
       +^k(1-_0)-1s^-2^l LZ̅^m+1φ_s,u+δ^k(1-_0)(+^k(1-_0)ln s)s^-3R^mμ_s,u
       +^2k(1-_0)-_0(δ+^k(1-_0))s^-3∫_t_0^sτ^-3/2R^mμ_τ,udτ
       +^(2k-1)(1-_0)-1s^-3√(∫_0^u F̃_1(s,u')du'),

where one has used (<ref>) and (<ref>) to estimate G_ L L^ Tφ_.


  * It follows from direct computations that

    ^l dZ̅^m{𝒜T̃^i( Lφ_i)T̃^i_4⋯T̃^i_kφ_i_4⋯φ_i_k(μ^-2 L^ν Tφ_ν)}_s,u
    ≲   ^(k-1)(1-_0)-1s^-5/2^lZ̅^m+1𝒜_s,u
    +^(k-1)(1-_0)-1s^-7/2^l_0Z̅^m_0ϕ_s,u
       +^k(1-_0)s^-5(^l_-1Z̅^m_-1x_s,u+s^l_0Z̅^m_0Ľ^1_s,u+s^l_0Z̅^m_0Ľ^2_s,u+s^l_0Z̅^m_0ϱ̌_s,u)
       +^k(1-_0)+1s^-5^l_1ℒ_Z̅^m_1^(R)π_T_s,u+^k(1-_0)s^-4^l_0Z̅^m_0μ_s,u
       +^(k-1)(1-_0)s^-5/2-ι√(Ẽ_1,≤ m+2)+^k(1-_0)+_0s^-7/2√(Ẽ_2,≤ m+2)
    ≲   δ^k(1-_0)-1/2(+^k(1-_0)ln s)s^-7/2+δ^(k-1)(1-_0)s^-5/2-ι√(Ẽ_1,≤ m+2)
       +δ^(k-1)(1-_0){δ^k(1-_0)(1+^k(1-_0)-1)ln^2 s+} s^-7/2√(Ẽ_2,≤ m+2)
       +^(2k-1)(1-_0)-1s^-3√(∫_0^u F̃_1(s,u')du')+(+^k(1-_0))δ^k(1-_0)s^-3R^mμ_s,u
       +(+^k(1-_0))^2k(1-_0)s^-7/2∫_t_0^sτ^-3/2R^mμ_τ,udτ.



  * It remains to estimate ℒ_Z̅^m(trχ dℰ) in ℒ_Z̅^m e^0
of (<ref>).
Since ℰ has an analogous expression as E in (<ref>), it follows from (<ref>) that

    δ^lℒ_Z̅^m (trχ dℰ)_s,u
    ≲   ^k(1-_0)-_0s^-5/2^l_0ℒ_Z̅^m_1χ̌_s,u+s^-1^l_1 dZ̅^m_1ℰ_s,u+^2k(1-_0)-_0s^-9/2ln s^l_1ℒ_Z̅^m_1 g_s,u
    ≲   δ^k(1-_0)-_0+1/2s^-3+δ^(k-1)(1-_0)s^-5/2-ι√(Ẽ_1,≤ m+2)+δ^k(1-_0)s^-3√(Ẽ_2,≤ m+2)
       +^k(1-_0)(δ+^k(1-_0))s^-3{R^mμ_s,u+δ^k(1-_0)
    -_0∫_t_0^sτ^-3/2R^mμ_τ,udτ}
       +^(2k-1)(1-_0)-1s^-3√(∫_0^u F̃_1(s,u')du').



Combining the estimates (<ref>)-(<ref>) with (<ref>) yields

    δ^lê^m_s,u
    ≲   δ^k(1-_0)+1/2-_0s^-3+δ^(k-1)(1-_0)s^-5/2-ι√(Ẽ_1,≤ m+2)
    +δ^k(1-_0)s^-3√(Ẽ_2,≤ m+2)
       +^k(1-_0)-1s^-2^l LZ̅^m+1φ_s,u+δ^k(1-_0)(+^k(1-_0)ln s)s^-3R^mμ_s,u
       +^2k(1-_0)-_0(δ+^k(1-_0))s^-3∫_t_0^sτ^-3/2R^mμ_τ,udτ
    +δ^k(1-ε_0) s^-2ln sδ^lF̂^m_s,u
       +^(k-1)(1-_0)(s^-1/2+^k(1-_0)-1)s^-3√(∫_0^u F̃_1(s,u')du').




Inserting (<ref>), (<ref>), (<ref>) and (<ref>) into (<ref>) gives

    δ^l   F̂^m_s,u≲δ^k(1-ε_0)+1/2-_0s^-2
    +δ^(k-1)(1-ε_0)s^-3/2-ι√(Ẽ_1,≤ m+2)
       +δ^k(1-ε_0) s^-2√(Ẽ_2,≤ m+2)+^k(1-_0)s^-2∫_t_0^s(+^k(1-_0)lnτ)τ^-1/2R^mμ_τ,udτ
       +^k(1-_0)-1s^-3/2-ι√(∫_0^uF̃_1,m+2(s,u')du').

In addition, dZ̅^mtrχ=F̂^m+ d Z̅^m E-12𝒞^m
holds due to the definition of F̂^m. Hence, one has by inequalities (<ref>), (<ref>) and (<ref>) that

    δ^l dZ̅^mtrχ_s,u
    +δ^l∇ℒ_Z̅^mχ̌_s,u
    ≲   δ^k(1-ε_0)+1/2-_0s^-2
    +δ^(k-1)(1-ε_0)s^-3/2-ι√(Ẽ_1,≤ m+2)
    +δ^k(1-ε_0) s^-2√(Ẽ_2,≤ m+2)
       +^k(1-_0)-1s^-3/2-ι√(∫_0^uF̃_1,m+2(s,u')du')+^k(1-_0)(+^k(1-_0)ln s)s^-2R^mμ_(s,u)
       +^k(1-_0)s^-2∫_t_0^s(+^k(1-_0)lnτ)τ^-1/2R^mμ_τ,udτ.



In summary, for any Z∈{ϱ L, T, R}, it follows from  (<ref>), (<ref>) and (<ref>) that

    δ^l dZ^mtrχ_s,u
    +δ^l∇ℒ_Z^mχ̌_s,u
    ≲   δ^k(1-ε_0)+1/2-_0s^-2
    +δ^(k-1)(1-ε_0)s^-3/2-ι√(Ẽ_1,≤ m+2)+δ^k(1-ε_0) s^-2√(Ẽ_2,≤ m+2)
       +^k(1-_0)-1s^-3/2-ι√(∫_0^uF̃_1,m+2(s,u')du')+^k(1-_0)(+^k(1-_0)ln s)s^-2R^mμ_(s,u)
       +^k(1-_0)s^-2∫_t_0^s(+^k(1-_0)lnτ)τ^-1/2R^mμ_τ,udτ.




	Here we emphasize that due to dZ^mtrχ̌= dZ^mtrχ
by (<ref>), (<ref>) gives also the L^2 estimate of dZ^mtrχ̌.






 §.§ Estimates on the derivatives of μ


Although the main idea in this subsection parallels that in <cit.>, in order to
close the energy estimate at the beginning of Section <ref> as ε_0 approaches ε_k^*,
we need to obtain more refined estimates for each term in the transport equation under
the general higher null condition (for instance, see (<ref>) below).




Similarly to the analysis in <cit.>, one can set

    Ẽ=   -12μ (G_ L L^γ+2G_T̃ L^γ)φ_γ
    +12μ^-1G_ L L^γ T Lφ_γ,
    ℰ̃=   -12G^γ_ L L Lφ_γ-G^γ_T̃ L Lφ_γ,
    F̃=   μ-Ẽ.

Then it holds that

    LF̃=   -2trχ̌F̃+(-2ϱ+ℰ̃)F̃
    +( d^Xtrχ)𝒬_X+ẽ

with

    𝒬_X=   - d_Xμ-μ G^γ_XT̃ Lφ_γ
    -μ G^γ_T̃ L d_Xφ_γ
    -μ G^γ_ L L d_Xφ_γ+ G^γ_X LTφ_γ

and

    ẽ=   2( d^Xμ) d_Xℰ̃+(-2trχ̌-2ϱ+ℰ̃)Ẽ-12 L(μ^-1G_ L L^γ)T Lφ_γ
       -12μ^-1G_ L L^γ{Tμ(φ_γ)+^(T)π_ L^X d_X Lφ_γ-12ϱ^2 Lφ_γ-12ϱT Lφ_γ+T H_γ}
       +f_1(φ,  L^1,  L^2, dx,ϱ x)(
    [  μ Lφ;  μ dφ;    Tφ; μφtrχ ])
    ([      φ^k-2φ; φ^k-1ϱ^-1 x; φ^k-1trχ̌ x ])
       +f_2(φ, L^1, L^2,ϱ d L^1,ϱ d L^2,ϱ x)φ^k-3(
    [ ϱ^-1φ^k+1;        dφ;   φ d L^1;   φ d L^2 ])
    (
    [      dφ;      Lφ; φ d L^1; φ d L^2 ])
    (
    [ μ Lφ; μ dφ;   Tφ ])
       +f_3(φ, L^1, L^2, dx) d^Xμ d_Xφ(
    [ φ^2(k-1) dφ; φ^2(k-1) Lφ;    φ^k-1trχ ])
    +f_4(φ, L^1, L^2, dx)(
    [    dTφ; μ d Lφ ])
    (
    [    φ^k-2 dφ; φ^k-1 d L^1; φ^k-1 d L^2 ]).


Analogously to the introduction of F^k in Subsection <ref>, one can set F̃^m=Z̅^mμ-Z̅^mẼ
with Z̅ being any vector field in {T,R}.  Then an induction argument yields that for m≥ 1,

    LF̃^m=   (-2trχ̌-2ϱ+ℰ̃)F̃^m+( d^XZ̅^ktrχ)𝒬_X+^(Z̅)π_ L^X( d_XF̃^m-1)+ẽ^m,

where

    ẽ^m=   ∑_(ℒ_Z̅^m_1^(Z̅)π_ L^X)( d_XF̃^m_2)_vanish when m=1+Z̅^m e
       +∑_{Z̅^m_1(trχ̌-2ϱ+ℰ̃)F̃^m_2+( d^XZ̅^m_2trχ)ℒ_Z̅^m_1𝒬_X}.

As (<ref>), it follows from (<ref>) and T^2φ_=T(μ_^0T̃^i Lφ_i
- L_T̃^iTφ_i+μT̃^ig_ j d^Xx^j d_Xφ_i) that

    G_ L L^ T^2φ_=f(φ,  L^1, L^2,xϱ,ϱ̌)(
    	[                          φ^kT^2φ;               φ^k-1T(μT̃^a Lφ_a);              φ^k-1T( L_T̃^a)Tφ_a; φ^k-1T(μT̃^a g_ j d^Xx^j d_Xφ_a);                     Ľ^1φ^k-1T^2φ;                     Ľ^2φ^k-1T^2φ;                      ϱ̌φ^k-1T^2φ;                   φ^k-2( Lϕ)T^2φ;          φ^k-2( d^Xx)( d_Xϕ)T^2φ ]).

