## The Boundary Layer for MHD Equations in a Plane-Parallel Channel

Wang Na,1, Wang Shu2

 基金资助: 国家自然科学基金.  11371042河北师范大学科技类博士（后）基金.  L2019B03

 Fund supported: the NSFC.  11371042the (Post) Doctor Fund of Science and Technology of Hebei Normal University.  L2019B03

Abstract

In this paper, we study the boundary layer problem for the incompressible MHD equations in a plane-parallel channel. Using the multiscale analysis and the careful energy method, we prove the convergence of the solution of viscous and diffuse MHD equations to that of the ideal MHD equations as the viscosity and magnetic diffusion coefficient tend to zero.

Keywords： Incompressible MHD equations ; Boundary layer ; Plane-parallel channel

Wang Na, Wang Shu. The Boundary Layer for MHD Equations in a Plane-Parallel Channel. Acta Mathematica Scientia[J], 2019, 39(4): 738-760 doi:

## 1 引言

$$$\left\{\begin{array}{ll} & {\partial_{t}u-\varepsilon_{1}\Delta u+u \cdot \nabla u-b\cdot \nabla b+\nabla p = F, }\\ & {\partial_{t}b-\varepsilon_{2}\Delta b+u\cdot\nabla b-b\cdot\nabla u = G, }\\ & {\nabla\cdot u = 0, \ \nabla\cdot b = 0, } \\ &{u|_{t = 0} = u_{0}, \ b|_{t = 0} = b_{0}, }\\ &{u|_{\partial\Lambda } = \alpha, \ b|_{\partial \Lambda} = \beta, } \end{array} \right.$$$

$$$u_{0} = (m_{1}(z), m_{2}(x, z), 0), \ b_{0} = (n_{1}(z), n_{2}(x, z), 0),$$$

$$$F = (F_{1}(t, z), F_{2}(t, x, z), 0), \ G = (G_{1}(t, z), G_{2}(t, x, z), 0),$$$

$$$u = (u_{1}(t, z), u_{2}(t, x, z), 0), \ b = (b_{1}(t, z), b_{2}(t, x, z), 0).$$$

$$$\left\{\begin{array}{ll} & {\partial_{t}u_{1}-\varepsilon_{1}\partial_{z}^{2}u_{1} = F_{1}, }\\ & {\partial_{t}u_{2}-\varepsilon_{1}(\partial_{x}^{2}u_{2}+\partial_{z}^{2}u_{2}) +u_{1}\partial_{x}u_{2}-b_{1}\partial_{x}b_{2} = F_{2}, }\\ & {\partial_{t}b_{1}-\varepsilon_{2}\partial_{z}^{2}b_{1} = G_{1}, }\\ & {\partial_{t}b_{2}-\varepsilon_{2}(\partial_{x}^{2}b_{2}+\partial_{z}^{2}b_{2}) +u_{1}\partial_{x}b_{2}-b_{1}\partial_{x}u_{2} = G_{2}.}\\ \end{array} \right.$$$

$$$\left\{\begin{array}{ll} & {u_{1}|_{t = 0} = m_{1}(z), \ u_{2}|_{t = 0} = m_{2}(x, z), }\\ & {b_{1}|_{t = 0} = n_{1}(z), \ b_{2}|_{t = 0} = n_{2}(x, z), }\\ & {u|_{z = 0} = \alpha^{0}, \ u|_{z = h} = \alpha^{h}, }\\ & {b|_{z = 0} = \beta^{0}, \ b|_{z = h} = \beta^{h}, } \end{array} \right.$$$

$$$\alpha^{i} = (\alpha_{1}^{i}(t), \alpha_{2}^{i}(t, x)), \beta^{i} = (\beta_{1}^{i}(t), \beta_{2}^{i}(t, x)), \ \ i = 0, h,$$$

