On admissible positions of transonic shocks for steady Euler flows in a 3-D axisymmetric cylindrical nozzle

Abstract

This paper concerns with the existence of transonic shocks for steady Euler flows in a 3-D axisymmetric cylindrical nozzle, which are governed by the Euler equations with the slip boundary condition on the wall of the nozzle and a receiver pressure at the exit. Mathematically, it can be formulated as a free boundary problem with the shock front being the free boundary to be determined. In dealing with the free boundary problem, one of the key points is determining the position of the shock front. To this end, a free boundary problem for the linearized Euler system will be proposed, whose solution gives an initial approximating position of the shock front. Compared with the 2-D case, new difficulties arise due to the additional 0-order terms and singularities along the symmetric axis. New observations and careful analysis will be done to overcome these difficulties. Once the initial approximation is obtained, a nonlinear iteration scheme can be carried out, which converges to a transonic shock solution to the problem.

Introduction

In this paper, we are concerned with the existence of steady transonic shocks, especially the position of the shock front, in a 3-D axisymmetric cylindrical nozzle (see Fig. 1.1). The steady flow in the nozzle is supposed to be governed by the Euler system which reads:

$$\mathrm{div}(\rho \mathbf{u})=0,$$

$$\mathrm{div}(\rho \mathbf{u}\otimes \mathbf{u})+\mathrm{\nabla }p=0,$$

$$\mathrm{div}(\rho (e+\frac{1}{2}|\mathbf{u}{|}^2+\frac{p}{\rho })\mathbf{u})=0,$$

where $\mathbf{x}:=(x_1,x_2,x_3)$ are the space variables, “div” is the divergence operator with respect to x, $\mathbf{u}={(u_1,u_2,u_3)}^T$ is the velocity field, ρ, p and e stand for the density, pressure, and the internal energy respectively. Moreover, the fluid is supposed to be a polytropic gas with the state equation

$$p=A(s){\rho }^{\gamma },$$

where s is the entropy, $\gamma >1$ is the adiabatic exponent, and $A(s)=(\gamma -1)\exp (\frac{s-s_0}{c_v})$ with $c_v$ the specific heat at constant volume.

It is well-known, in the Euler system, the equation (1.3) can be replaced by the Bernoulli's law below:

$$\mathrm{div}(\rho \mathbf{u}B)=0,$$

where the Bernoulli constant B is given by, with $i=\frac{\gamma p}{(\gamma -1)\rho }$ being the enthalpy,

$$B=\frac{1}{2}|\mathbf{u}{|}^2+i.$$

For Euler flows, a shock front is a strong discontinuity of the distribution functions of the state parameters $U_{\mathcal{N}}={(u_1,u_2,u_3,p,\rho )}^T$ of the fluid. Let ${\mathrm{\Gamma }}_{\mathrm{s}}^{\mathcal{N}}:=\{x_1={\psi }_{\mathcal{N}}(x^{\prime })\}$, with $x^{\prime }:=(x_2,x_3)$, be the position of a shock front, then the following Rankine-Hugoniot (R-H) conditions should be satisfied:

$$[(1,-{\mathrm{\nabla }}_{x^{\prime }}{\psi }_{\mathcal{N}}(x^{\prime }))\cdot \rho \mathbf{u}]=0,$$

$$[((1,-{\mathrm{\nabla }}_{x^{\prime }}{\psi }_{\mathcal{N}}(x^{\prime }))\cdot \rho \mathbf{u})\mathbf{u}]+{(1,-{\mathrm{\nabla }}_{x^{\prime }}{\psi }_{\mathcal{N}}(x^{\prime }))}^T[p]=0,$$

$$[(1,-{\mathrm{\nabla }}_{x^{\prime }}{\psi }_{\mathcal{N}}(x^{\prime }))\cdot \rho \mathbf{u}B]=0,$$

where $[\cdot ]$ denotes the jump of the quantity across the shock front ${\mathrm{\Gamma }}_{\mathrm{s}}^{\mathcal{N}}$, and ${\mathrm{\nabla }}_{x^{\prime }}:=({\partial }_{x_2},{\partial }_{x_3})$. Moreover, the entropy condition $[p]>0$ should also hold, which means that the pressure increases across ${\mathrm{\Gamma }}_{\mathrm{s}}^{\mathcal{N}}$.