Thus, ^l|F̃^m(t_0,u,ϑ)|≲δ^k(1-ε_0)-1-_0 holds by (<ref>).
In addition, by virtue of Propositions <ref>, <ref> and Corollary <ref>, one can apply (<ref>) again to obtain

    ^lZ̅^m(μ^-1G_ L L^ T^2φ_)_s,u
    ≲   ^k(1-_0)-1s^-1^lTZ^m+1φ_s,u+^k(1-_0)-_0s^-5/2^l_1ℒ_Z̅^m_1^(R)π_T_s,u
       +^k(1-_0)-2s^-1^l_0(Z̅^m_0Ľ^1_s,u+Z̅^m_0Ľ^2_s,u+Z̅^m_0ϱ̌_s,u+Z^m_0φ_s,u)
       +^(k-1)(1-_0)-2s^-3/2^l_0Z^m_0ϕ_s,u+^(k+1)(1-_0)-2s^-5/2^l_-1Z̅^m_-1x_s,u
       +^k(1-_0)-1s^-2^l_0ℒ_Z̅^m_0 g_s,u+^k(1-_0)-1-_0s^-3/2^l_0Z̅^m_0μ_s,u
    ≲   ^k(1-_0)-1/2-_0s^-1+^k(1-_0)-1s^-1{(1+^k(1-_0)-1)s^-ι√(Ẽ_1,≤ m+2)+√(Ẽ_2,≤ m+2)}
       +^2k(1-_0)-1-_0(+^k(1-_0))s^-1∫_t_0^sτ^-3/2R^mμ_τ,udτ
       +^(2k-1)(1-_0)-2s^-1√(∫_0^uF̃_1,m+2(s,u')du').

Thus, it follows from the definition of Ẽ in (<ref>) and (<ref>) that

    ^lZ̅^mẼ_s,u
    ≲   ^k(1-_0)-1/2-_0s^-1+^k(1-_0)-1s^-1{(1+^k(1-_0)-1)s^-ι√(Ẽ_1,≤ m+2)+√(Ẽ_2,≤ m+2)}
       +^2k(1-_0)-1-_0(+^k(1-_0))s^-1∫_t_0^sτ^-3/2R^mμ_τ,udτ
       +^(2k-1)(1-_0)-2s^-1√(∫_0^uF̃_1,m+2(s,u')du').

Replacing R^mμ by F̃^m+R^mẼ in (<ref>) and using Gronwall's inequality yield

    ^lZ̅^mẼ_s,u
    ≲   ^k(1-_0)-1/2-_0s^-1+^k(1-_0)-1s^-1{(1+^k(1-_0)-1)√(Ẽ_1,≤ m+2)+√(Ẽ_2,≤ m+2)}
       +^2k(1-_0)-1-_0(+^k(1-_0))s^-1∫_t_0^sτ^-3/2^lF̃^m_τ,udτ
       +^(2k-1)(1-_0)-2s^-1√(∫_0^uF̃_1,m+2(s,u')du').

Set F(s,u,ϑ)=ϱ^2F̃^m(s,u,ϑ)-ϱ_0^2F̃^m(t_0,u,ϑ) in (<ref>).
Then applying (<ref>) yields

    δ^lϱ^3/2F̃^m_s,u
    ≲   δ^k(1-ε_0)-1/2-_0+∫_t_0^sτ^3/2δ^lẽ^m_τ,udτ+δ^k(1-ε_0)-1∫_t_0^sτ^1/2δ^l dZ̅^mtrχ_τ,udτ.

The last term on the right hand side above can be estimated by (<ref>) and (<ref>)
as follows

    δ^k(1-ε_0)-1∫_t_0^sτ^1/2^l dZ̅^mtrχ_τ,udτ
    ≲   δ^2k(1-ε_0)-1/2-_0+δ^(2k-1)(1-ε_0)-1√(Ẽ_1,≤ m+2)+δ^2k(1-ε_0)-1√(Ẽ_2,≤ m+2)
    
    +   ^2k(1-_0)-1{∫_t_0^sτ^-1(+^k(1-_0)lnτ)^lF̃^m_τ,udτ+^-1√(∫_0^uF̃_1,m+2(s,u')du')}.


Next, one estimates ẽ^m. By (<ref>), one starts with Z̅^mẽ. Note that ẽ in (<ref>) contains
the following term

    L(μ^-1G_ L L^)T^2φ_
    
    =   -μ^-2( Lμ)G_ L L^ T^2φ_+2μ^-1G_iβ^( L L^i_(<ref>)) L^β(T^2φ_)+μ^-1(_φ_νG_β^)( Lφ_ν) L^ L^β(T^2φ_),

where

    (_φ_νG_β^)( Lφ_ν) L^ L^β(T^2φ_)
    
    =   -k(k-1)m_'m_ββ'g^'β',ν_3…_kφ__3⋯φ__k( Lφ_ν) L^ L^β(T^2φ_)+f(φ, L^1, L^2)φ^k-1( Lφ)T^2φ.

It follows from Lφ_ν=_ν^0 L^ Lφ_- L_νT̃^i Lφ_i+g_ν j d^Xx^j( L^ d_Xφ_), T^2φ_=T(μ_^0T̃^i Lφ_i- L_T̃^iTφ_i+μT̃^ig_ j d^Xx^j d_Xφ_i) and (<ref>) that

    (_φ_νG_β^)( Lφ_ν) L^ L^β(T^2φ_)=f(φ, Ľ^1,Ľ^2,xϱ,ϱ̌)(
    [                        φ^k-1( Lφ)T^2φ;               φ^k-2( Lφ)T(μT̃^a Lφ_a);              φ^k-2( Lφ)T( L_T̃^a)Tφ_a; φ^k-2( Lφ)T(μT̃^a g_ j d^Xx^j d_Xφ_a);                     Ľ^1φ^k-2( Lφ)T^2φ;                     Ľ^2φ^k-2( Lφ)T^2φ;                      ϱ̌φ^k-2( Lφ)T^2φ;                   φ^k-3( Lϕ)( Lφ)T^2φ;          φ^k-3( d^Xx)( d_Xϕ)( Lφ)T^2φ;                    φ^k-2( L^ Lφ_)T^2φ;             φ^k-2 d^Xx( L^ d_Xφ_)T^2φ ]).

Inserting (<ref>) into (<ref>), utilizing (<ref>) and (<ref>) to derive L^ Lφ_= L^2ϕ+f(φ, L^1, L^2)φ^k(
[        Lφ; d^Xφ d_Xx ]) and L^ d_Xφ_= d_X Lϕ-trχ d_Xϕ+f(φ, L^1, L^2)φ^k(
[ d_Xx Lφ;    d_Xφ ]), one then can get with the help of Propositions <ref>, <ref>, Corollary <ref>, (<ref>) and (<ref>) that

    ^lZ̅^m( L(μ^-1G_ L L^)T^2φ_)_s,u
    ≲   ^k(1-_0)-_0s^-3/2^l_1Z̅^m_1(μ^-1G_ L L^ T^2φ_)_s,u+^k(1-_0)-1-_0s^-5/2^l_0Z̅^m_0μ_s,u
       +^k(1-_0)-1s^-2^lTZ^m+1φ_s,u+^k(1-_0)-_0s^-7/2^l_1ℒ_Z̅^m_1^(R)π_T_s,u
       +^k(1-_0)-2s^-2^l_0(Z̅^m_0Ľ^1_s,u+Z̅^m_0Ľ^2_s,u+Z̅^m_0ϱ̌_s,u+Z^m_0φ_s,u)
       +^(k-1)(1-_0)-2s^-5/2^l_0Z^m_0ϕ_s,u+^(k+1)(1-_0)-2s^-7/2^l_-1Z̅^m_-1x_s,u
       +^(k-1)(1-_0)-2s^-1/2^l_1(Z̅^m_1 L^2ϕ_s,u+ dZ̅^m_1 Lϕ_s,u)
       +^k(1-_0)-1s^-3^l_0ℒ_Z̅^m_0 g_s,u+^k(1-_0)-1s^-2^l_1ℒ_Z̅^m_1χ̌_s,u
    ≲   ^k(1-_0)-1/2-_0s^-2+^(k-1)(1-_0)-1s^-3/2-ι√(Ẽ_1,≤ m+2)+^k(1-_0)-1s^-2√(Ẽ_2,≤ m+2)
       +^k(1-_0)-1(+^k(1-_0))s^-2^lF̃^m_s,u+^(2k-1)(1-_0)-2s^-2√(∫_0^uF̃_1,m+2(s,u')du')
       +^2k(1-_0)-1-_0(+^k(1-_0))s^-2∫_t_0^sτ^-3/2^lF̃^m_τ,udτ.

Note that (<ref>) and (<ref>) imply that Z̅^mẽ contains the term
12μ^-1(G_ L L^γ Tφ_γ)Z̅^m Ttrχ̌, which equals 12μ^-1(G_ L L^γ Tφ_γ)Z̅^mμ+⋯ by (<ref>).
Thus,
it follows from Propositions <ref>, <ref>, (<ref>), (<ref>) and (<ref>) that

    ^lZ̅^mẽ_s,u
    ≲   ^k(1-_0)-1-_0(s^-3^l_0Z̅^m_0μ_s,u+s^-3/2^l_1ℒ_Z̅^m_1χ̌_s,u)+s^-1^l_1Z̅^m_1Ẽ_s,u
       +^l_1Z̅^m_1( L(μ^-1G_ L L^)T^2φ_)_s,u+s^-1^l_1Z̅^m_1(μ^-1G_ L L^ T^2φ_)_s,u
       +^k(1-_0)-1s^-3^l_1ℒ_Z̅^m_1 g_s,u+^k(1-_0)-1s^-2^l_1ℒ_Z̅^m_1^(T)π_ L_s,u
       +^k(1-_0)-1(^k(1-_0)-1+1)s^-3(^l_0Z̅^m_0Ľ^1_s,u+^l_0Z̅^m_0Ľ^2_s,u+s^-1^l_-1Z̅^m_-1x_s,u)
       +^2k(1-_0)-_0s^-7/2ln s^l_1ℒ_Z̅^m_1^(T)π_s,u+^k(1-_0)-_0s^-3/2^l_1Z̅^m_1μ_s,u
       +^(k-1)(1-_0)-1s^-3/2-ι√(Ẽ_1,≤ m+2)+^k(1-_0)-1s^-2√(Ẽ_2,≤ m+2)
       +^k(1-_0)s^-5(^k(1-_0)-1ln s+1)^l_1ℒ_Z̅^m_1^(R)π_T_s,u
    ≲   ^k(1-_0)-1/2-_0s^-2+^(k-1)(1-_0)-1s^-3/2-ι√(Ẽ_1,≤ m+2)+^k(1-_0)-1s^-2√(Ẽ_2,≤ m+2)
       +^k(1-_0)-_0s^-3/2^lF̃^m_s,u+^2k(1-_0)-1-_0(+^k(1-_0))s^-2∫_t_0^sτ^-3/2^lF̃^m_τ,udτ
       +^(2k-1)(1-_0)-2s^-2√(∫_0^uF̃_1,m+2(s,u')du').

Therefore, it holds that

    δ^lẽ^m_s,u
    ≲   ^k(1-_0)-1/2-_0s^-2+^(k-1)(1-_0)-1s^-3/2-ι√(Ẽ_1,≤ m+2)+^k(1-_0)-1s^-2√(Ẽ_2,≤ m+2)
       +^k(1-_0)-_0s^-32^lF̃^m_s,u+^2k(1-_0)-1-_0(+^k(1-_0))s^-2∫_t_0^sτ^-3/2^lF̃^m_τ,udτ
       +^(2k-1)(1-_0)-2s^-2√(∫_0^uF̃_1,m+2(s,u')du').

Substituting (<ref>) and (<ref>)  into (<ref>) and then applying Gronwall's inequality yield

    δ^lF̃^m_s,u≲   δ^k(1-ε_0)-1/2-_0s^-1+δ^(k-1)(1-ε_0)-1s^-1/2-ι√(Ẽ_1,≤ m+2)
    
    +^k(1-_0)-1   s^-1√(Ẽ_2,≤ m+2)+^(2k-1)(1-_0)-2s^-1√(∫_0^uF̃_1,m+2(s,u')du')

and further

    δ^lZ̅^mμ_s,u≲   δ^k(1-ε_0)-1/2-_0s^-1+δ^(k-1)(1-ε_0)-1s^-1/2-ι√(Ẽ_1,≤ m+2)
    
    +^k(1-_0)-1   s^-1√(Ẽ_2,≤ m+2)+^(2k-1)(1-_0)-2s^-1√(∫_0^uF̃_1,m+2(s,u')du')

with the help of (<ref>).