## 2 近似解的构造

$\rho(z)$的定义,可以很容易得到当$z\in[0, h]$时, $\rho(z)\rho(h-z)\equiv0$.接下来构造方程组(1.7)的近似解

$$$\left\{\begin{array}{ll} {u_{1}^{app}(t, z) = u_{1}^{0}(t, z)+\rho(z)\varphi_{1}^{0, 0}(t, \frac{z}{\sqrt{\varepsilon^{*}}}) +\rho(h-z)\varphi_{1}^{0, h}(t, \frac{h-z}{\sqrt{\varepsilon^{*}}}), }\\ {u_{2}^{app}(t, x, z) = u_{2}^{0}(t, x, z)+\rho(z)\varphi_{2}^{0, 0}(t, x, \frac{z}{\sqrt{\varepsilon^{*}}}) +\rho(h-z)\varphi_{2}^{0, h}(t, x, \frac{h-z}{\sqrt{\varepsilon^{*}}}), }\\ {b_{1}^{app}(t, z) = b_{1}^{0}(t, z)+\rho(z)\psi_{1}^{0, 0}(t, \frac{z}{\sqrt{\varepsilon^{*}}}) +\rho(h-z)\psi_{1}^{0, h}(t, \frac{h-z}{\sqrt{\varepsilon^{*}}}), }\\ {b_{2}^{app}(t, x, z) = b_{2}^{0}(t, x, z)+\rho(z)\psi_{2}^{0, 0}(t, x, \frac{z}{\sqrt{\varepsilon^{*}}}) +\rho(h-z)\psi_{2}^{0, h}(t, x, \frac{h-z}{\sqrt{\varepsilon^{*}}}).} \end{array} \right.$$$

## 3 边界层函数

令$w_{1}(t, Z) = \varphi_{1}^{0, 0}(t, Z)-(\alpha_{1}^{0}(t)-u_{1}^{0}|_{z = 0}){\rm e}^{-Z}$,由方程组(3.1)的第一个方程可得

$$$\left\{\begin{array}{ll} &{\partial_{t}w_{1}-\partial_{Z}^{2}w_{1} = (\alpha_{1}^{0}-u_{1}^{0}|_{z = 0}-\partial_{t}\alpha_{1}^{0}+F_{1}|_{z = 0}){\rm e}^{-Z}, }\\ &w_{1}|_{Z = 0} = w_{1}|_{Z = +\infty} = 0, \\ & w_{1}|_{t = 0} = 0.\\ \end{array} \right.$$$

$\begin{eqnarray} && \frac{1}{2}\frac{\rm d}{{\rm d}t}\|w_{1}\|_{L^{2}(0, +\infty)}^{2}+\|\partial_{Z}w_{1}\|_{L^{2}(0, +\infty)}^{2}\\ & = &\int_{0}^{+\infty}(\alpha_{1}^{0}-u_{1}^{0}|_{z = 0}-\partial_{t}\alpha_{1}^{0}+F_{1}|_{z = 0}){\rm e}^{-Z}w_{1}{\rm d}Z\\ & \leq&(|\alpha_{1}^{0}|+|u_{1}^{0}|_{z = 0}|+|\partial_{t}\alpha_{1}^{0}|+|F_{1}|_{z = 0}|)\|{\rm e}^{-Z}\|_{L^{2}(0, +\infty)} \|w_{1}\|_{L^{2}(0, +\infty)}\\ & \leq & 2\|w_{1}\|^{2}_{L^{2}(0, +\infty)}+\frac{1}{4}(|\alpha_{1}^{0}|^{2}+|u_{1}^{0}|_{z = 0}|^{2}+|\partial_{t}\alpha_{1}^{0}|^{2}+|F_{1}|_{z = 0}|^{2}). \end{eqnarray}$