Given a flat cylindrical nozzle:

$$\mathcal{N}:=\{(x_1,x_2,x_3)\in {\mathbb{R}}^3:0<x_1<L,0\le x_2^2+x_3^2<1\},$$

with the entrance ${\mathcal{N}}_0$, the exit ${\mathcal{N}}_L$, and the wall ${\mathcal{N}}_w$ being

$${\mathcal{N}}_0:=\mathcal{N}\cap \{(x_1,x_2,x_3)\in {\mathbb{R}}^3:x_1=0\},{\mathcal{N}}_L:=\mathcal{N}\cap \{(x_1,x_2,x_3)\in {\mathbb{R}}^3:x_1=L\},{\mathcal{N}}_w:=\mathcal{N}\cap \{(x_1,x_2,x_3)\in {\mathbb{R}}^3:x_2^2+x_3^2=1\},$$

it is well-known that there may exist plane normal shocks in it with the shock front being a plane perpendicular to the $x_1$-axis (see Fig. 1.1). Let $x_1={\overline{x}}_{\mathrm{s}}$ with ${\overline{x}}_{\mathrm{s}}\in (0,L)$ be the position of the plane shock front, ${\overline{U}}_{{\mathcal{N}}_{-}}:=({\overline{q}}_{-},0,0,{\overline{p}}_{-},{\overline{\rho }}_{-})$ be the state of the uniform supersonic flow ahead of it, and ${\overline{U}}_{{\mathcal{N}}_{+}}:=({\overline{q}}_{+},0,0,{\overline{p}}_{+},{\overline{\rho }}_{+})$ be the state of the uniform subsonic flow behind it. (In this paper, the subscript “−” will be used to denote the parameters ahead of the shock front and the subscript “+” behind the shock front.) Then the R-H conditions (1.6)-(1.8) yield

$$\{\begin{array}{cc}\hfill & [\overline{\rho }\overline{q}]={\overline{\rho }}_{+}{\overline{q}}_{+}-{\overline{\rho }}_{-}{\overline{q}}_{-}=0,\hfill \\ \hfill & [\overline{p}+\overline{\rho }{\overline{q}}^2]=({\overline{p}}_{+}+{\overline{\rho }}_{+}{\overline{q}}_{+}^2)-({\overline{p}}_{-}+{\overline{\rho }}_{-}{\overline{q}}_{-}^2)=0,\hfill \\ \hfill & [\overline{B}]={\overline{B}}_{+}-{\overline{B}}_{-}=0.\hfill \end{array}$$

Then direct computations yield that

$${\overline{p}}_{+}=((1+{\mu }^2){\overline{M}}_{-}^2-{\mu }^2){\overline{p}}_{-},$$

$${\overline{q}}_{+}={\mu }^2({\overline{q}}_{-}+\frac{2}{\gamma -1}\frac{{\overline{c}}_{-}^2}{{\overline{q}}_{-}}),$$

$${\overline{\rho }}_{+}=\frac{{\overline{\rho }}_{-}{\overline{q}}_{-}}{{\overline{q}}_{+}}=\frac{{\overline{\rho }}_{-}{\overline{q}}_{-}^2}{{\overline{c}}_{⁎}^2},$$

where $c=\sqrt{\frac{\gamma p}{\rho }}$ is the sonic speed, $M=\frac{q}{c}$ is the Mach number, and

$${\overline{c}}_{⁎}^2:={\overline{q}}_{+}{\overline{q}}_{-}={\mu }^2({\overline{q}}_{-}^2+\frac{2}{\gamma -1}{\overline{c}}_{-}^2),{\mu }^2=\frac{\gamma -1}{\gamma +1},{\overline{M}}_{-}^2=\frac{{\overline{q}}_{-}^2}{{\overline{c}}_{-}^2}.$$

Remark 1.1

By applying the entropy condition $[\overline{p}]>0$, the equation (1.11) yields that ${\overline{M}}_{-}>1$, that is, ${\overline{q}}_{-}>{\overline{c}}_{⁎}$. Then, since ${\overline{q}}_{+}{\overline{q}}_{-}={\overline{c}}_{⁎}^2$, it is obvious that ${\overline{q}}_{+}<{\overline{c}}_{⁎}$, which implies that ${\overline{M}}_{+}<1$. That is, the flow is supersonic ahead of the shock front, and is subsonic behind it.