For the other cases containing at least one ϱ L in Z^m, one can make use of the
commutators [ϱ L,Z̅] and [ϱ L,] and subsequently
utilize the transport equation (<ref>)
to obtain  the related L^2 norm estimates.
Therefore, we eventually arrive at

    δ^lZ^mμ_s,u≲   δ^k(1-ε_0)-1/2-_0s^-1
    +δ^(k-1)(1-ε_0)-1s^-1/2-ι√(Ẽ_1,≤ m+2)
       +^k(1-_0)-1s^-1√(Ẽ_2,≤ m+2)+^(2k-1)(1-_0)-2s^-1√(∫_0^uF̃_1,m+2(s,u')du').

And hence, it follows from (<ref>) and (<ref>) that

    δ^l dZ^mtrχ_s,u
    +δ^l∇ℒ_Z^mχ̌_s,u
    ≲   δ^k(1-ε_0)+1/2-_0s^-2
    +δ^(k-1)(1-ε_0)s^-3/2-ι√(Ẽ_1,≤ m+2)+δ^k(1-ε_0) s^-2√(Ẽ_2,≤ m+2)
       +^k(1-_0)-1s^-3/2-ι√(∫_0^uF̃_1,m+2(s,u')du').






§ ESTIMATES FOR THE ERROR TERMS

After all the preparations for the optimal L^2 estimates on the related quantities in Sections <ref> and <ref>, we are ready to
handle the error terms δ∫_D^s, u|Φ·LΨ| and
∫_D^s, uϱ^2ι|Φ· LΨ| in (<ref>), and then complete the final energy estimates for .
Although the main strategy is analogous to that in <cit.>, due to the slow time decay of the
solutions to the 2D wave equations
and the requirement on the optimal smallness exponent of short pulse data, we still give  all the details since many precise estimates
in Sections <ref>-<ref> derived from the higher order null condition (<ref>) will be applied (for instance,
one can see (<ref>), (<ref>) and Remark <ref> below for details).


For Ψ=Z^m+1_ in (<ref>), then the corresponding Φ is just Φ_^m+1 which has been explicitly
given in (<ref>).
One can now deal with each term in (<ref>) as follows.



 §.§ Treatment on J_1^m+1


This subsection deals with the term J_1^m+1 in (<ref>).
First, expand Z^m+1 as Z_m+1Z_m⋯ Z_2Z_1 with Z_i∈{ϱ L,T,R} and set φ_γ^n=Z_n⋯ Z_1φ_γ,    n≥ 1,
φ_γ,    n=0.
 
  
  * Due to form of Φ_γ^m+1, it is necessary to estimate the derivatives of μ𝒟^^(Z)C_γ^n_, (0≤ n≤ m). To this end, one treats first ^(Z)D_γ,1^n and ^(Z)D_γ,3^n in (<ref>), which do not contain the top order derivatives of φ_γ.
      In fact, substituting (<ref>)-(<ref>) in Appendix A into (<ref>) and (<ref>) yields directly

    ^(T)D_γ,1^n=   (Tμ) L^2φ_γ^n+μ( d_Xμ+2μζ_X) d^X Lφ_γ^n+12tr^(T)π( LLφ_γ^n+12trχLφ_γ^n)
       +( d_Xμ+2μζ_X) d^XLφ_γ^n-(Tμ)φ_γ^n
    +12μtr^(T)πφ_γ^n,
    ^(T)D_γ,3^n=   {trχ Tμ+14(μtrχ+tr^(T)π)tr^(T)π-12| dμ|^2-μζ_X( d^Xμ)} Lφ_γ^n
       +(12 Lμ-μtrχ)( d_Xμ+2μζ_X) d^Xφ_γ^n,
    ^(ϱ L)D_γ,1^n=   (2-μ+ϱ Lμ) L^2φ_γ^n-2ϱ( d_Xμ+2μζ_X) d^X Lφ_γ^n+ϱtrχ( LLφ_γ^n+12trχLφ_γ^n)
       -(ϱ Lμ-ϱμtrχ̌)φ_γ^n,
    ^(ϱ L)D_γ,3^n=   trχ{2-μ+ϱ Lμ+12ϱtr^(T)π+12ϱμtrχ} Lφ_γ^n+2ϱtrχ(2μζ^X+ d^Xμ) d_Xφ_γ^n,
    ^(R)D_γ,1^n=   (Rμ) L^2φ_γ^n-^(R)π_LX d^X Lφ_γ^n+12tr^(R)π( LLφ_γ^n+12trχLφ_γ^n)-^(R)π_ LX d^XLφ_γ^n
       -(Rμ)φ_γ^n
    +12μtr^(R)πφ_γ^n,
    ^(R)D_γ,3^n=   {trχ Rμ+14(tr^(T)π+μtrχ)tr^(R)π+12 d^Xμ^(R)π_ LX} Lφ_γ^n
    +{trχ^(R)π_TX
       +tr^(R)π(μζ_X+12 d_Xμ)+(-12 Lμ+μtrχ+12tr^(T)π)^(R)π_ LX} d^Xφ_γ^n.


Note that it follows from (<ref>), (<ref>), and (<ref>) that each of ^(Z)D_γ,1^n contains the factor LLφ_γ^n+12trχLφ_γ^n which can be estimated by using (<ref>) as explained after (<ref>).
It follows easily from the expression of J_1^m+1 that at most
(m-n)- order derivatives appear in ^(Z)D_γ,i^n (i=1,3).
 Thus, the L^2 norms of all terms in J_1^m+1 can be estimated by
the corresponding L^∞ estimates in Subsection <ref>
and the related L^2 estimates in Proposition <ref>, which do not depend on the estimates of top order derivatives
in Section <ref>.
Therefore,


    δ^2l+1|∫_D^s, u∑_p=1^m(Z_m+1+^(Z_m+1)λ)…(Z_m+2-p
    +^(Z_m+2-p)λ)^(Z_m+1-p)D_γ,1^m-p·Lφ_γ^m+1|
    ≲   δ^2l+1∫_t_0^s∑_p=1^m(Z_m+1+^(Z_m+1)λ)…(Z_m+2-p
    +^(Z_m+2-p)λ)^(Z_m+1-p)D_γ,1^m-p_τ,u
               ·LZ^m+1φ_γ_τ,udτ
    ≲   δ^6-6ε_0+∫_t_0^sτ^-1-ιẼ_1,≤ m+2(τ,u)dτ
    +δ∫_t_0^sτ^-1-ιẼ_2,≤ m+2(τ,u)dτ+^-1∫_0^uF̃_1,m+2(s,u')du'.

Similarly,

    δ^2l+1|∫_D^s, u∑_p=1^m(Z_m+1+^(Z_m+1)λ)…(Z_m+2-p
    +^(Z_m+2-p)λ)^(Z_m+1-p)D_γ,3^m-p·LZ^m+1φ_γ|
    ≲   δ^4-2ε_0+^2(k+1)(1-_0 )+∫_t_0^sτ^-2Ẽ_1,≤ m+2(τ,u)dτ+δ∫_t_0^sτ^-2lnτẼ_2,≤ m+2(τ,u)dτ
       +^-1∫_0^uF̃_1,m+2(s,u')du'.


The terms in δ^2l∫_D^s, uϱ^2ι|Φ· LΨ|
related to the integrand factors ^(Z)D_γ,i^n (i=1,3)  can also be estimated as follows

    δ^2l∫_D^s, u|∑_p=1^mϱ^2ι(Z_m+1+^(Z_m+1)λ)…(Z_m+2-p
    +^(Z_m+2-p)λ)(^(Z_m+1-p)D_γ,1^m-p
                   +^(Z_m+1-p)D_γ,3^m-p) Lφ_γ^m+1|
    ≲   δ^2l+1∫_D^s, uϱ^2ι{∑_p=1^m(Z_m+1+^(Z_m+1)λ)…(Z_m+2-p
    +^(Z_m+2-p)λ)(^(Z_m+1-p)D_γ,1^m-p
           +^(Z_m+1-p)D_γ,3^m-p)}^2+δ^2l-1∫_D^s, uϱ^2ι| LZ^m+1φ_γ|^2
    ≲   δ^6-6ε_0+^4-2_0+∫_t_0^sτ^-2Ẽ_1,≤ m+2(τ,u)dτ+δ∫_t_0^sτ^-4+2ιln^4τẼ_2,≤ m+2(τ,u)dτ
       +δ^-1∫_0^uF̃_1,m+2(s,u')du'.



  * We now estimate the terms involving ^(Z)D_γ,2^n (0≤ n≤ m) in J_1^m+1. Note that in the special case n=0, j equals m in J_1^m+1, and the order of the top derivatives in ^(Z)D_γ,2^0 is m, so that ^(Z)D_γ,2^0 contains terms involving the (m+1)^th order derivatives of the deformation tensor.
This prevents one from using Proposition <ref> to estimate the L^2 norm of ^(Z)D_γ,2^0
directly since the L^2 norm of the (m+1)^th order derivatives of the deformation tensor can be controlled only by Ẽ_1,≤ m+3(s,u) and Ẽ_2,≤ m+3(s,u) in the energy estimates
(note that Ẽ_1,≤ m+3(s,u) and Ẽ_2,≤ m+3(s,u) cannot be absorbed directly by the energies
Ẽ_1,≤ m+2(s,u) and Ẽ_2,≤ m+2(s,u) on the left hand side of the resulting energy inequality by (<ref>)).
Thus, we will
examine the expression of ^(Z)D_γ,2^n carefully and apply the estimates in Section <ref>
to handle the top order derivatives of trχ and μ.
Indeed, it follows from direct computations that

    ^(T)D_γ,2^n=   { L Tμ+14L(tr^(T)π)+∇_X(12μ d^Xμ+μ^2ζ^X)} Lφ_γ^n+{14 L(tr^(T)π)
       +12∇_X( d^Xμ+2μζ^X)}Lφ_γ^n+{12ℒ_ L(2μ^2ζ_X+μ d_Xμ)- d_XTμ
       +12ℒ_L( d_Xμ+2μζ_X)+12 d_X(μtr^(T)π)} d^Xφ_γ^n,
    ^(ϱ L)D_γ,2^n=   { L(ϱ Lμ-μ)+12L(ϱtrχ̌)-ϱ∇_X( d^Xμ+2μζ^X)} Lφ_γ^n+12 L(ϱtrχ̌)Lφ_γ^n
       -{ℒ_ L(ϱ d_Xμ+2ϱμζ_X)+ d_X(μ+ϱ Lμ)-ϱ d_X(μtrχ)} d^Xφ_γ^n,
    ^(R)D_γ,2^n=   { LRμ
    -12∇^X^(R)π_LX+14L(tr^(R)π)} LΨ_^n+{14 L(tr^(R)π)
       -12∇^X^(R)π_ LX}LΨ_γ^n+{-12ℒ_ L^(R)π_LX-12ℒ_L^(R)π_ LX
    - d_XRμ
       +12 d_X(μtr^(R)π)} d^XΨ_γ^n.



It is emphasized that special attentions are needed to handle the terms with underlines, wavy lines, boxes, or braces
in the (<ref>)-(<ref>). In ^(T)D_γ,2^n,
due to 12ℒ_L d_Xμ= d_XTμ+12μ d_X Lμ by L=μ L+2T, the corresponding underline part becomes

    - d_XTμ+12ℒ_L d_Xμ=12μ d_X Lμ,

which can be treated by using (<ref>). It follows from (<ref>) and (<ref>) that all terms with wavy lines in (<ref>)-(<ref>) contain ℒ_Lχ̌, which can be written as 2∇^2μ+μℒ_Lχ̌+⋯ by (<ref>). Note that the terms in μℒ_Lχ̌+⋯
may be estimated by (<ref>) and Proposition <ref> while L^2 norm of the terms corresponding to ∇^2μ
may be treated by (<ref>). Meanwhile,
one can use (<ref>) and (<ref>) to estimate those terms with boxes and braces respectively.