$\begin{eqnarray} && \|w_{1}\|_{L^{\infty}(0, T;L^{2}(0, +\infty))}^{2}+\|\partial_{Z}w_{1}\|_{L^{2}((0, T)\times(0, +\infty))}^{2}\\ & \leq & C(T)(\|\alpha_{1}^{0}\|^{2}_{L^{2}(0, T)}+\|u_{1}^{0}|_{z = 0}\|^{2}_{L^{2}(0, T)}+\|\partial_{t}\alpha_{1}^{0}\|^{2}_{L^{2}(0, T)} +\|F_{1}|_{z = 0}\|^{2}_{L^{2}(0, T)})\\ & \leq & C(T)(\|\alpha_{1}^{0}\|^{2}_{H^{1}(0, T)}+|m_{1}(0)|^{2} +\|F_{1}|_{z = 0}\|^{2}_{L^{2}(0, T)})\\ & \leq & C(T)(\|\alpha_{1}^{0}\|^{2}_{H^{1}(0, T)}+\|m_{1}\|_{L^{\infty}(0, h)}^{2} +\|F_{1}\|^{2}_{L^{2}(0, T;L^{\infty}(0, h))})\\ & \leq & C(T)(\|\alpha_{1}^{0}\|^{2}_{H^{1}(0, T)}+\|m_{1}\|_{H^{1}(0, h)}^{2} +\|F_{1}\|^{2}_{L^{2}(0, T;H^{1}(0, h))}), \end{eqnarray}$

$\begin{eqnarray} && \frac{1}{2}\frac{\rm d}{{\rm d}t}\|\partial_{Z}w_{1}\|_{L^{2}(0, +\infty)}^{2}+\|\partial_{Z}^{2}w_{1}\|_{L^{2}(0, +\infty)}^{2}\\ & = &\int_{0}^{+\infty}(\alpha_{1}^{0}-u_{1}^{0}|_{z = 0}-\partial_{t}\alpha_{1}^{0}+F_{1}|_{z = 0}){\rm e}^{-Z}(-\partial_{Z}^{2}w_{1}){\rm d}Z\\ & \leq&(|\alpha_{1}^{0}|+|u_{1}^{0}|_{z = 0}|+|\partial_{t}\alpha_{1}^{0}|+|F_{1}|_{z = 0}|)\|{\rm e}^{-Z}\|_{L^{2}(0, +\infty)} \|\partial_{Z}^{2}w_{1}\|_{L^{2}(0, +\infty)}\\ & \leq & \frac{1}{2}\|\partial_{Z}^{2}w_{1}\|^{2}_{L^{2}(0, +\infty)}+(|\alpha_{1}^{0}|^{2}+|u_{1}^{0}|_{z = 0}|^{2}+|\partial_{t}\alpha_{1}^{0}|^{2}+|F_{1}|_{z = 0}|^{2}). \end{eqnarray}$

$\begin{eqnarray} && \|\partial_{Z}w_{1}\|_{L^{\infty}(0, T;L^{2}(0, +\infty))}^{2}+\|\partial_{Z}^{2}w_{1}\|_{L^{2}((0, T)\times(0, +\infty))}^{2}\\ & \leq & 4(\|\alpha_{1}^{0}\|^{2}_{L^{2}(0, T)}+\|u_{1}^{0}|_{z = 0}\|^{2}_{L^{2}(0, T)}+\|\partial_{t}\alpha_{1}^{0}\|^{2}_{L^{2}(0, T)} +\|F_{1}|_{z = 0}\|^{2}_{L^{2}(0, T)})\\ & \leq & C(\|\alpha_{1}^{0}\|^{2}_{H^{1}(0, T)}+\|m_{1}\|_{H^{1}(0, h)}^{2} +\|F_{1}\|^{2}_{L^{2}(0, T;H^{1}(0, h))}). \end{eqnarray}$

