Zero Mach Number Limit of the Compressible Primitive Equations: Ill-prepared Initial Data

In the work, we consider the zero Mach number limit of compressible primitive equations in the domain $\mathbb{R}^2 \times 2\mathbb{T}$ or $\mathbb{T}^2 \times 2\mathbb{T}$. We identify the limit equations to be the primitive equations with the incompressible condition. The convergence behaviors are studied in both $\mathbb{R}^2 \times 2\mathbb{T}$ and $\mathbb{T}^2 \times 2\mathbb{T}$, respectively. This paper takes into account the high oscillating acoustic waves and is an extension of our previous work by X. Liu and E.S. Titi, Arch. Rational Mech. Anal., 238, 705-747, 2020.


Zero Mach number limit
As an example of multi-scale analysis, the zero Mach number limit problem for compressible flows has been an important classical problem in the study of hydrodynamic equations. Pioneered by Klainerman and Majda in [19,20], it is shown that the solutions of inviscid compressible Euler equations converge to that of inviscid incompressible Euler equations for the isentropic flows in R d and T d , d ∈ Z + . The initial data are well-prepared and almost incompressible. The result is further studied in bounded domains for nonisentropic flows by Schochet in [38,39]. In [43], the author establishes the zero Mach number limit with general (ill-prepared) initial data in R d . As pointed out in [33], in comparison to the case when the initial data are wellprepared, the time derivatives are no longer uniformly bounded with respect to the Mach number when the system is complemented with ill-prepared initial data. This leads to high frequency acoustic waves with large amplitude.
We have studied the zero Mach number limit of compressible primitive equations with well-prepared initial data in [29]. In this work, we are considering such a singular limit problem with ill-prepared initial data.
To present the general idea of singular limit problems with ill-prepared initial data, consider the following equation of the unknown function G, (1.1) in the kernel of operator L(∂ x ), which is called the non-singular part of the equation, it follows that ∂ t P ker(L(∂x)) G = P ker(L(∂x)) N (G) is uniformly bounded in H s ′ space for some s ′ ∈ Z + ∪ {0} with respect to ε. On the other hand, G − P ker(L(∂x)) G satisfies ∂ t (G − P ker(L(∂x)) G) + 1 ε L(∂ x )(G − P ker(L(∂x)) G) = N (G) − P ker(L(∂x)) N (G), (1.2) which has wave packet solutions with fast oscillations as ε → 0 + . Consequently, G − P ker(L(∂x)) G is oscillating at high frequency. Moreover, as ε → 0 + , the H s norm preserving property of the anti-symmetric operator L(∂ x ) implies that the amplitude of the oscillations does not vanish in general. Instead, the limit of the oscillatory part is driven by certain PDEs.
To resolve the singular limit problem of (1.1) as ε → 0 + , one has to study the interactions of the non-singular part and the oscillatory part in the nonlinearity N (G) in order to identify the limit equations. Such a method of studying the singular limit problem of (1.1) is developed by Schochet in [37] for hyperbolic PDEs with applications to the incompressible limit problem of Euler equations, nonlinear wave equations and the theory of weakly nonlinear geometric optics. Later, this method is further developed for some parabolic equations by Gallagher in [16]. We remark here that, if equation (1.1) is complemented with well-prepared initial data, the amplitude of oscillations is small, and consequently, the interaction between the non-singular part and the oscillatory part is much weaker.
In the study of zero Mach number limit of hydrodynamic equations for isentropic flows, if one writes the equations in the form of (1.1), the corresponding kernel of L(∂ x ) consists the solenoidal velocity field. The equations corresponding to the fast oscillation equations (1.2) are referred to as the acoustic wave equations. Using these terminologies, the theorem developped by Ukai in [43] is basically showing that the acoustic waves decay to zero as ε → 0 + in R d . Such a fact is the consequence of the Strichartz estimates for linear wave equations, as pointed out by Desjardins and Grenier in [10] (see, e.g., [17,18,25] about the Strichartz estimate). In T d , Lions and Masmoudi study the resonance of the high frequency osciallations and show that in the sense of distribution, the solutions to the compressible Navier-Stokes equations converge to that to the incompressible Navier-Stokes equations as the Mach number goes to zero in [27]. Later in [31], Masmoudi studies the incompressible, inviscid limit, with low Mach number and large Reynolds number, of compressible Navier-Stokes equations and identities the equations in both R d and T d . We refer readers, for further developments, to [6][7][8]40].
Additionally, we would like to mention that when considering the nonisentropic flows, the corresponding operator L(∂ x ) in (1.1) has coefficients depending on space and time variables. This causes non-trivial difficulties to study the resonances between the oscillations (see, e.g., [1-3, 33, 34]). Moreover, when taking into account the stratification effect of gravity, the compressible Navier-Stokes equations with gravity may converge to the Oberbeck-Boussinesq equations or the anelastic equations, depending on the strength of the gravity effect. We refer readers, for more discussions of related topics, to [4, 9, 11-15, 32, 35, 36, 44]. Also, for more multi-scale analysis, we refer readers to [21][22][23][24]30].