On the other hand, it is noticed that there are some terms whose factors are the derivatives of the deformation tensors
with respect to L, for example, 14 L(tr^(T)π) Lφ_γ^n appears in (<ref>). In fact, these terms are not “bad" in the sense that the derivatives of L
for the involved deformation tensors are
 actually equipped with the “good" quantities in terms of (<ref>) and (<ref>) after examining
 each term in (<ref>) and (<ref>) in Appendix A.

In summary, using (<ref>)-(<ref>), one can eventually arrive at

    δ^2l+1|∫_D^s, u∑_p=1^m(Z_m+1+^(Z_m+1)λ)…(Z_m+2-p
    +^(Z_m+2-p)λ)^(Z_m+1-p)D_γ,2^m-p·Lφ_γ^m+1|
    ≲   ∫_t_0^s(^1-_0τ^-1/2^lZ^mμ_τ,u+^-_0τ^1/2^l dZ^mtrχ_τ,u+⋯)√(E_2,m+2(τ,u))dτ
    ≲   δ^2(k+1)(1-ε_0)-2_0+∫_t_0^sτ^-1-ιẼ_1,≤ m+2(τ,u)dτ
    +δ∫_t_0^sτ^-1-ιẼ_2,≤ m+2(τ,u)dτ
       +^2k(1-_0)-2_0-1∫_0^uF̃_1,m+2(s,u')du'

and

    δ^2l∫_D^s, u∑_p=1^m|ϱ^2ι(Z_m+1+^(Z_m+1)λ)…(Z_m+2-p
    +^(Z_m+2-p)λ)^(Z_m+1-p)D_γ,2^m-p· Lφ_γ^m+1|
    ≲   δ^2(k+1)(1-ε_0)-2_0+∫_t_0^sτ^-2Ẽ_1,≤ m+2(τ,u)dτ
    +δ∫_t_0^sτ^-3+2ιln^2τẼ_2,≤ m+2(τ,u)dτ
       +δ^-1∫_0^uF̃_1,m+2(s,u')du'.




It should be pointed out that the main reason for getting the precise improvements of the L^2 norm of Z^m+1μ in Proposition <ref> is to
obtain the optimal estimates of Z^mμ and dZ^mtrχ in  (<ref>) and (<ref>), respectively. These optimal estimates lead to the smallness factor ^2(k+1)(1-_0)-2_0 in the right hand sides of (<ref>) and (<ref>). Note that ^2(k+1)(1-_0)-2_0<δ^2-2_0 holds
due to _0<_k^*. Based on this, one can close the energy estimates in Section <ref> (see (<ref>) for details).





 §.§ Treatments on J_2^m+1 and Φ_^1

This subsection deals with J_2^m+1 and Φ_^1 in (<ref>).
Note that J_2^m+1 and Φ_^1 do not contain the top order derivatives of trχ and μ. Therefore, according to Proposition <ref> and the expressions
of ^(Z)D_, j^n in (<ref>)-(<ref>) and (<ref>)-(<ref>), one can obtain

    δ^2l+1∫_D^s,u|∑_p=1^3^(Z_m+1)D_γ,p^m·Lφ_γ^m+1|
    ≲   δ^2(k+1)(1-ε_0)+∫_t_0^sτ^-1-ιln^2τẼ_1,≤ m+2(τ,u)dτ
    +δ∫_t_0^sτ^-1-ιẼ_2,≤ m+2(τ,u)dτ
       +δ^-1∫_0^uF̃_1,m+2(s,u')du'.

and

    δ^2l∫_D^s,u|∑_p=1^3ϱ^2ι^(Z_m+1)D_γ,p^m· Lφ_γ^m+1|
    ≲   δ^2(k+1)(1-ε_0)+∫_t_0^sτ^-2ln^2τẼ_1,≤ m+2(τ,u)dτ
    +δ∫_t_0^sτ^-4+2ιln^2τẼ_2,≤ m+2(τ,u)dτ
       +δ^-1∫_0^uF̃_1,m+2(s,u')du'.


In addition, Φ_^0=μ_gφ_ is given explicitly in (<ref>).
Then it follows from this, Proposition <ref> and (<ref>) that

    δ^2l+1∫_D^s,u|(Z_m+1+^(Z_m+1)λ)⋯(Z_1
    +^(Z_1)λ)Φ_γ^0·Lφ_γ^m+1|
    ≲   δ^2(k+1)(1-ε_0)+∫_t_0^sτ^-1-ιẼ_1,≤ m+2(τ,u)dτ
    +δ∫_t_0^sτ^-1-ιẼ_2,≤ m+2(τ,u)dτ
       +δ^-1∫_0^uF̃_1,m+2(s,u')du'

and

    δ^2l∫_D^s,uϱ^2ι|(Z_m+1+^(Z_m+1)λ)⋯(Z_1
    +^(Z_1)λ)Φ_γ^0· Lφ_γ^m+1|
    ≲   δ^2(k+1)(1-ε_0)+∫_t_0^sτ^-2Ẽ_1,≤ m+2(τ,u)dτ
    +δ∫_t_0^sτ^-4+2ιẼ_2,≤ m+2(τ,u)dτ
       +δ^-1∫_0^uF̃_1,m+2(s,u')du'.





§ GLOBAL ESTIMATES IN A_2



Following the idea of <cit.>, we are now ready to prove the global estimates on the smooth solution ϕ to the equation (<ref>) with (<ref>),  (<ref>) and (<ref>)-(<ref>) near C_0 when _0<_k^*. To this end, one has to estimate the solution in C_2 as in <cit.>. Compared with <cit.>, to obtain the behavior of the solution for _0 approaching _k^*, one needs the more precise estimates on all related quantities.

By substituting (<ref>)-(<ref>) and (<ref>)-(<ref>) into (<ref>), one gets from
Gronwall's inequality and 0<_0<_k^* that

    δẼ_2,≤ 2N-4(s,u)+δ F_2,≤ 2N-4(s,u)+Ẽ_1,≤ 2N-4(s,u)+F_1,≤ 2N-4(s,u)
    ≲δ^2-2ε_0.

This, together with (<ref>) and (<ref>), yields that for m≤ 2N-5,

    δ^lZ^mϕ_s,u≲δ^5/2-ε_0.


Based on (<ref>) and (<ref>), one can close the bootstrap
assumptions (⋆) in Subsection <ref> by using the analogous Sobolev-type embedding formula which follows form Proposition 18.10 in <cit.>.

 For any function f∈ H^2(S_s,u), under the assumptions (⋆), if δ>0 is small, then
 
    f_L^∞(S_s,u)≲1s^1/2∑_p≤ 1R^p f_L^2(S_s, u).





It follows from (<ref>)-(<ref>) that for 0<_0<_k^* and m≤ 2N-6,

    δ^l|Z^mφ_γ|≲δ^ls^1/2∑_p≤ 1R^p Z^mφ_γ_L^2(S_s, u)(<ref>)≲δ^1/2s^1/2(√(E_1,≤ 2N-4)+√(E_2,≤ 2N-4))
    ≲δ^1-ε_0 s^-1/2;

while for m≤ 2N-7,

    δ^l|Z^mϕ|≲δ^1/2s^-1/2δ^l∑_p≤ 1( LR^p Z^mϕ_s,u
    + LR^p Z^mϕ_s,u)≲δ^2-ε_0s^-1/2.

Replacing (⋆) by (<ref>)-(<ref>) in Section <ref> and following all procedures there yield the following
improved estimates over Proposition <ref>: for m≤ 2N-9,

    |ℒ_ Z^mχ̌|≲δ^k(1-ε_0)-ls^-2,  |ℒ_Z^m+1 dx^j|≲δ^-l,   |ℒ_Z^m^(T)π_ L|≲δ^k(1-ε_0)-1-ls^-1,
       |Z^m+1Ľ^j|+|ℒ_Z^m^(R)π|+|ℒ_Z^m^(R)π_ L|+|Z^m+1ϱ̌|≲δ^k(1-ε_0)-ls^-1ln s,
        |ℒ_Z^mR|+|Z^m+1υ|≲δ^k(1-ε_0)-lln s,  |Z^m+1μ|≲δ^k(1-ε_0)-ε_0-ls^-(k-1)/2,
        |ℒ_Z^m^(T)π|≲δ^k(1-ε_0)-1-ls^-1+s^-l-1,  |ℒ_Z^m^(R)π_T|≲(δ^k(1-_0)-l
    +δ^2k(1-_0)-1-l)s^-1ln s,

where l is the number of T as before.

Since the estimates in (<ref>) and (<ref>) are independent of the constant M in (⋆),
the bootstrap assumptions (⋆) are proved. Therefore, the global estimates  on the solution
ϕ to (<ref>) with (<ref>), (<ref>) and (<ref>)
in the domain D^s,4δ (see Figure <ref> in Subsection <ref>) are established.

In addition, one can further update the estimates on ϱ̌ (see (<ref>)), υ (see (<ref>)) and g_ijĽ^iø^j, which will be used to obtain the more precise smallness orders and time decay rates for the L^∞ norms of ϕ under the actions of Γ∈{L,L,Ø_ij} in A_2
and further be taken as the boundary values on C_2 to study the global Goursat problem
of (<ref>) in B_2.


	In D^s,4, the quantities Ľ^i and ϱ̌ have the following estimates:
	
    δ^l|Z^mϱ̌|
    	+δ^l|Z^m(g_ijĽ^iø^j)|≲δ^(k+1)(1-_0)s^-1,  δ^l|Z^m(g_βø^ø^β)|≲δ^(k+1)(1-ε_0)s^-3/2,

	where l is the number of T in Z^m and m≤ 2N-9.


	On the initial time t_0,
	
    μ=1√((g^0iø_i)^2+g^ijø_iø_j),
       L̃^i=-g^0i(-g^0jø_j+√((g^0jø_j)^2+g^abø_aø_b))+g^ijø_j.

	Due to g^β=m^β+g^β,_1⋯_k__1⋯__k+O(^k+1), then it
follows from (<ref>) that
	
    Ľ^i|_t_0
    	=   {(-g^0i,_1⋯_k-12g^ab,_1⋯_kø_aø_bø^i+g^ij,_1⋯_kø_j)__1⋯__k
    +f(ø,)^k+1}|_t_0.

In addition, by _t=-12ø_0L+12L, _i=-12ø_iL+12ø_i L+1rø^i_⊥Ø
	and (<ref>), one has
	
    g_ijĽ^iø^j|_t_0   ={(-Lϕ2)^k( g^0i,_1⋯_kø_0ø_i+12g^ij,_1⋯_kø_iø_j)ø__1⋯ø__k+f(φ,ø)(
    	[     ^k+1;   ^k-1Lϕ; ^k-11rØϕ ])}|_t_0
       =f(φ,ø)(
    	[     ^k+1;   ^k-1Lϕ; ^k-11rØϕ ])|_t_0.

	On the other hand, it follows from (<ref>) that
	
    υ|_t_0=   {-g_0jϵ_i^jx^i-g_ajĽ^aϵ_i^jx^i-(g_aj-m_aj)ø^aϵ_i^jx^i}|_t_0
    
    	=   ϵ_i^jx^i{(-g^0j,γ_1,⋯_k+g^aj,_1,⋯_kø_a)__1⋯__k-Ľ^j+f(φ,ø)^k+1}|_t_0
    
    	=   rf(φ,ø)^k+1|_t_0.

	For any Z̅∈{T=-μ(g^0i+ L^i)_i, R=Ø-υT̃}, (<ref>) and (<ref>) imply
	
    δ^l|Z̅^m(g_ijĽ^iø^j)|_t_0≲δ^(k+1)(1-_0).