$\begin{eqnarray} &&\frac{1}{2}\frac{\rm d}{{\rm d}t}(\|\partial_{Z}w_{2}\|_{L^{2}(\Omega_{\infty})}^{2}+\|\partial_{Z}w_{3}\|_{L^{2}(\Omega_{\infty})}^{2}) +\|\partial_{Z}^{2}w_{2}\|_{L^{2}(\Omega_{\infty})}^{2}+\|\partial_{Z}^{2}w_{3}\|_{L^{2}(\Omega_{\infty})}^{2} \\& = &\int_{\Omega_{\infty}}[\Gamma_{1}(-\partial_{Z}^{2}w_{2})+\Gamma_{2}(-\partial_{Z}^{2}w_{3})]{\rm d}x{\rm d}Z \\&&+\int_{\Omega_{\infty}}(-\varphi_{1}^{0, 0}\partial_{x}u_{2}^{0}|_{z = 0}+\psi_{1}^{0, 0}\partial_{x}b_{2}^{0}|_{z = 0})(-\partial_{Z}^{2}w_{2}){\rm d}x{\rm d}Z \\&&+\int_{\Omega_{\infty}}(-\varphi_{1}^{0, 0}\partial_{x}b_{2}^{0}|_{z = 0}+\psi_{1}^{0, 0}\partial_{x}u_{2}^{0}|_{z = 0})(-\partial_{Z}^{2}w_{3}){\rm d}x{\rm d}Z \\&&+\int_{\Omega_{\infty}}(u^{0}_{1}|_{z = 0}+\varphi_{1}^{0, 0})\partial_{x}w_{2}\partial_{Z}^{2}w_{2}{\rm d}x{\rm d}Z +\int_{\Omega_{\infty}}(u^{0}_{1}|_{z = 0}+\varphi_{1}^{0, 0})\partial_{x}w_{3}\partial_{Z}^{2}w_{3}{\rm d}x{\rm d}Z \\&&-\int_{\Omega_{\infty}}(b^{0}_{1}|_{z = 0}+\psi_{1}^{0, 0})(\partial_{x}w_{3}\partial_{Z}^{2}w_{2} +\partial_{x}w_{2}\partial_{Z}^{2}w_{3}){\rm d}x{\rm d}Z \\ &\leq&(\|\partial_{t}\alpha_{2}^{0}\|_{L^{2}(0, L)}+\|F_{2}|_{z = 0}\|_{L^{2}(0, L)}+\|\alpha_{2}^{0}\|_{L^{2}(0, L)} +\|u_{2}^{0}|_{z = 0}\|_{L^{2}(0, L)}\\&&+|u_{1}^{0}|_{z = 0}|\|\partial_{x}\alpha_{2}^{0}\|_{L^{2}(0, L)} +\|\varphi_{1}^{0, 0}\|_{L^{\infty}(0, +\infty)}\|\partial_{x}\alpha_{2}^{0}\|_{L^{2}(0, L)} \\&&+\|\varphi_{1}^{0, 0}\|_{L^{\infty}(0, +\infty)}\|\partial_{x}u_{2}^{0}|_{z = 0}\|_{L^{2}(0, L)} +|b_{1}^{0}|_{z = 0}|\|\partial_{x}\beta_{2}^{0}\|_{L^{2}(0, L)} \\&&+\|\psi_{1}^{0, 0}\|_{L^{\infty}(0, +\infty)}\|\partial_{x}\beta_{2}^{0}\|_{L^{2}(0, L)} +\|\psi_{1}^{0, 0}\|_{L^{\infty}(0, +\infty)}\|\partial_{x}b_{2}^{0}|_{z = 0}\|_{L^{2}(0, L)})\\&& \times\|{\rm e}^{-Z}\|_{L^{2}(0, +\infty)} \|\partial_{Z}^{2}w_{2}\|_{L^{2}(\Omega_{\infty})} +(\|\partial_{t}\beta_{2}^{0}\|_{L^{2}(0, L)}+\|G_{2}|_{z = 0}\|_{L^{2}(0, L)}+\|\beta_{2}^{0}\|_{L^{2}(0, L)} \\&&+\|b_{2}^{0}|_{z = 0}\|_{L^{2}(0, L)}+|u_{1}^{0}|_{z = 0}|\|\partial_{x}\beta_{2}^{0}\|_{L^{2}(0, L)} +\|\varphi_{1}^{0, 0}\|_{L^{\infty}(0, +\infty)}\|\partial_{x}\beta_{2}^{0}\|_{L^{2}(0, L)} \\&&+\|\varphi_{1}^{0, 0}\|_{L^{\infty}(0, +\infty)}\|\partial_{x}b_{2}^{0}|_{z = 0}\|_{L^{2}(0, L)} +|b_{1}^{0}|_{z = 0}|\|\partial_{x}\alpha_{2}^{0}\|_{L^{2}(0, L)} +\|\psi_{1}^{0, 0}\|_{L^{\infty}(0, +\infty)}\\&&\times\|\partial_{x}\alpha_{2}^{0}\|_{L^{2}(0, L)} +\|\psi_{1}^{0, 0}\|_{L^{\infty}(0, +\infty)}\|\partial_{x}u_{2}^{0}|_{z = 0}\|_{L^{2}(0, L)})\|{\rm e}^{-Z}\|_{L^{2}(0, +\infty)} \|\partial_{Z}^{2}w_{3}\|_{L^{2}(\Omega_{\infty})}\\&&+(\|\partial_{x}u_{2}^{0}|_{z = 0}\|_{L^{2}(0, L)}\|\varphi_{1}^{0, 0}\|_{L^{2}(0, +\infty)} +\|\partial_{x}b_{2}^{0}|_{z = 0}\|_{L^{2}(0, L)}\|\psi_{1}^{0, 0}\|_{L^{2}(0, +\infty)}) \|\partial_{Z}^{2}w_{2}\|_{L^{2}(\Omega_{\infty})} \\&&+(\|\partial_{x}b_{2}^{0}|_{z = 0}\|_{L^{2}(0, L)}\|\varphi_{1}^{0, 0}\|_{L^{2}(0, +\infty)} +\|\partial_{x}u_{2}^{0}|_{z = 0}\|_{L^{2}(0, L)}\|\psi_{1}^{0, 0}\|_{L^{2}(0, +\infty)}) \|\partial_{Z}^{2}w_{3}\|_{L^{2}(\Omega_{\infty})}\\&&+(|u_{1}^{0}|_{z = 0}|+\|\varphi_{1}^{0, 0}\|_{L^{\infty}(0, +\infty)}) (\|\partial_{x}w_{2}\|_{L^{2}(\Omega_{\infty})}\|\partial_{Z}^{2}w_{2}\|_{L^{2}(\Omega_{\infty})}\\&&+ \|\partial_{x}w_{3}\|_{L^{2}(\Omega_{\infty})}\|\partial_{Z}^{2}w_{3}\|_{L^{2}(\Omega_{\infty})}) +(|b_{1}^{0}|_{z = 0}|+\|\psi_{1}^{0, 0}\|_{L^{\infty}(0, +\infty)}) \\ &&\times(\|\partial_{x}w_{3}\|_{L^{2}(\Omega_{\infty})}\|\partial_{Z}^{2}w_{2}\|_{L^{2}(\Omega_{\infty})}+ \|\partial_{x}w_{2}\|_{L^{2}(\Omega_{\infty})}\|\partial_{Z}^{2}w_{3}\|_{L^{2}(\Omega_{\infty})}), \end{eqnarray}$