The compressible primitive equations
As mentioned before, we aim at studying the low Mach number limit of the compressible primitive equations. As discussed in our previous work [29], this is part of the justification of the PE diagram (see Figure 1 in [29]). We refer readers, for more background of the compressible primitive equations, to [29].
where P (ρ ε ) = ρ γ ε , S(v ε ) = µ(∇ h v ε + ∇ h v ⊤ ε ) + (λ − µ)div h v ε I 2 represent the pressure for isentropic flows and the viscous stress tensor for Newtonian flows, respectively. Here, we assume that µ, λ > 0 and γ > 1. We consider Ω h = R 2 or T 2 in this paper, where T 2 represents the periodic domain with period 1 in both directions in R 2 . (CPE) is complemented with the stress-free and non-permeable boundary conditions: ∂ z v ε z=0,1 = 0, w ε z=0,1 = 0. (BC-CPE) Hereafter, we have and will use ∇ h , div h and ∆ h to represent the horizontal gradient, the horizontal divergence, and the horizontal Laplace operator, respectively; that is, Notice, (CPE) with (BC-CPE) is invariant with respect to the following symmetry: v ε and w ε are even and odd, respectively, in the z-variable.
Owing to such symmetry, in order to study the limit system of (CPE) as ε → 0 + , it suffices to consider the following system: in Ω h × 2T, (1.3) subject to the periodic boundary condition and symmetry (SYM). Here 2T is the periodic domain with period 2 in R.
We remark that the restrictions of solutions to (1.3) in Ω h × [0, 1] solve (CPE) with (BC-CPE), provided that the solutions exist and are regular enough.
We can rewrite (1.3) 2 as, provided that ρ ε > 0, Also from the continuity equation (1.3) 1 , one can derive We consider ρ γ−1 ε with perturbations around the constant state given by ρ γ−1 ε =ρ γ−1 ∈ (0, ∞). Then we define the perturbation variable ξ ε by . Without loss of generality, we takeρ ≡ 1. Hence (1.3) can be written as complemented with periodic initial data with symmetry (SYM). In fact, (1.4) is already in the form of (1.1). In this work, we study the asymptotic limit of (1.4) as ε → 0 + . In fact, we will demonstrate that the existence time of strong solutions to (1.4) has a uniform lower bound, independent of ε ∈ (0, ε 0 ) for some small ε 0 ∈ (0, 1). In addition, in the sense of distribution, the limit equations of (1.4), as ε → 0 + , are the primitive equations (2.9) with the incompressible condition, below. Also, the associated acoustic wave equations for the three dimensional system (1.4) are only two dimensional (see (4.20), below). Consequently, we are able to adopt the strategy of studying the acoustic wave equations for the compressible Navier-Stokes equations in R 2 and T 2 to investigate the oscillatory part of the equations. This is done in section 4, below. The rest of this work is organized as follows. In the next section, we will introduce some notations as well as some function spaces. Also, the main theorems in this work are stated in both Ω h = R 2 and Ω h = T 2 . Next, in section 3, we establish the uniform local well-posedness of strong solutions to system (1.4). This is done with uniform a priori estimates in section 3.1, a local existence theorem in section 3.2 and a continuity argument in section 3.3. In section 4, we first identity the primitive equations, i.e., (2.9), below, as the limit of system (1.4) as ε → 0 + in the sense of distribution. Then in section 4.1, we argue that the acoustic waves decay to zero in the case when Ω h = R 2 ; in section 4.2, we study the oscillation equations and identify the limit equations of oscillations in the case when Ω h = T 2 . Consequently, we conclude the compactness in both cases as stated in the main theorems, below.