	Under the action of L, the term ϱ g_ijĽ^iø^j can be written as
	
    L(ϱ g_ijĽ^iø^j)=(G_ij^γ Lφ_γ)ϱĽ^iø^j
    	+ϱ rg_ij L(ϱĽ^i)( L^j-Ľ^j)+ϱ r g_ijĽ^i(Ľ^j-ø^jø_aĽ^a),

	where
	
    g_ij L(ϱĽ^i) L^j=
    	-12ϱ G_ L L^γ Lφ_γ+f(φ, L^1, L^2)ϱ^2k-1(
    	[        L; d_X d^Xx ])

	follows from (<ref>), (<ref>), and g_ij( d^Xx^i) L^j=-g_0j( d^Xx^j).
	In addition, (<ref>) implies
	
    G_ L L^γ Lφ_γ=   
    	-(_φ_γg^κλ) L_κ L_λ L^ν_γφ_ν
    
    	=   -(_φ_0g^κλ) L_κ L_λ( L^2ϕ-( L L^i)φ_i)+((_φ_γg^κλ) L_κ L_λ L_γ)T̃^i Lφ_i
       -(_φ_γg^κλ) L_κ L_λ g_γ j( d^Xx^j)( d_X Lϕ-( d_X L^i)φ_i),

	where ( L L^i)_i=O(|^k Lφ|+|^k dφ|) and ( d_X L^i)φ_i=trχ d_Xϕ+O(|^k Lφ|+|^k dφ|) due to (<ref>) and (<ref>) respectively, and   (_φ_γg^κλ) L_κ L_λ L_γ satisfies (<ref>). Substituting (<ref>)-(<ref>) into (<ref>) and using (<ref>)-(<ref>) lead to
	
    ^l |Z̅^m L(ϱ g_ijĽ^iø^j)|≲^(k+1)(1-_0)s^-3/2,   m≤ 2N-9.

	Therefore, for m≤ 2N-9, it follows from (<ref>) and (<ref>) that
	
    δ^l| LZ̅^m(ϱ g_ijĽ^iø^j)|≲   δ^l∑_p_1+p_2=m-1|(ℒ_Z̅^p_1^(Z̅)π_ L^X) d_XZ̅^p_2(ϱ g_ijĽ^iø^j)|_vanish when m=0+δ^l|Z̅^m L(ϱ g_ijĽ^iø^j)|
    ≲   δ^(k+1)(1-_0)s^-3/2.

	This, together with (<ref>), yields
	
    δ^l|Z̅^m(g_ijĽ^iø^j)|≲δ^(k+1)(1-_0)s^-1,   m≤ 2N-9.

	
	If there is at least one ϱ L in Z^m (m≤ 2N-8), namely, Z^m=Z^p_1(ϱ L)Z̅^p_2
	with p_1+p_2=m-1, then
	
    (ϱ L)Z̅^p_2(g_ijĽ^iø^j)=∑_q_1+q_2=p_2-1ϱZ̅^q_1^(Z̅)π_ L^X· d_XZ̅^q_2(g_ijĽ^iø^j)_vanish when p_2=0+ϱZ̅^p_2(ϱ^-1 L(ϱ g_ijĽ^iø^j)-ϱ^-1g_ijĽ^iø^j).

	With an induction argument on the number of ϱ L in Z^p_1, it follows from the expression above, (<ref>) and (<ref>) that
	
    |Z^p_1(ϱ L)Z̅^p_2(g_ijĽ^iø^j)|
    ≲δ^(k+1)(1-_0)-ls^-1.

	Therefore, for any vector Z∈{ϱ L,T,R}, when m≤ 2N-9, by (<ref>) and (<ref>), one can get
	
    δ^l|Z^m(g_ijĽ^iø^j)|≲δ^(k+1)(1-_0)s^-1.

	
	
	In order to improve the estimate (<ref>) on ϱ̌, one needs to check the numerator in (<ref>).
	Due to __p=__p^0 Lϕ- L__pT̃^i_i+g__pi d_Xx^i d^Xϕ
and g^00=-1, then
	
    1-g_ijø^iø^j+2g_ijŤ^iø^j=1-g_ijø^iø^j-2g_0iø^i-2g_ijĽ^iø^j
    
    	=   g^ij,_1⋯_kø_iø_j__1⋯__k+2 g^0i,_1⋯_kø_0ø_i__1⋯__k-2g_ijĽ^iø^j+f(φ,ø)^k+1
    
    	=   (-1)^k g^β,_1⋯_kø_ø_β L__1⋯ L__k(T̃^i_1_i_1)⋯(T̃^i_k_i_k)-2g_ijĽ^iø^j+f(φ,ø)(
    	[           ^k+1;       φ^k-1 Lϕ; ^k-1 d^Xx d_Xϕ ])
    
    	=   -2g_ijĽ^iø^j+f(φ, L^1, L^2,xϱ,ϱ̌)(
    	[           ^k+1;       φ^k-1 Lϕ; ^k-1 d^Xx d_Xϕ;         Ľ^1φ^k;         Ľ^2φ^k;           ϱ̌^k ]).

	Substituting (<ref>) into (<ref>) and applying (<ref>) yield that for m≤ 2N-9,
	
    δ^l|Z^mϱ̌|≲δ^(k+1)(1-_0)s^-1.
	
	Since g_00=-1+f()^k+1, then g_βø^ø^β=-1+2g_0iø^i+g_ijø^iø^j+f(,ø)^k+1.
This, together with the first identity in (<ref>), shows that
	
    g_βø^ø^β
    	=f(φ, L^1, L^2,xϱ,ϱ̌)(
    	[           ^k+1;       φ^k-1 Lϕ; ^k-1 d^Xx d_Xϕ;         Ľ^1φ^k;         Ľ^2φ^k;           ϱ̌^k ]),

	which implies ^l |Z^m(g_βø^ø^β)|≲^(k+1)(1-_0)s^-3/2.



	It follows from (<ref>) that ϱ̌=rϱ-1=O(δ^(k+1)(1-_0)s^-1). This
shows that the distance between C_0 and C_4δ on the hypersurface Σ_t is 4δ
	+O(δ^(k+1)(1-_0)). Thus, the characteristic
	surface C_u (0≤ u≤ 4δ) is almost straight with the error O(δ^(k+1)(1-_0)) from
	the corresponding light conic surface due to ^(k+1)(1-_0)=o() for _0<_k^*.



	In D^s,3, for m≤ 2N-8, it holds that
	
    δ^l|Z^mυ|≲δ^(k+1)(1-_0),

	where l is the number of T in Z^m.


	Write g_β=m_β+g_β^_1⋯_k__1⋯__k+h_β() in (<ref>).
Then (<ref>) implies
	
    υ=   -g_0jϵ_i^jx^i-g_ajĽ^aϵ_i^jx^i-g_ajϵ_i^jx^ax^iϱ
    
    	=   ϱ{-(g_0j^γ_1⋯_k+g_ja^_1⋯_kx^aϱ)ϵ_i^jx^iϱ__1⋯__k
    -m_jaĽ^aϵ_i^jx^iϱ}+ϱ f(φ,xϱ)(
    	[  ^k+1; Ľ^1^k; Ľ^2^k ]).

	Let
	
    ℱ=-(g_0j^γ_1⋯_k+g_ja^_1⋯_kx^aϱ)ϵ_i^jx^iϱ__1⋯__k-m_jaĽ^aϵ_i^jx^iϱ.

	Then
	
    Tℱ=   -k(g_0j^γ_1⋯_k+g_ja^_1⋯_kx^aϱ)
    ϵ_i^jx^iϱ(T__1)__2⋯__k-m_jaϵ_i^j((T L^a)x^iϱ+Ľ^aT(x^iϱ))
       -(g_0j^γ_1⋯_kT(x^iϱ)+g_ja^_1⋯_kT(x^ax^iϱ^2))
    ϵ_i^j__1⋯__k+m_jaT(x^aϱ)ϵ_i^jx^iϱ
    
    	=   -k(g_0j^γ_1⋯_k+g_ja^_1⋯_kx^aϱ)ϵ_i^jx^iϱ(T__1)
    __2⋯__k-m_jaϵ_i^j(T L^a)x^iϱ
       +1ϱ f(,Ľ^1,Ľ^2,μ,xϱ)([  ^k; Ľ^1; Ľ^2; μ-1 ]),

	where T(x^aϱ)=-μϱg^0a-μϱĽ^a+(1-μ)x^aϱ^2 has been used.
	Note that there is a factor T L^a in (<ref>) containing the term -G_X L^γ(Tφ_γ) d^Xx^a
due to (<ref>). It follows from the definition of G_X L^γ in (<ref>)
and (<ref>) that
	
    -G_X L^γ(Tφ_γ) d^Xx^a=-G_iβ^γ( d_Xx^i· d^Xx^a) L^β(Tφ_γ)
    
    	=   -G_aβ^γ L^β(Tφ_γ)-G_ L L^(Tφ_γ)T̃^a
    	+G_0β^γ L^βT̃^a(Tφ_γ)+f(φ, L^i)^2k-1Tφ_γ
    
    	=   -k(g_a0^_2⋯_k+g_aj^_2⋯_kx^jϱ+g_0β^_2⋯_k L^βx^aϱ)__2⋯__k(Tφ_)-G_ L L^(Tφ_γ)T̃^a
       +f(φ, L^1, L^2)([      ^k; ^k-1Ľ^1; ^k-1Ľ^2 ])Tφ.

	Examining (<ref>) carefully and using (<ref>), one can get
	
    T L^a=   -k(g_a0^_2⋯_k+g_aj^_2⋯_kx^jϱ+g_0β^_2⋯_k L^βx^aϱ)__2⋯__k(Tφ_)
       +f(φ, L^1, L^1)([           ^k(Tφ);      ^k-1Ľ^1(Tφ);      ^k-1Ľ^2(Tφ);    G_ L L^(Tφ_γ);        d^Xμ d_Xx;        ^k-1(μ L); ^k-1(μ d^X d_Xx) ]).

	For m≤ 2N-8, substitute (<ref>) into (<ref>) to yield
	
    δ^l|Z^mTℱ|≲δ^k(1-_0)-_0s^-1,

	where (<ref>) and (<ref>) have been used.
	Thus,
	
    δ^l|TZ^mℱ|≤δ^l|[T,Z^m]ℱ|+δ^l|Z^mTℱ|≲δ^k(1-_0)-_0s^-1,

	which, together with (<ref>), implies
	
    δ^l|(_t-_r)Z^mℱ|≤δ^l|TZ^mℱ|+^l| L Z^mℱ|+δ^l| dZ^mℱ|≲δ^k(1-_0)-_0s^-1.

 Integrating (<ref>) along integral curves of _t-_r yields
	
    δ^l|Z^mℱ|≲δ^(k+1)(1-_0)s^-1,     m≤ 2N-8.

	It follows from (<ref>), (<ref>) and (<ref>) that for m≤ 2N-8,
	
    δ^l|Z^mυ|≲δ^(k+1)(1-_0).






Now we are ready to obtain the more precise orders of smallness of  and time decay rates for
the L^∞ norms of ϕ under the actions of Γ∈{L,L,Ø}.
Recall the standard vectors {L,L,Ø} defined in the end of Section <ref>.
It follows from (<ref>) and (<ref>) that

    L= L-μ^-1ø^ L_ T+ø^ g_ j( d^Xx^j)X,
       L= L+μ^-1(ø^i L_i- L_0)T+(g_0j d^Xx^j-ø^i g_ij d^Xx^j)X,
       Ω=R+μ^-1υ T.