$\begin{eqnarray} &&\|\partial_{Z}w_{2}\|^{2}_{L^{\infty}(0, T;L^{2}(\Omega_{\infty}))}+\|\partial_{Z}w_{3}\|^{2}_{L^{\infty}(0, T;L^{2}(\Omega_{\infty}))} \\ &&+\|\partial_{Z}^{2}w_{2}\|^{2}_{L^{2}((0, T)\times\Omega_{\infty})} +\|\partial_{Z}^{2}w_{3}\|^{2}_{L^{2}((0, T)\times\Omega_{\infty})} \leq C_{4}, \end{eqnarray}$

$\begin{eqnarray} &&\|\partial_{x}\partial_{Z}w_{2}\|^{2}_{L^{\infty}(0, T;L^{2}(\Omega_{\infty}))}+\|\partial_{x}\partial_{Z}w_{3}\|^{2}_{L^{\infty}(0, T;L^{2}(\Omega_{\infty}))} \\ &&+\|\partial_{x}\partial_{Z}^{2}w_{2}\|^{2}_{L^{2}((0, T)\times\Omega_{\infty})} +\|\partial_{x}\partial_{Z}^{2}w_{3}\|^{2}_{L^{2}((0, T)\times\Omega_{\infty})} \leq C_{5}, \end{eqnarray}$