Preliminaries and main theorems
In this work, we denote by as the average and the fluctuation of any function f = f (x, y, z) over the zvariable. We use ∂ h to denote the horizontal derivatives, i.e., ∂ h ∈ {∂ x , ∂ y }. ∂ t and ∂ z denote the time derivative and the vertical derivative, respectively.
For any function f , ∂ g f is sometimes denoted as f g , for g ∈ {t, x, y, z, h}. Similar notations are also adopted for higher order derivatives.
As in [26], we introduce the following function spaces. Denote by Then with respect to the L 2 -inner product, are the spaces of test functions and Sobolev functions in Ω h × 2T, defined similarly for s ∈ Z + . We define the projection operators P σ , P τ in the following. Let u ∈ C ∞ 0 (Ω h × 2T; R 2 ). Consider the elliptic problem Then P σ , P τ are the projection operators from C ∞ 0 to C ∞ σ , C ∞ τ , respectively. Also, by a density argument, P σ , P τ can act on functions in L 2 , H s , s ∈ (0, ∞). In particular, the standard elliptic estimates yield, provided that Thus P σ u ∈ H s σ , P τ u ∈ H s τ , u = P σ u + P τ u and with the compatibility conditions: Also, in the case when Ω h = R 2 , the far field condition is also imposed. We denote the constant M ∈ (0, ∞) satisfying to be the bound of initial data (ξ 0 , v 0 ). Now we describe our main theorems in this work. The first theorem is stating that the H 2 norms of the solutions to (1.4) are uniformly bounded with respect to ε, provided that ε is small enough.

Uniform stability
In this section, we will establish the uniform local existence of solutios to (1.4) with respect to ε ∈ (0, ε 0 ) for some ε 0 ∈ (0, ∞) small enough. This is done via a series of a priori estimates, a local well-posedness theorem and a continuity argument. To simplify the presentation, in this section, we shorten the notations by dropping the subscript ε. That is, we denote

A priori estimates
We first establish some a priori estimates, which are independent of ε. Indeed, for ε small enough, the a priori estimates obtained in this subsection allow us to establish a uniform existence time in subsection 3.3. We remind readers system (1.4): (1.4) We will show the following: . Then the following inequality holds: for any δ ∈ (0, 1) with corresponding C δ ≃ δ −1 , where H 1 (·), H 2 (·) are smooth and bounded functions of the arguments. Also, H 1 (0) = 0, H 2 (0) = 0. Moreover, with the same notations, below, we have the inequalities: In order to establish such a priori estimates, we first represent the vertical velocity w in terms of (ξ, v). In fact, after averaging (1.4) 1 in the z-variable, one has and consequently, after comparing the above equation with (1. Then the following representations of the vertical velocity and its derivatives hold (recall that α = 1/(γ − 1)): In the following, we separate the proof of Proposition 1 in three parts: estimates on the horizontal derivatives; estimates on the vertical derivatives; and estimates on the time derivatives.
Estimates on the horizontal derivatives where we have denoted by After taking the L 2 -inner product of (3.13) 2 with v hh in Ω h × 2T, it holds d dt On the other hand, after applying integration by parts and substituting (3.13) 1 , one has Then after writing e εξ εξ t |ξ hh | 2 d x, one has d dt (3.14) Now we will estimate the right-hand side of (3.14). After applying integration by parts, I 1 can be written as Then it follows, after substituting (3.9) and (3.10), that where we have applied the Minkowski, Hölder, Sobolev embedding, and Young inequalities. Hence we have Next, to estimate I 3 , I 4 , I 5 , applying the Hölder, Sobolev embedding, and Young inequalities yields, In order to estimate I 6 , I 7 , after substituting (1.4) 1 and applying integration by parts, it holds Therefore, one has where we have applied the Hölder, Sobolev embedding and Young inequalities. Similarly, I 2 can be estimated as following: Therefore, (3.14) implies, after summing up the estimates above with where are two regular functions of the arguments and H 1 (0) = H 2 (0) = 0. We will adopt the same notations for functions with such properties in this work. After taking the L 2 -inner product of (1.4) 2 , and the horizontal derivative of (1.4) 2 with v, v h , respectively, repeating similar arguments as above yields the following estimate: Estimates on the vertical derivatives Again, applying ∂ z to (3.19) yields, Next, we take the L 2 -inner produce of (3.20) with v zz . After applying integration by parts, one has d dt where After applying the Hölder, Sobolev embedding and Young inequalities, it holds, where we have substituted (3.9) and (3.11) into I 10 . Therefore, we have d dt Next, we establish the estimate of v hz . Apply ∂ h to (3.19). It follows Take the L 2 -inner product of (3.23) with v hz and apply integration by parts in the resultant equation. It follows, d dt where After substituting (3.12), (3.9) in I 13 and (3.10) in I 14 , applying the Hölder, Sobolev embedding, Minkowski and Young inequalities yields, Therefore, summing up the estimates above with ∂ h ∈ {∂ x , ∂ y } leads to, d dt Similarly, after taking the L 2 -inner product of (3.19) with v z and performing estimates as above, one has, d dt