Remark 1.2

For each ${\overline{x}}_{\mathrm{s}}\in (0,L)$, it is obvious that $({\overline{U}}_{{\mathcal{N}}_{+}};{\overline{U}}_{{\mathcal{N}}_{-}};{\overline{x}}_{\mathrm{s}})$ gives a plane transonic shock solution to the steady Euler system (1.1)-(1.3) in the following sense: the position of the shock front is ${\overline{\mathrm{\Gamma }}}_{\mathrm{s}}^{\mathcal{N}}:=\{x_1={\overline{x}}_{\mathrm{s}},x_2^2+x_3^2\le 1\}$, the state of the fluid $U_{\mathcal{N}}\equiv {\overline{U}}_{{\mathcal{N}}_{-}}$ within the region between ${\mathcal{N}}_0$ and ${\overline{\mathrm{\Gamma }}}_{\mathrm{s}}^{\mathcal{N}}$ in the nozzle $\mathcal{N}$, and the state of the fluid $U_{\mathcal{N}}\equiv {\overline{U}}_{{\mathcal{N}}_{+}}$ within the region between ${\overline{\mathrm{\Gamma }}}_{\mathrm{s}}^{\mathcal{N}}$ and ${\mathcal{N}}_L$. Therefore, as the state ${\overline{U}}_{{\mathcal{N}}_{-}}$ is given, the state of the flow behind the shock front ${\overline{U}}_{{\mathcal{N}}_{+}}$ is uniquely determined by (1.11)-(1.13), while the position of the plane shock front could be arbitrary in the flat nozzle $\mathcal{N}$.

Based on the above steady plane normal shock solutions in a flat cylindrical nozzle, this paper is going to study the existence of transonic shocks in an axisymmetric 3-D nozzle, which is a small perturbation of a flat cylindrical nozzle, with a pressure condition at the exit.

Let $(z,r,\varpi )$ be the cylindrical coordinate in ${\mathbb{R}}^3$ such that

$$(x_1,x_2,x_3)=(z,r\cos \varpi ,r\sin \varpi ),$$

and the perturbed nozzle is axisymmetric with respect to $x_1$-axis:

$$\tilde{\mathcal{N}}:=\{(z,r,\varpi )\in {\mathbb{R}}^3:0<z<L,0<r<1+\phi (z)\},$$

where, with $\sigma >0$ sufficiently small and $\mathrm{\Theta }(\cdot )$ a given function defined on $[0,L]$,

$$\phi (z):=\underset{0}{\overset{z}{\int }}\tan (\sigma \mathrm{\Theta }(\tau ))\mathrm{d}\tau .$$

In this paper, we further assume that the states of the fluid in the nozzle are also axisymmetric with respect to $x_1$-axis such that $U_{\mathcal{N}}$ are independent of ϖ, and

$$-u_2\sin \varpi +u_3\cos \varpi =0.$$

Then direct computations yield that the 3-D steady Euler equations (1.1)-(1.3) are reduced to

$$\{\begin{array}{c}{\partial }_z(r\rho u)+{\partial }_r(r\rho v)=0,\hfill \\ \hfill \\ {\partial }_z(\rho uv)+{\partial }_r(p+\rho v^2)+\frac{\rho v^2}{r}=0,\hfill \\ \hfill \\ {\partial }_z(p+\rho u^2)+{\partial }_r(\rho uv)+\frac{\rho uv}{r}=0,\hfill \\ \hfill \\ {\partial }_z(r\rho uB)+{\partial }_r(r\rho vB)=0,\hfill \end{array}$$

where

$$u=u_1,v=u_2\cos \varpi +u_3\sin \varpi .$$

Hence, it suffices to determine $U:=(\theta ,p,q,s)$, with $\theta =\arctan \frac{v}{u}$ the flow angle and $q=\sqrt{u^2+v^2}$ the magnitude of the flow velocity, as functions of variables $(z,r)$.