The identity

    ø^ L_=g_βø^ø^β+g_0iĽ^i+g_ijø^iĽ^j+ϱ̌(g_0iø^i+g_ijø^iø^j)

and (<ref>) imply that for m≤ 2N-9,

    δ^l|Z^m(ø^ L_)|≲δ^(k+1)(1-ε_0)s^-1

holds in D^s,4.
Therefore, collecting (<ref>), (<ref>), (<ref>) with (<ref>) and (<ref>) yields that in D^s,3,

    |Γ^mφ_γ|≲δ^1-ε_0-ls^-1/2,
        |Γ^mϕ|≲δ^2-ε_0-ls^-1/2,   m≤ 2N-8,

where Γ∈{(s+r)L,L,Ω} and l is the number of L in Γ^m.


For any point P∈C̃_2δ={(t,x):t≥ t_0,t-|x|=2δ}, there is an integral curve of
L across this point, and the corresponding initial point is denoted by P_0(t_0,x_0) on Σ_t_0 with |x_0|=1.
From (<ref>), |L^pL^qØ^cϕ(P_0)|≲δ^2-ε_0 holds.
Next we derive an improved estimate on L^pL^qØ^cϕ at the point P.
It is emphasized here that the condition of 0<_0<_k^* plays a key role
in the following proposition.


	For any (t,x)∈C̃_2δ, when N is large enough, it holds that
	
    |L^pL^qØ^cϕ(t,x)|≲δ^2-ε_0t^-1/2-q.



According to (<ref>) in Appendix B and (<ref>), one has

    | L(r^1/2^mØ^cLϕ)|≲δ^2-ε_0-mt^-2+δ^(k+1)(1-ε_0)-ε_0-m t^-3/2,

where ∈{_t,_r}.
Integrating (<ref>) along integral curves of L and using the estimate (<ref>) on initial data at t_0,
one then can get
that on C̃_2δ,

    |^mØ^cLϕ|≲δ^2-ε_0-mt^-1/2+δ^(k+1)(1-ε_0)-ε_0-mt^-1/2, m+c≤ 2N-10.

Next we construct a sequence of functions inductively as follows:

    a_0(ϵ)   =2-ϵ,
    
    a_1(ϵ)   =(k+1)(1-ϵ)-ϵ,
    
    a_n+1(ϵ)   =(k+2)a_n(ϵ)-1,     n≥ 1.

Then for n≥ 1,

    a_n+1(ϵ)=(k+2)^na_1(ϵ)-(k+2)^n-1k+1.

Let ϵ(n) and ϵ̅(n) be the solutions to a_n(ϵ)=a_0(ϵ) and ka_n(ϵ)+1-ϵ=a_n+1(ϵ) respectively, so that

    ϵ(n)   = kk+1-1(k+2)^n-1,
    ϵ̅(n)   = kk+1- k(k+1)(2(k+2)^n-1).

Thus,

    lim_n→∞ϵ(n)=lim_n→∞ϵ̅(n)=kk+1,
       ϵ(n)<ϵ̅(n)<ϵ(n+1).

For any fixed number _0∈ (0,_k^*), there exists an n∈ℕ such that ϵ(n-1)<_0≤ϵ(n)
with the convention that ϵ(0)=0.

	
  * When n=1, i.e., 0<_0≤ϵ(1), (k+1)(1-_0)-_0≥ 2-_0 holds.
It follows from (<ref>) that on C̃_2δ,
	
    |^mØ^cLϕ|≲δ^2-ε_0-mt^-1/2, m+c≤ 2N-10.

	
  * When n=2 and _0≤ϵ̅(1), one has (k+1)(1-_0)-_0< 2-_0 and 2-_0<1-_0+ka_1(_0)≤ a_2(_0).
Then (<ref>) implies that on C̃_2δ,
	
    |^mØ^cLϕ|≲^a_1(_0)-mt^-1/2, m+c≤ 2N-10.

	Using (<ref>) and (<ref>) in Appendix B again, then one has that for m+c≤ 2N-11,
	
    |L(r^1/2^mØ^cLϕ)|≲^2-_0-mt^-2+^1-_0+ka_1(_0)-mt^-2+^a_2(_0)-mt^-3/2≲^2-_0-mt^-3/2,

	which implies that on C̃_2δ,
	
    |^mØ^cLϕ|≲δ^2-ε_0-mt^-1/2, m+c≤ 2N-11.

	When n=2 and ϵ̅(1)<_0≤ϵ(2), one has (k+1)(1-_0)-_0< 2-_0 and 2-_0≤ a_2(_0)<1-_0+ka_1(_0). Exactly similar to the case for n=2 and _0≤ϵ̅(1), one has
that on C̃_2δ,
		
    |^mØ^cLϕ|≲δ^2-ε_0-mt^-1/2, m+c≤ 2N-11.

	
  * For general n≥ 3, we now focus on the case of j∈ℕ∩ [1,n-1] and _0>ϵ(j).
First, it follows from _0>ϵ(1) and (<ref>) that on C̃_2δ,
		
    |^mØ^cLϕ|≲^a_1(_0)-mt^-1/2, m+c≤ 2N-10.
	
Then, an induction argument on j can be used to show that on C̃_2δ,

    |^mØ^cLϕ|≲^a_j(_0)-mt^-1/2

for m+c≤ 2N-9-j (1≤ j≤ n-1). In fact, if (<ref>) holds true for all j with 1≤ j≤ j_0≤ n-2, then when m+c≤ 2N-10-j_0, it holds that
	
    |L(r^1/2^mØ^cLϕ)|≲^2-_0-mt^-2+^1-_0-m+ka_j_0(_0)t^-2+^a_j_0+1(_0)-mt^-3/2.

	It follows from _0>ϵ(j_0+1)>ϵ̅(j_0) that a_j_0+1(_0)<2-_0 and a_j_0+1(_0)<1-_0+ka_j_0(_0)
hold, and hence |L(r^1/2^mØ^cLϕ)|≲^a_j_0+1(_0)-mt^-3/2, which implies (<ref>).
Thus, on C̃_2δ,
	
    |^mØ^cLϕ|≲^a_n-1(_0)-mt^-1/2, m+c≤ 2N-8-n.

	By (<ref>) and (<ref>) in Appendix B, we obtain that for m+c≤ 2N-9-n,
	
    |L(r^1/2^mØ^cLϕ)|≲^2-_0-mt^-2+^1-_0+ka_n-1(_0)-mt^-2+^a_n(_0)-mt^-3/2.

	Since ϵ(n-1)<_0≤ϵ(n), then 2-_0≤ a_n(_0) and 2-_0<ka_n-1(_0)+1-_0
hold. Therefore, integrating (<ref>) along integral curves of L yields that on C̃_2δ,
	
    |^mØ^cLϕ|≲^2-_0-mt^-1/2, m+c≤ 2N-9-n.


Consequently, for any n∈ℕ and m+c≤ 2N-9-n, it holds that

    |^mØ^cLϕ|_C̃_2≲^2-_0-mt^-1/2,
       |L^mØ^cLϕ|_C̃_2≲^2-_0-mt^-3/2.

With an induction argument on the power of L, one can show
that for c+m+2p≤ 2N-7-n and on C̃_2δ,

    |^mØ^cL^pϕ|≲δ^2-ε_0-mt^-1/2,
       |L^mØ^cL^pϕ|≲δ^2-ε_0-mt^-3/2.



Based on (<ref>), we now assume that for some positive integer q_0, when 1≤ q≤ q_0 and q-1+c+m+2p≤ 2N-7-n,
there holds that on C̃_2δ,

    |L^q^mØ^cL^pϕ|≲δ^2-ε_0-mt^-1/2-q.

For q_0+c+m+2p≤ 2N-7-n and p≥ 1, by (<ref>), (<ref>) and the assumption (<ref>),
it follows from (<ref>) in Appendix B  that on C̃_2δ,

    δ^m|L^q_0+1^mØ^cL^pϕ|=δ^m| L^q_0^mØ^cL^p-1(LLϕ)|
    ≲   δ^2-ε_0t^-3/2-q_0
    +∑_c_1+m_1≤ c+m+1δ^m_1|L^q_0+1^m_1Ø^c_1L^≤ p-1ϕ|t^-1.

By reducing the power of L gradually, one can eventually obtain that on C̃_2δ,

    δ^m|L^q_0+1^mØ^cL^pϕ|≲   δ^2-ε_0t^-3/2-q_0+∑_c_1+m_1≤ c+m+pδ^m_1|L^q_0+1^m_1Ø^c_1ϕ|t^-1
    ≲   δ^2-ε_0t^-3/2-q_0.

This shows that (<ref>) holds for any positive q_0 by induction.

Then (<ref>) follows by setting m=0 in (<ref>) and (<ref>) together with the second estimate in (<ref>).





§ GLOBAL ESTIMATES IN B_2 AND THE PROOF OF THEOREM <REF>



In this section, we will derive the global estimates on the solution ϕ to the equation (<ref>) in B_2.
Note that although the general procedure is exactly same as the last section in <cit.>, yet
due to the requirements of the precise orders of smallness of  and the time decay rates  together
with the higher order null condition (<ref>), some details are still given here.

Set

    D_t:={(t̅,x): t̅-|x|≥ 2δ, t_0≤t̅≤ t},

see Figure <ref> below. As explained in <cit.>, different from <cit.> and <cit.>, the solution
ϕ to (<ref>) in D_t may be large here due to its initial data on time t_0 (see Theorem <ref>).
By the way, when δ>0 is small, we know that the L^∞ norms of ϕ and its derivatives are small on the
boundary C̃_2δ of D_t (see (<ref>)).

		



As in <cit.>, we use the following modified Klainerman-Sobolev Lemma (see <cit.>)
to solve the Goursat problem here.


	For any function f(t,x)∈ C^∞(ℝ^1+2), t≥ 1, (t,x)∈ D_T=
	{(t,x): t-|x|≥ 2δ, t_0≤ t≤ T}, the following inequalities hold:
	
    | f(t,x)|≲∑_i=0^2 t^-1δ^(i-1)𝚜Γ̅^if(t,·)_L^2(r≤ t/2), |x|≤14t,

	
    | f(t,x)|≲| f(t,B_t^x)|+∑_a≤ 1,|β|≤ 1t^-1/2Ω^a^β f(t,·)_L^2(t/4≤ r≤ t-2δ), |x|≥14t,

	where Γ̅∈{S,H_i,Ω}, (t,B_t^x) is the intersection point of
	the boundary C̃_2δ and the ray crossing (t,x) which emanates from (t,0),
	and 𝚜 is the any nonnegative constant in (<ref>).



The following inequality will also be needed (see <cit.>).


	For f(t,x)∈ C^∞(ℝ^1+2) and t≥ 1, it holds that for 1≤t̅≤ t-2δ,
	
    f(t,·)1+t-|·|_L^2(t̅≤ |x|≤ t-2δ)≲ t^1/2f(t,B_t^·)_L^∞(t̅≤ |x|≤ t-2δ)+ f(t,·)_L^2(t̅≤ |x|≤ t-2δ).






We intend to use the energy method to prove global estimates of the solution ϕ to (<ref>) in B_2.
Set

    E_p,l(t)=∑_Γ̃∈{,S,H_1,H_2}{Γ̃^pΩ^lϕ(t,·)_L^2(Σ_t∩ D_t)^2+∑_i=1^2∬_D_t|Z̃_iΓ̃^pΩ^lϕ|^2(t',x)1+(t'-|x|)^3/2dxdt'},

where Z̃_i=ω_i_t+_i, i=1,2 and p+l≤ 4. As in <cit.>,
the introduction of the term ∑_i=1^2∬_D_t|Z̃_iΓ̃^pΩ^lϕ|^2(t',x)1+(t'-|x|)^3/2dxdt'
in (<ref>) is motivated by the works on global existence of small data solutions to the 3D nonlinear wave equations
satisfying the weak null condition in <cit.>.