Estimates on the time derivatives
Now we establish the final pieces of Proposition 1. After multiplying (1.4) 1 with εe εξ and averaging the resulting equation in the z-variable, one has Then applying the triangle inequality and the Sobolev embedding inequality in (3.27) leads to (3.28) Meanwhile, we have, after applying ∂ h to (3.27), Thus it holds, Consequently, (3.28) and (3.30) imply (3.2).
On the other hand, after applying the projection operator P σ (defined in (2.2)) to (1.4) 2 , we have the following: (3.31) In order to estimate the L 2 and H 1 norms of ∂ t P σ v, we apply the Hölder and Sobolev embedding inequalities as follows: Here ∂ ∈ {∂ x , ∂ y , ∂ z } denotes the spatial derivatives. To estimate the L 2 and H 1 norms of w∂ z v, we first substitute the identities in (3.8), (3.9), (3.10) to w, ∂ z w, ∂ h w, respectively, and write down the following: Therefore, after applying the Hölder, Minkowski, and Sobolev embedding inequalities, the following estimates hold: (3.33) Combining the above estimates, together with (2.3), we have shown (3.3) and (3.4).
In addition, notice that ∂ t ξ = ε −1 e −εξ ∂ t e εξ . From (3.2), one has On the other hand, (1.4) 2 , (3.32), and (3.33) yield where in the last inequality we have used the fact that This finishes the proof of Proposition 1.