Moreover, under the cylindrical coordinate, the position of a shock front which is axisymmetric with respect to $x_1$-axis can be denoted as $\{z={\psi }_{\tilde{\mathcal{N}}}(r)\}$, and the Rankine-Hugoniot conditions (1.6)-(1.8) across it become

$$[\rho u]-{\psi }_{\tilde{\mathcal{N}}}^{\prime }[\rho v]=0,$$

$$[\rho uv]-{\psi }_{\tilde{\mathcal{N}}}^{\prime }[p+\rho v^2]=0,$$

$$[p+\rho u^2]-{\psi }_{\tilde{\mathcal{N}}}^{\prime }[\rho uv]=0,$$

$$[B]=0.$$

In the $(z,r)$-space, let

$$\tilde{\mathcal{N}}:=\{(z,r)\in {\mathbb{R}}^2:0<z<L,0<r<1+\phi (z)\},$$

denotes the domain of the nozzle with the boundaries

$$E_0=\{(z,r)\in {\mathbb{R}}^2:z=0,0<r<1+\phi (0)\},W_0=\{(z,r)\in {\mathbb{R}}^2:0<z<L,r=0\},E_L=\{(z,r)\in {\mathbb{R}}^2:z=L,0<r<1+\phi (L)\},W_{\phi }=\{(z,r)\in {\mathbb{R}}^2:0<z<L,r=1+\phi (z)\},$$

in which $E_0$ is the entry, $E_L$ is the exit, $W_{\phi }$ is the nozzle wall, and $W_0$ denotes the symmetry axis of the nozzle. In this paper, we are going to determine the steady flow pattern with a single shock front in $\tilde{\mathcal{N}}$, satisfying the slip boundary condition on the nozzle wall $W_{\phi }$, for given supersonic states at the entry $E_0$ and given pressure condition at the exit $E_L$. The problem is formulated as a free boundary problem described in detail below.

The Free Boundary Problem FBPC

Let $\alpha \in (\frac{1}{2},1)$. Given ${\overline{U}}_{-}:=(0,{\overline{p}}_{-},{\overline{q}}_{-},{\overline{s}}_{-})$, $P_{\mathrm{e}}\in {\mathcal{C}}^{2,\alpha }({\overline{\mathbb{R}}}_{+})$, $\mathrm{\Theta }\in {\mathcal{C}}^{2,\alpha }([0,L])$, try to determine the states of the fluid U in the nozzle with a single shock front ${\mathrm{\Gamma }}_{\mathrm{s}}^{\tilde{\mathcal{N}}}:=\{z={\psi }_{\tilde{\mathcal{N}}}(r)\}$ (see Fig. 1.2) such that:

• The nozzle domain $\tilde{\mathcal{N}}$ is divided by ${\mathrm{\Gamma }}_{\mathrm{s}}^{\tilde{\mathcal{N}}}$ into two parts:

  $${\tilde{\mathcal{N}}}_{-}=\{(z,r)\in {\mathbb{R}}^2:0<z<{\psi }_{\tilde{\mathcal{N}}}(r),0<r<1+\phi (z)\},$$

  $${\tilde{\mathcal{N}}}_{+}=\{(z,r)\in {\mathbb{R}}^2:{\psi }_{\tilde{\mathcal{N}}}(r)<z<L,0<r<1+\phi (z)\},$$

  where ${\tilde{\mathcal{N}}}_{-}$ denotes the region of the supersonic flow ahead of the shock front, and ${\tilde{\mathcal{N}}}_{+}$ is the region of the subsonic flow behind it;

• In ${\tilde{\mathcal{N}}}_{-}$, the states of the fluid $U=U_{-}(z,r)$, which satisfies the Euler system (1.17), given supersonic state at the entry of the nozzle

  $$U_{-}={\overline{U}}_{-},\text{on}{E}_0,$$

  and the slip boundary condition on the wall of the nozzle

  $${\theta }_{-}=\sigma \mathrm{\Theta }(z),\text{on}{W}_{\phi }\cap \stackrel{‾}{{\tilde{\mathcal{N}}}_{-}};$$

• In ${\tilde{\mathcal{N}}}_{+}$, the states of the fluid $U=U_{+}(z,r)$, which satisfies the Euler system (1.17), the slip boundary condition on the wall of the nozzle

  $${\theta }_{+}=\sigma \mathrm{\Theta }(z),\text{on}{W}_{\phi }\cap \stackrel{‾}{{\tilde{\mathcal{N}}}_{+}},$$

  and given pressure at the exit of the nozzle

  $$p_{+}=p_{\mathrm{e}}(L,r):={\overline{p}}_{+}+\sigma P_{\mathrm{e}}(r),\text{on}{E}_L;$$