Thanks to
the estimate (<ref>) on Σ_t_0, one makes the following bootstrap assumptions:

For t≥ t_0, there exists a uniform constant M_0 such that

    E_0,l(t)≤M_0^2δ^4-2ε_0,     E_1,l(t)≤M_0^2δ^176-2ε_0,     E_2,l(t)≤M_0^2δ^1-2ε_0,
       E_3,l(t)≤M_0^2δ^-1-2ε_0,     E_4,l(t)≤M_0^2δ^-3-2ε_0.


According to Lemma <ref> and the assumptions in (<ref>), similarly to the last section of <cit.>, one can establish
the following L^∞ estimates.


	Under the assumptions (<ref>), when δ>0 is small, it holds that
	
    |Ω^≤ 2ϕ|≲ M_0δ^5/4-ε_0t^-1/2,       |Γ̃Ω^≤ 1ϕ|≲ M_0δ^11/24-ε_0t^-1/2,
       |Γ̃^2ϕ|≲ M_0δ^-1/2-ε_0t^-1/2

	and
		
    | ZΩ^≤ 2ϕ|≲ M_0δ^5/4-ε_0t^-3/2(1+t-r),       | ZΓ̃Ω^≤ 1ϕ|≲ M_0δ^11/24-ε_0t^-3/2(1+t-r),
       | ZΓ̃^2ϕ|≲ M_0δ^-1/2-ε_0t^-3/2(1+t-r),

	where Z∈{ Z_1, Z_2}.



	Following the proof of <cit.>, one can choose s=34 for ZØ^≤ 2ϕ, s=2324
		for Z̃Γ̃Ø^≤ 1ϕ and s=1 for Z̃^2ϕ in (11.11) of <cit.>
to obtain (<ref>)-(<ref>), respectively. The details are omitted here.


As in the proof of <cit.>, it follows from Proposition <ref> that

	Under the same assumptions in Proposition <ref>, it holds that
	
    |^2Ω^≤ 1ϕ|≲ M_0δ^11/24-ε_0t^-1/2(1+t-r)^-1,
       |^2Γ̃ϕ|≲ M_0δ^-1/2-ε_0t^-1/2(1+t-r)^-1

	and
	
    |Z̃Ω^≤ 1ϕ|≲ M_0δ^11/24-ε_0t^-3/2,
       |Z̃Γ̃ϕ|≲ M_0δ^-1/2-ε_0t^-3/2.




We now derive the global weighted energy estimates on ϕ in B_2. As in <cit.>, after choosing
the smooth ghost weight function W=e^2(1+t-r)^-1/2 introduced first in <cit.>,
then direct computations show
 
    W(_tv)g^β_β^2v
    
     =   12_t{W((g^00(_t v)^2-g^ij_iv_jv))}+_i{W((g^0i(_tv)^2
     +g^ij_tv_jv))}
       +(-(_ig^0i)(_tv)^2-(_ig^ij)_tv_jv+12(_tg^ij)_iv_jv)W
       +W(1+t-r)^3/2(12g^00(_tv)^2-12g^ij(_iv)(_jv)-g^0iø_i(_tv)^2-g^ijø_i(_tv)(_jv)).

Due to _i= Z_i-ø_i_t, then
 
    -(_ig^0i)(_tv)^2-(_ig^ij)_tv_jv+12(_tg^ij)_iv_jv
    
     =   -k{g^0i,_1⋯_k(_i__1ϕ)(_tv)^2+g^ij,_1⋯_k(_i__1ϕ)(_tv)(_jv)
       -12g^ij,_1⋯_k(_t__1ϕ)(_iv)(_jv)}(__2ϕ)⋯(__kϕ)+f(ϕ)(ϕ)^k^2ϕ( v)^2
    
     =   f(ø, ϕ)([        ^2ϕ(ϕ)^k( v)^2;       Zϕ(ϕ)^k-1( v)^2; ^2ϕ( Zϕ)(ϕ)^k-2( v)^2;     ^2ϕ(ϕ)^k-1( Zv) v ]).

 Note that integrating (<ref>) over D_t and using the integration by parts give rise to a term on the lateral
 boundary C̃_2δ as
 
    {-12(g^00(_t v)^2-g^ij_iv_jv)+(g^0i(_tv)^2+g^ij_tv_jv)ø_i}W
    
     =   {12(Lv)^2+12(1rØ v)^2}W+f(ø, ϕ)W([   (ϕ)^k+1( v)^2; Zϕ(ϕ)^k-1( v)^2;    (ϕ)^k( Zv) v ]).

This can be controlled on C̃_2δ by

    {(Lv)^2+(1rØ v)^2+δ^k(2-ε_0)t^-3/2( v)^2}W

with the help of (<ref>) and Z̃_i=ω^iL+ω^i_⊥rØ, where one has neglected the
unimportant constant coefficients in (<ref>).


Integrating (<ref>) on D_t and
utilizing the estimates in Proposition <ref> and Corollary <ref>,
 then one can get by (<ref>)-(<ref>) and Gronwall's inequality that for small δ>0,

    ∫_Σ_t∩ D_tW((_tv)^2+|∇ v|^2)+∬_D_t∑_i|Z̃_iv|^2(1+τ-r)^3/2W
    ≲   ∫_Σ_t_0∩ D_tW((_τ v)^2+|∇ v|^2)+∬_D_tW| _τ vg^β_β^2v|
       +∫_C̃_2δ∩ D_tW{(Lv)^2+(1rØ v)^2
    +δ^k(2-ε_0)τ^-3/2| v|^2}.



To close the bootstrap assumptions (<ref>), one sets v=Γ̃^pΩ^lϕ (p+l≤ 4) in (<ref>).
 By (<ref>), (LΓ̃^pΩ^lϕ)^2+(1rØΓ̃^pΩ^lϕ)^2
 +δ^k(2-ε_0)t^-3/2|Γ̃^pΩ^lϕ|^2≲δ^4-2ε_0t^-5/2 holds
 on C̃_2δ. Therefore,

    ∫_C̃_2δ∩ D_tW{(Lv)^2+(1rØ v)^2+δ^k(2-ε_0)τ^-3/2| v|^2}≲δ^4-2ε_0.

In addition, on the initial hypersurface Σ_t_0∩ D_t,
one has |Γ̃^pΩ^lϕ|≲δ^2-p-ε_0
for 0≤ p≤ 4-l by (<ref>). Hence, (<ref>) gives that

    E_0,l(t)≲δ^4-2ε_0+∬_D_t| (_τΩ^lϕ)(g^β_β^2Ω^lϕ)|,
       l≤ 4,
       E_p,l(t)≲δ^5-2p-2ε_0+∬_D_t| (_τΓ̃^pΩ^lϕ)(g^β_β^2Γ̃^pΩ^lϕ)|,   1≤ k≤ 4-l.


It remains to estimate ∬_D_t|(_τΓ̃^pΩ^lϕ)(g^β_β^2Γ̃^pΩ^lϕ)|
in the right hand sides of (<ref>)-(<ref>) and further obtain the global estimates of ϕ in B_2.


	When δ>0 is small, (<ref>) in B_2 holds true.


	Acting the operator Γ̃^kΩ^l on (<ref>) and commuting it with g^β_β^2 yield
	
    |g^β_β^2   Γ̃^pΩ^lϕ|≲∑_{| Z^p_1Ø^l_1ϕ|·|^p_2Ø^l_2ϕ|⋯|^p_kØ^l_kϕ|·|^2Γ̃^p_0Ø^l_0ϕ|
       +|^p_1Ø^l_1ϕ|⋯|^p_kØ^l_kϕ|·| Z^p_0Ø^l_0ϕ|}+∑_|^q_1Ø^j_1h^β|·|^2^q_0Ø^j_0ϕ|,

		which can be treated case by case as follows.



0.1 true cm
	
  * The case of p=0 and l≤ 4
0.1 true cm
	
		
  * If l_0≥max{l_1,⋯,l_k}, then l_1,⋯,l_k≤ 2 and l_0≤ 3. Then by Proposition <ref>,
		
    | ZØ^l_1ϕ|·|Ø^l_2ϕ|⋯|Ø^l_kϕ|·|^2Ø^l_0ϕ|≲ M_0^k^k(5/4-_0)t^-2(1+t-r)|^2Ø^l_0ϕ|,
       |Ø^l_1ϕ|⋯|Ø^l_kϕ|·| ZØ^l_0ϕ|≲ M_0^k^k(5/4-_0)t^-1| ZØ^l_0ϕ|.

		In addition, due to (1+t-r)|^2Ø^l_0ϕ|≲|Ø^l_0ϕ|+|ØØ^l_0ϕ| and | ZØ^l_0ϕ|≲ t^-1|Ø^l_0ϕ|+t^-1|ØØ^l_0ϕ|, one has by the assumptions (<ref>) and _0∈(0,_k^*) that

    ∬_D_t|Ω^lϕ|{| Z^p_1Ø^l_1ϕ|·|^p_2Ø^l_2ϕ|⋯|^p_kØ^l_kϕ|·|^2Γ̃^p_0Ø^l_0ϕ|
                   +|^p_1Ø^l_1ϕ|⋯|^p_kØ^l_kϕ|·| Z^p_0Ø^l_0ϕ|}
    ≲   ∫_t_0^tτ^-2E_0,≤ 4(τ)dτ+(M_0^5/4-_0)^2k∫_t_0^tτ^-2E_1,≤ 3(τ)dτ
    ≲   ∫_t_0^tτ^-2E_0,≤ 4(τ)dτ+^4-2_0.


0.2 true cm
		
  * If l_0<max{l_1,⋯,l_k}, then l_0≤ 1. It follows from Corollary <ref> that
		
    |^2Ω^l_0ϕ|≲ M_0δ^11/24-ε_0t^-1/2(1+t-r)^-1,
       | ZØ^l_0ϕ|≲ M_0δ^11/24-ε_0t^-3/2.

		Similarly, with the help of (<ref>) one can get
		
    ∬_D_t|Ω^lϕ|{| Z^p_1Ø^l_1ϕ|·|^p_2Ø^l_2ϕ|⋯|^p_kØ^l_kϕ|·|^2Γ̃^p_0Ø^l_0ϕ|
                   +|^p_1Ø^l_1ϕ|⋯|^p_kØ^l_kϕ|·| Z^p_0Ø^l_0ϕ|}
    ≲   M_0^kδ^5(k-1)4+1124-kε_0{∬_D_tτ^-1|Ω^lϕ|·| ZØ^≤ lϕ1+τ-r|+∬_D_tτ^-2|Ø^≤ lϕ|^2}
    ≲   ∫_t_0^tτ^-2E_0,≤ 4(τ)dτ+^4-2_0.

		
  * If j_0≥ j_1, then j_1≤ 2 and j_0≤ 3. Similarly to (<ref>), one can show
		
    ∬_D_t|Ω^lϕ|·|Ø^j_1h^β|·|^2Ø^j_0ϕ|
    ≲   (M_0^5/4-_0)^k+1∫_t_0^tτ^-(k+1)/2√(E_0,≤ 4(τ)E_1,≤ 3(τ))dτ
    ≲   ∫_t_0^tτ^-3/2E_0,≤ 4(τ)dτ+^4-2_0.

		
  * When j_0<j_1, then it holds that
		
    ∬_D_t|Ω^lϕ|·|Ø^j_1h^β|·|^2Ø^j_0ϕ|
    ≲   (M_0^5/4-_0)^k(M_0^11/24-_0)∫_t_0^tτ^-(k+1)/2E_0,≤ 4(τ)dτ≲∫_t_0^tτ^-3/2E_0,≤ 4(τ)dτ.

	
Inserting (<ref>) and (<ref>) into (<ref>) yields

    E_0,l(t)≲δ^4-2ε_0,   l≤ 4.