Local-in-time a priori estimates and local well-posedness
In this subsection, we aim at establishing the following proposition: Then for some positive constant ε 0 ∈ (0, 1) small enough, any ε ∈ (0, ε 0 ), there exists T ε ∈ (0, ∞) such that there exists a unique strong solution (ξ, v) Moreover, there exist positive constants C 0 , C 1 independent of ε, and Proposition 2 can be shown by applying the Banach fixed point theorem. In the following, without going into too much details, we will only sketch the proper steps to construct this local strong solution.
Sketch of constructing strong solutions. Let ξ ′ , v ′ be regular enough functions. Consider the following linear system associated with (1.4): where w ′ is given by Then for (ξ ′ , v ′ ) with (ξ ′ , v ′ ) t=0 = (ξ 0 , v 0 ) and satisfying the same regularity and bounds as in (3.35) and (3.36), there is a unique solution to system (3.37) with initial data (ξ, v) t=0 = (ξ 0 , v 0 ), after applying the standard existence theory for linear hyperbolic and parabolic equations. Moreover, similar a priori estimates as in our previous work [28] show that the solution (ξ, v) satisfies the same regularity and bounds of norms in (3.35) and (3.36). We define the following function framework in order to apply the Banach fixed point theorem.
Consider the function space For any (ξ, v) ∈ Y Tε , define the norm Then (Y Tε , · Y Tε ) is a complete metric space. In addition, we define the to the solution to system (3.37) (3.42) with trivial extension outside the set X Tε . Then we show that, For ε 0 ∈ (0, 1) small enough and any ε ∈ (0, ε 0 ), there exists T ε ∈ (0, ∞), such that T : Y Tε → Y Tε is a contraction mapping in X Tε . Notice, once this is proved, then the Banach fixed point theorem implies that we have a unique strong solution to (1.4) with ε, T ε as described above.
In the rest of this subsection, we focus on showing that for ε ∈ (0, ε 0 ) with ε 0 small enough, and some T ε ∈ (0, ∞) depending on ε, we have: (2) T is a contraction mapping in X Tε with respect to the topology of Y Tε ; i.e., for any ( for some q ∈ (0, 1).
We will only show the corresponding a priori estimates to show (1) and (2). Proof of (1): Estimates of ξ. First, we shall derive the estimates of ξ from (3.37) 1 . We shall only show the highest order estimates. After applying ∂ hh to (3.37) 1 , we have Then after taking the L 2 -inner product of (3.44) with 2ξ hh in Ω h , it follows Similar arguments also hold for ξ h L 2 and ξ L 2 , and therefore we have Then applying the Grönwall inequality and the Hölder inequality yields 45) where we have chosen T ε sufficiently small in the last inequality and C 0 ∈ (1, ∞).
On the other hand, from (3.37) 1 , we have (3.46) Estimates of v. Next we shall present the estimates of v. Similarly, we will only sketch the highest order estimates. After applying ∂ hh to (3.37) 2 , one obtains (3.47) Then after taking the L 2 -inner product of (3.47) with 2v hh in Ω h × 2T, it follows, after applying integration by parts, where we have substituted the identity which is obtained by taking ∂ z to (3.38), Similarly as in the last section, from (3.38), we have Then following similar arguments as before, one can derive, for any δ ∈ (0, 1) and fixed ε ∈ (0, 1), We remind readers that we have been using {H i } i=1,2,3 to denote regular functions of the arguments with property H i (0) = 0. Details are listed below, for readers' reference: Similar estimates also hold for v hz L 2 , v zz L 2 , v h L 2 , v z L 2 , v L 2 . Then we have arrived at the estimate 50) where we have chosen ε ∈ (0, ε 0 ), with ε 0 small enough such that εα ξ ′ H 2 ≤ εα(C 0 M 0 ) 1/2 ≤ log 2, and thus 1/2 ≤ e −εαξ ′ < 2. (3.51) Next, we choose ε 0 , δ small enough such that Then for ε ∈ (0, ε 0 ), after applying the Grönwall inequality to (3.50), we have (3.53) Now we let δ small enough such that Then for fixed ε 0 , δ satisfying (3.52) and (3.54), and ε ∈ (0, ε 0 ), let T ε small enough, depending on ε, δ, such that (3.55) Then (3.53) yields (3.56) Here we have required C 1 to be sufficiently large such that On the other hand, from (3.37) 2 , we have after applying similar arguments as in (3.32) and (3.33). Then we have the following (3.58) This finishes the proof of (1). Proof of (2): Denote by (ξ 12 , v 12 ) : . Then (ξ 12 , v 12 ) satisfies the following system: in Ω h × 2T, 59) with (ξ 12 , v 12 ) t=0 = 0. Then after taking the L 2 -inner product of (3.59) 1 with 2ξ 12 in Ω h , one has, for any δ ∈ (0, 1) with corresponding C δ ≃ δ −1 , where we have applied the Hölder, Sobolev embedding and Young inequalities. Therefore, applying the Grönwall inequality in the above inequality yields, (3.60) On the other hand, after taking the L 2 -inner product of (3.59) 2 with 2v 12 in Ω h × 2T and applying integration by parts, one has where we have substituted (3.38) for w ′ i , i = 1, 2. Consequently, after applying similar arguments as before, we have the following estimates of the terms on the right-hand side: for any δ ∈ (0, 1) and some constant C δ ≃ δ −1 , where H(·), as before, is a regular function of the arguments. Consequently, one has Notice that C 0 , C 1 are independent of ε. For ε 0 , δ ∈ (0, 1) small enough and ε ∈ (0, ε 0 ], applying the Grönwall inequality in the above inequality yields