• On the shock front ${\mathrm{\Gamma }}_{\mathrm{s}}^{\tilde{\mathcal{N}}}$, the Rankine-Hugoniot conditions (1.19)-(1.22) hold for the states $(U_{-},U_{+})$;

• Finally, on the axis ${\mathrm{\Gamma }}_2$, under the assumption of axisymmetric, both $U_{-}$ and $U_{+}$ satisfy

  $${\theta }_{-}=0,{\partial }_r(p_{-},q_{-},s_{-})=0,{\partial }_r^2{\theta }_{-}=0,\text{on}{W}_0\cap \stackrel{‾}{{\tilde{\mathcal{N}}}_{-}},$$

  and

  $${\theta }_{+}=0,{\partial }_r(p_{+},q_{+},s_{+})=0,{\partial }_r^2{\theta }_{+}=0,\text{on}{W}_0\cap \stackrel{‾}{{\tilde{\mathcal{N}}}_{+}}.$$

Remark 1.3

$\alpha \in (\frac{1}{2},1)$ is a sufficient condition, which is needed to establish a prior estimates of the solution. One is referred to Section 3 for details.

Remark 1.4

The condition (1.31) guarantees that the compatibility conditions hold for the supersonic solution (see Lemma 4.1 in Section 4). The condition (1.32) will be verified in Section 3 - Section 6.

This paper will deal with the problem FBPC and establish the existence of the transonic shock solution in the 3-D axisymmetric nozzle by showing the following theorem.

Theorem 1.5

Assume that

$$\mathit{\Theta }(z)>0,\mathit{\text{for any}}z\in (0,L),$$

and

$$\mathit{\Theta }(0)={\mathit{\Theta }}^{\prime }(0)={\mathit{\Theta }}^{″}(0)=0.$$

Let

$$\mathcal{R}(z):=\underset{0}{\overset{L}{\int }}\mathit{\Theta }(\tau )\mathrm{d}\tau -\dot{k}\underset{0}{\overset{z}{\int }}\mathit{\Theta }(\tau )\mathrm{d}\tau ,$$

$${\mathcal{P}}_{\mathrm{e}}:=2\frac{1-{\overline{M}}_{+}^2}{{\overline{\rho }}_{+}^2{\overline{q}}_{+}^3}\underset{0}{\overset{1}{\int }}t{P}_{\mathrm{e}}(t)\mathrm{d}t,$$

with $\dot{k}:=[\overline{p}](\frac{\gamma -1}{\gamma {\overline{p}}_{+}}+\frac{1}{{\overline{\rho }}_{+}{\overline{q}}_{+}^2})>0$, such that

$${\mathcal{R}}^{⁎}:=\underset{z\in (0,L)}{\sup }\mathcal{R}(z)=\underset{0}{\overset{L}{\int }}\mathit{\Theta }(\tau )\mathrm{d}\tau ,{\mathcal{R}}_{⁎}:=\underset{z\in (0,L)}{\inf }\mathcal{R}(z)=(1-\dot{k})\underset{0}{\overset{L}{\int }}\mathit{\Theta }(\tau )\mathrm{d}\tau .$$

Then, if

$${\mathcal{R}}_{⁎}<{\mathcal{P}}_{\mathrm{e}}<{\mathcal{R}}^{⁎},$$

then there exists a sufficiently small constant ${\sigma }_0>0$, such that for any $0<\sigma \le {\sigma }_0$, there exists at least a transonic shock solution $(U_{-},U_{+};{\psi }_{\tilde{\mathcal{N}}}(r))$ to the free boundary problem FBPC.

Remark 1.6

In Theorem 1.5, the assumption (1.33) is imposed in order for the simplicity of the presentation of this paper. The existence of the transonic shock solutions can also be established for general $\mathrm{\Theta }(\cdot )$ if (1.38) holds and there exists $z_{⁎}\in (0,L)$ such that

$$\mathcal{R}(z_{⁎})={\mathcal{P}}_{\mathrm{e}},\text{and }\mathrm{\Theta }(z_{⁎})\ne 0.$$

Actually, for general $\mathrm{\Theta }(\cdot )$, similar as the existence of transonic shock solutions in a 2-D nozzle established in [9], there may exist more than one transonic shock solutions to the free boundary problem FBPC.