0.1 true cm


  * The case p+l≤ 4 and p≥ 1

0.1 true cm

For p+l≤ 4 (p≥ 1), as in the above case, one can make use of Proposition <ref> and Corollary <ref> to estimate the
related integrals in (<ref>) for max{p_1+l_1,⋯,p_k+l_k}≤ p_0+l_0 and q_1+j_1≤ q_0+j_0 (p_1+l_1,⋯,p_k+l_k≤2, p_0+l_0≤3, q_1+j_1≤ 2 and q_0+j_0≤ 3) and the rest terms (p_0+l_0≤ 1 or q_0+j_0≤ 1). In the whole process,
one can use repeatedly the bootstrap assumptions <ref> and the restricted condition ε_0∈(0,_k^*) for k≥ 2.


Consequently, by Grownwall's inequality, one can arrive at

    E_0,l(t)≲δ^4-2ε_0,     E_1,l(t)≲δ^17/6-2ε_0,     E_2,l(t)≲δ^1-2ε_0,
       E_3,l(t)≲δ^-1-2ε_0,     E_4,l(t)≲δ^-3-2ε_0.

Since the constants in (<ref>) are all independent of M_0, the bootstrap
assumptions  (<ref>) are proved as long as (<ref>) holds for the time t=t_0
by the continuous induction argument.



We are now ready to prove Theorem <ref>.

By Theorem <ref>, the local existence of the smooth solution ϕ to (<ref>) with (<ref>),
(<ref>) and
(<ref>)-(<ref>)  has been obtained.
On the other hand,  the global estimates of ϕ in A_2
and B_2  have been established in Section <ref> and Section <ref> respectively.
Then it follows from the local well-posedness of the smooth solution to (<ref>) with (<ref>) (see <cit.> or <cit.>)
and the continuous induction argument
that ϕ∈ C^∞([1,+∞)×ℝ^2) exists globally.
In addition, |ϕ|≲δ^1-ε_0t^-1/2 follows from (<ref>) in Appendix B, (<ref>) and
the first inequality in (<ref>). Meanwhile, |ϕ|≲δ^1-ε_0t^1/2 holds true by integration
on ϕ from the conic surface C_0.
Thus Theorem <ref> is proved.








§ COMPUTATIONS ON THE DEFORMATION TENSOR AND COMMUTATORS RELATIONS



Let ^(V)π_UX:=^(V)π_UX for U∈{ L, L,X}.
As in <cit.> for 3D case or in <cit.> for 4D case,
it follows from direct computations that


	
  * for V=T,
	
    ^(T)π_ L L=0,  ^(T)π_TT̃=2Tμ,  ^(T)π_ LT=-Tμ,  ^(T)π_TX=0,
       ^(T)π_ LX=-2μζ_X- d_Xμ,  ^(T)π_XX=2μσ_XX;

	
	
  * for V= L,
	
    ^( L)π_ L L=0,  ^( L)π_TT̃=2 Lμ,  ^( L)π_ LT=- Lμ,  ^( L)π_ LX=0,
       ^( L)π_TX=2μζ_X+ d_Xμ,  ^( L)π_XX=2χ_XX;

	
	
  * for V=R,
	
    ^(R)π_ L L=0,  ^(R)π_TT̃=2Rμ,  ^(R)π_ LT=-Rμ,
       ^(R)π_ LX=-R^Xχ̌_XX-υ{G_ LT̃^γ d_Xφ_γ+12G_T̃T̃^γ d_Xφ_γ-G_XT̃^γ Lφ_γ}
                 +12R^XG_XX^γ Lφ_γ+ϵ_i^jg_jaĽ^i d_Xx^a,
       ^(R)π_TX=μR^Xχ̌_XX+υ d_Xμ+μ G_X L^γ Rφ_γ-12μ R^XG_XX^γ Lφ_γ
                 +μ G_XT̃^γ Rφ_γ+υ{G_XT̃^γ Tφ_γ-12μ G_T̃T̃^γ d_Xφ_γ}+μϵ_i^j g_jaŤ^i d_Xx^a,
       ^(R)π_XX=2υχ_XX+υ{2G_X L^γ d_Xφ_γ
    +2G_XT̃^γ d_Xφ_γ-G_XX^γ Lφ_γ}
                 +G_XX^γ Rφ_γ
    +2ϵ_i^jǧ_ja( d_Xx^a) d_Xx^i,

where ǧ_ia:=g_ia-m_ia, and the definitions of Ť^i, Ľ^i are given in (<ref>).	






In addition, the following two results hold for commutators,
which are given in Lemma 4.10, 8.9 and 8.11 of <cit.>.


	It holds that
	
    [ L, R]   =^(R)π_ L^XX,  
    	[ L, T]=^(T)π_ L^XX,  
    	[T,R]=^(R)π_T^XX.




For any vector field Z∈{ L,T,R},

(a) if f is a smooth function, then
 
    ([∇^2,ℒ_Z]f)_XX   
     =12∇_X(tr^(Z)π) d_Xf,
    
     [, Z]f   =^(Z)π^XX∇_X^2f
     +12(∇_X^(Z)π^XX) d_Xf;


(b) if ξ is a one-form on S_s, u, then

    ([∇_X,ℒ_Z]ξ)_X=12∇_X(tr^(Z)π)ξ_X;


(c) if  ξ is a (0,2)-type tensor on S_s, u, then

    ([∇_X,ℒ_Z]ξ)_XX   =∇_X(tr^(Z)π)ξ_XX,
    
    ([∇_X,ℒ_Z]∇ξ)_XXX   =32∇_X(tr^(Z)π)∇_Xξ_XX.







§ THE PROOF  OF THEOREM <REF>


In this Appendix, we utilize the energy method and the special structure of the initial data (<ref>)
together with (<ref>) and
(<ref>)-(<ref>)
to prove Theorem <ref>.
Although the procedure is similar
to that in <cit.>,
some details are required to be given due to the general form of the quasilinear wave
equation (<ref>) with the higher order null condition structure here.


	 Denote by Z_g any fixed vector field in {S, H_i, i=1,2}.
Then analogously to the proof of (2.5) in <cit.>, one can have that for 1≤ t≤ t_0, ν∈ℕ_0^3
	and N_0∈ℕ_0 with N_0≥ 3,
	
    |^κØ^pZ_g^νϕ|≤δ^2-ϵ_0-|κ|   (|κ|+p+|ν|≤ N_0,   |ν|≤ 1).

	

In addition, by L=S+ø^iH_it+r, it can be shown that for |κ|+p+m≤ 2N_0-3 and m≤ 1,

    |L^m^κØ^pϕ(t,x)|≲∑_|ν|≤ 1|Z_g^ν^κØ^pϕ(t,x)|≲δ^2-|κ|-ε_0.


		



Now we start to improve the L^∞ estimate of ϕ(t_0,x) on some special space domains. To this end,
one rewrites (<ref>) as

    LLϕ=12rLϕ-12rLϕ+1r^2Ω^2ϕ+g^β,
    	γ_1,⋯,γ_k(_γ_1ϕ)⋯(_γ_kϕ)^2_βϕ+h^β(ϕ)^2_βϕ.


Applying the operator L^ιØ^p to both sides of (<ref>), where ∈{_t,_r} and ι∈ℕ_0^2, then one can obtain an expression of LL^2^ιØ^pϕ by using LL^2^ιØ^p=L^ιØ^pLL
and a direct computation.
This, together with (<ref>), yields that for |ι|+p≤ 2N_0-6,

    |L(L^2^ιØ^pϕ)|≲δ^1-ε_0-|ι|+δ^(k+1)(1-_0)-1-|ι|.

Due to the vanishing property of ϕ on {(t, x): t≥ 1, t=r},
integrating (<ref>) along integral curves of L yields that for (t,r)∈ D^1
(see Figure <ref>) and for |ι|+p≤ 2N_0-6,

    |L^2^ιØ^pϕ(t,x)|≲δ^2-ε_0-|ι|+δ^(k+1)(1-_0)-|ι|.

Note that the null condition (<ref>) implies

    g^β,γ_1⋯γ_k_γ_1ϕ⋯_γ_kϕ_β^2ϕ=   f_1(ø)Lϕ(ϕ)^k-1^2ϕ
    +f_2(ø)r^-1Øϕ(ϕ)^k-1^2ϕ
       +f_3(ø)(ϕ)^kLϕ+f_4(ø)r^-1(ϕ)^kØϕ,

where  f_i (i=1,2,3,4) are some smooth functions.
Using the equation for LL^2^ιØ^pϕ again, together with (<ref>) and
(<ref>), gives |L(L^2^ιØ^pϕ) |≲δ^1-ε_0-|ι| for 0<ε_0<_k^* and |ι|+p≤2N_0-7, which yields that for (t,x)∈ D^1,

    |L^2^ιØ^pϕ(t,x)|≲δ^2-ε_0-|ι|,  |ι|+p≤2N_0-7.

Then an induction argument and  (<ref>) show that for (t,x)∈ D^1,
 
    |LL^m^ιØ^pϕ(t,x)|≲δ^2-|ι|-ε_0,   3m+|ι|+p≤ 2N_0-4,

which implies (<ref>) due to _i=ø^i_r+ø_⊥^irØ.





Similarly, by the expression of LL^n^ιØ^pϕ, integrating it
along integral curves of L yields that for r∈ [1-3δ,1+δ] (see Figure <ref>),

    |L^n^ιØ^pϕ(t_0,x)|≲δ^2-|ι|-ε_0,   2n+|ι|+p≤ 2N_0-3,

which implies (<ref>).
In addition, it follows from (<ref>) that

    |Ø^pϕ|_D^2≲^3-_0, p≤ 2N_0-4

after integrating LØ^kϕ along integral curves of L in D^2.
Furthermore, using _t=12(L+L),
_i=ø^i2(L-L)+ø^i_⊥rØ,
(<ref>), (<ref>), (<ref>) and (<ref>), one concludes that for |κ|+p≤ 2N_0-5
and r∈ [1-3δ, 1+δ],

    |^κØ^pϕ(t_0,x)|≲δ^3-|κ|-ε_0,

and then (<ref>) follows.


Next we prove the estimate (<ref>). It is already shown that on the surface Σ_t_0, when r∈[1-2δ,1+δ] and 2n+|κ|+p≤ 2N_0-3, |L^n^κØ^pϕ|≲δ^2-ε_0-|κ| holds. Then |LL^n^κØ^pϕ|≲δ^2-ε_0-|κ| is derived by (<ref>) and a direct computation.
Meanwhile, when r∈[1-2δ,1+δ] and 3m+|κ|+p≤ 2N_0-4,
one has |L^1+m^κØ^pϕ|≲δ^2-ε_0-|κ|.
Furthermore, one can obtain

    |L^mL^n^κØ^pϕ|≲δ^2-ε_0-|κ|  for 3m+2n+|κ|+p≤ 2N_0-3 and n≥ 1.

Indeed, by an induction argument, it suffices to prove (<ref>) for 3(m+1)+2n+|κ|+p≤ 2N_0-3 and n≥ 1.
It follows from (<ref>) and the assumption (<ref>) that

    |L^m+1L^n^ιØ^pϕ|=|L^mL^n-1^ιØ^p(LLϕ)|
    ≲   δ^2-ε_0-|ι|+∑_[    |ι_1|+p_1≤|ι|+p+1; |ι_1|≤|ι|+1,p_1≤ p+1 ]δ^|ι_1|-|ι||L^m+1L^≤ n-1^ι_1Ø^p_1ϕ|.

Applying the same argument as for (<ref>) to treat |L^m+1L^≤ n-1^ι_1Ø^p_1ϕ|,
one can reduce the number of L gradually to get

    |L^m+1L^n^ιØ^pϕ|   ≲δ^2-ε_0-|ι|+∑_[   |ι_1|
    +p_1≤|ι|+p+n; |ι_1|≤|ι|+n,p_1≤ n+p ]δ^|ι_1|-|ι||L^m+n^ι_1Ø^p_1ϕ|
       ≲δ^2-ε_0-|ι|,

which yields (<ref>). Then (<ref>) is  deduced through utilizing (<ref>) with κ=0.
Therefore, the proof of Theorem <ref> is completed.








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