Uniform stability
In this section, we will show that the existence time T ε of the strong solutions constructed in Propositive 2 is uniform in ε provided ε ∈ (0, ε 1 ) for some ε 1 ∈ (0, ε 0 ). In order to show this, it suffices to show that there is a positive constant ε 1 such that the H 2 -norms of (ξ, v) remain bounded in a time interval (0, T ) with T ∈ (0, ∞) independent of ε, provided ε ∈ (0, ε 1 ) with ε 1 ∈ (0, 1) small enough. We perform a continuity argument in the following. Recall that we are given initial data (ξ 0 , v 0 ) ∈ H 2 (Ω h × 2T), and a positive constant M ∈ (0, ∞) satisfying Denote by Then it is obvious that From Proposition 2, there is a strong solution (ξ, v) satisfying (3.35) and (3.36) in the time interval [0, T ε ], for some T ε ∈ (0, ∞). Then for any t ∈ [0, T ε ], we have where C 0 , C 1 are given in Proposition 2. We remind readers that T ε depends on M 0 , M 1 and ε. On the other hand, consider ε 1 satisfying where H 1 is as in the right-hand side of (3.1). Then the a priori estimate in (3.1) implies that, for ε ∈ (0, ε 1 ) and t ∈ [0, T ε ], Hence after substituting (2.6) and choosing Notice that T is independent of ε once ε ∈ (0, ε 1 ). Also from (3.65), we have (3.66) Therefore, supposed that T ε < T , we will have Then by setting (ξ(T ε ), v(T ε )) as the new initial data, applying Proposition 2 again, estimate (3.63) holds in [0, 2T ε ]. Thus the arguments between (3.63) and (3.65) hold with T ε replaced by 2T ε , without needing to choose the smallness of ε 1 and T . Then estimate (3.65) holds in the time interval [0, min{2T ε , T }]. Repeat such arguments n times, n ∈ Z + , until nT ε ≥ T . This extends the existence time of the local strong solution (ξ, v) to (1.4) to T > 0, which is independent of ε provided ε ∈ (0, ε 1 ) with ε 1 given as above. Consequently, we conclude that: (3.67)
Proof. Indeed (4.31) is a direct consequence of rescaling the temporal variable and replacing φ 0 with (I − ∆ h ) s/2 φ 0 in (4.29). In order to show (4.32), notice that after applying the Minkowski and Hölder inequalities, one has Then by employing the Fubini theorem, it holds This finishes the proof of (4.32).

The case when Ω h = T 2 : the fast oscillation
We will establish the convergence behavior of U ε in the case when Ω h = T 2 in this subsection. Indeed, we shall investigate the fast oscillations of the acoustic waves as ε → 0 + . This is motivated by [37] (see also [16]).

The oscillation equations and the convergence of oscillations
To begin with, (4.20) 2 can be written as (4.33) Moreover, for any u ∈ (D ′ (T 2 ×2T)) 2 , consider u(x, y) = 1 0 u dz ∈ (D ′ (T 2 )) 2 as a function on T 2 × 2T. One has P τ u = P τ u. Then one has, after applying integration by parts and substituting (3.9), where we have used the facts that 1 0 P σ v ε dz = 0, v ε = P σ v ε and that P τ v ε , ξ ε are independent of the z-variable.
On the other hand, notice that (4.20) 1 can be written as Additionally, integrating (4.35) in T 2 yields Consequently, after combining equations (4.33), (4.34), (4.35), (4.36), we have the following system of oscillations: where In the following, denote by 2 will be referred to as the first and the second components of L, respectively.

The limit equations of oscillations
In order to identify the limit equations of (4.40) and (4.43) as ε → 0 + , we first introduce the Fourier representation of the operators defined in (4.41).
Notice that (4.27) implies T 2 V o ε dxdy = T 2 U o ε dxdy = 0. It suffices to study in the space consisting of functions in D ′ (T 2 ) 3 with zero average.
where, hereafter, · c represents the complex conjugate.
The limit equations. Now we have prepared enough to identify the limit equations of equations (4.40) and (4.43) as ε → 0 + .
We rewrite equation (4.40) in the following fashion: (4.68) Then, from the regularity in (4.2) and (4.50), the weak convergence of ∂ t V o ε in (4.51), and the convergence of operators in (4.63), (4.64), (4.65), (4.66) and (4.67), as ε → 0 + , the left-hand side of (4.68) converges in the sense of distribution to Indeed, one can replace v p , V o with their finite dimensional Fourier truncations on the left-hand side of (4.68), and similar estimates as in (4.48) of the operators for such truncations imply that the actions of the operators on the remainders are bounded by certain norms of the remainders uniformly in ε.
Therefore, one only has to investigate, (4.70) where we have substituted the representation (4.58), and it is represented, using the relation (4.59) and (4.60), Together with the norm preserving property (4.44) and the fact that  This finishes the proof of (4.69). Therefore, we have identified the limit equation of (4.40):