In dealing with the free boundary problem FBPC, one of the key difficulties is determining the position of the shock front. However, there is no information on it since the problem FBPC is going to be solved near the steady plane normal shock solutions and, as pointed out in Remark 1.2, the position of the shock front can be arbitrary in the flat nozzle. This difficulty also arises for 2-D transonic shock problem in an almost flat nozzle and Fang and Xin successfully overcome it in [9] by designing a free boundary problem of the linearized 2-D Euler system based on the background normal shock solution, which provides information on the position of the shock front as long as it is solvable: the free boundary can be regarded as an initial approximating position of the shock front. It turns out that this idea also works for the 3-D axisymmetric case studied in this paper, and an initial approximating position of the shock front can be obtained by solving the free boundary problem of the linearized 3-D axisymmetric Euler system based on the plane normal shocks (see Section 4 for details). Different from the problem for the 2-D case in [9], there exist additional 0-order terms and singularities along the symmetric axis in the linearized Euler system for the 3-D axisymmetric case. These differences will bring new difficulties in solving the free boundary problem and determining the position of the free boundary. They need further observations and careful analysis which will be done in this paper. Once the initial approximation of the transonic shock solution is obtained, nonlinear iteration process similar as in [9] can be executed which converges to a transonic shock solution to the problem FBPC.

The study on gas flows with shocks in a nozzle plays a fundamental role in the operation of turbines, wind tunnels and rockets. Thanks to steady efforts made by many mathematicians, there have been plenty of results on it from different viewpoints and for different models, for instance, see [1], [2], [3], [4], [5], [6], [7], [9], [16], [17], [21], [22], [23], [24], [25] and references therein. For steady multi-dimensional flows with shocks in a finite nozzle, in order to determine the position of the shock front, Courant and Friedrichs pointed out in [7] that, without rigorous mathematical analysis, additional conditions are needed to be imposed at the exit of the nozzle and the pressure condition is preferred among different possible options. From then on, many mathematicians have been working on this issue and there have been many substantial progresses. In particular, Chen-Feldman proved in [3] the existence of transonic shock solutions in a finite flat nozzle for multi-dimensional potential flows with given potential value at the exit and an assumption that the shock front passes through a given point. Later in [2], with given vertical component of the velocity at the exit, Chen-Chen-Song established the existence of the shock solutions for the 2-D steady Euler flows. See also [19] for a recent result for the 3-D axisymmetric case. Both existence results are established under the assumption that the shock front passes through a given point, which is employed to deal with the same difficulty as the problem in this paper that the position of the shock front of the unperturbed shock solutions can be arbitrary in the nozzle. Without such an artificial assumption, recently in [9], Fang-Xin establish the existence of the transonic shock solutions in an almost flat nozzle with the pressure condition at the exit, as suggested by Courant-Friedrichs in [7]. It is interesting that the results in [9] indicate that, for a generic nozzle and given pressure condition at the exit, there may exist more than one shock solutions, that is, there may exist more than one admissible positions of the shock front. It should be noted that, in a diverging nozzle which is an expanding angular sector, the position of the shock front can be uniquely determined by the pressure condition at the exit under the assumption that the flow states depend only on the radius (see [7]). And the structural stability of this shock solution under small perturbation of the nozzle boundary as well as the pressure condition at the exit has been established for the 2-D case in a series of papers [15], [17] by Li-Xin-Yin and in [6] by Chen. See also [20], [21] for a recent advance towards the 3-D axisymmetric case.

The paper is organized as follows. In Section 2, the problem FBPC is reformulated by a modified Lagrange transformation, introduced by Weng-Xie-Xin in [21], which straightens the stream line without the degeneracy along the symmetric axis. Then the free boundary problem for the linearized Euler system is described, which serves to determine the initial approximation. Finally, the main theorem to be proved is given. In Section 3, we shall establish a well-posed theory for boundary value problem of the elliptic sub-system of the linearized Euler system at the subsonic state behind the shock front. It turns out that there exists a solution to the problem if and only if a solvability condition is satisfied for the boundary data. This solvability condition will be employed to determine the position of the free boundary. In Section 4, we prove the existence of the initial approximation by applying the theorem proved in Section 3. Then a nonlinear iteration scheme will be described, starting from the initial approximation, in Section 5. Finally, in Section 6, the nonlinear iteration scheme will be verified to be well-defined and contractive, which concludes the proof for the main theorem.