$\begin{eqnarray} &&\frac{1}{2}\frac{\rm d}{{\rm d}t}(\|\langle Z\rangle^{l}w_{2}\|_{L^{2}(\Omega_{\infty})}^{2}+\|\langle Z\rangle^{l}w_{3}\|_{L^{2}(\Omega_{\infty})}^{2}) +\|\langle Z\rangle^{l}\partial_{Z}w_{2}\|_{L^{2}(\Omega_{\infty})}^{2}+\|\langle Z\rangle^{l}\partial_{Z}w_{3}\|_{L^{2}(\Omega_{\infty})}^{2}\\& = &\int_{\Omega_{\infty}}(\Gamma_{1}w_{2}+\Gamma_{2}w_{3})\langle Z\rangle ^{2l}{\rm d}x{\rm d}Z +\int_{\Omega_{\infty}}(-\varphi_{1}^{0, 0}\partial_{x}u_{2}^{0}|_{z = 0}+\psi_{1}^{0, 0}\partial_{x}b_{2}^{0}|_{z = 0})\langle Z\rangle ^{2l}w_{2}{\rm d}x{\rm d}Z\\&&+\int_{\Omega_{\infty}}(-\varphi_{1}^{0, 0}\partial_{x}b_{2}^{0}|_{z = 0}+\psi_{1}^{0, 0}\partial_{x}u_{2}^{0}|_{z = 0})\langle Z\rangle ^{2l}w_{3}{\rm d}x{\rm d}Z \\&&-l\int_{\Omega_{\infty}}[\partial_{Z}(w_{2})^{2}+\partial_{Z}(w_{3})^{2}]Z(1+Z^{2})^{l-1}{\rm d}x{\rm d}Z \\&\leq&(\|\partial_{t}\alpha_{2}^{0}\|_{L^{2}(0, L)}+\|F_{2}|_{z = 0}\|_{L^{2}(0, L)}+\|\alpha_{2}^{0}\|_{L^{2}(0, L)} +\|u_{2}^{0}|_{z = 0}\|_{L^{2}(0, L)}\\&&+|u_{1}^{0}|_{z = 0}|\|\partial_{x}\alpha_{2}^{0}\|_{L^{2}(0, L)} +\|\varphi_{1}^{0, 0}\|_{L^{\infty}(0, +\infty)}\|\partial_{x}\alpha_{2}^{0}\|_{L^{2}(0, L)} +\|\varphi_{1}^{0, 0}\|_{L^{\infty}(0, +\infty)}\\&& \times\|\partial_{x}u_{2}^{0}|_{z = 0}\|_{L^{2}(0, L)} +|b_{1}^{0}|_{z = 0}|\|\partial_{x}\beta_{2}^{0}\|_{L^{2}(0, L)} +\|\psi_{1}^{0, 0}\|_{L^{\infty}(0, +\infty)}\|\partial_{x}\beta_{2}^{0}\|_{L^{2}(0, L)} \\&&+\|\psi_{1}^{0, 0}\|_{L^{\infty}(0, +\infty)}\|\partial_{x}b_{2}^{0}|_{z = 0}\|_{L^{2}(0, L)})\|\langle Z\rangle ^{l} {\rm e}^{-Z}\|_{L^{2}(0, +\infty)} \|\langle Z\rangle ^{l}w_{2}\|_{L^{2}(\Omega_{\infty})} \\&&+(\|\partial_{t}\beta_{2}^{0}\|_{L^{2}(0, L)}+\|G_{2}|_{z = 0}\|_{L^{2}(0, L)}+\|\beta_{2}^{0}\|_{L^{2}(0, L)} +\|b_{2}^{0}|_{z = 0}\|_{L^{2}(0, L)}\\&&+|u_{1}^{0}|_{z = 0}|\|\partial_{x}\beta_{2}^{0}\|_{L^{2}(0, L)} +\|\varphi_{1}^{0, 0}\|_{L^{\infty}(0, +\infty)}\|\partial_{x}\beta_{2}^{0}\|_{L^{2}(0, L)} +\|\varphi_{1}^{0, 0}\|_{L^{\infty}(0, +\infty)}\\&& \times\|\partial_{x}b_{2}^{0}|_{z = 0}\|_{L^{2}(0, L)} +|b_{1}^{0}|_{z = 0}|\|\partial_{x}\alpha_{2}^{0}\|_{L^{2}(0, L)} +\|\psi_{1}^{0, 0}\|_{L^{\infty}(0, +\infty)}\|\partial_{x}\alpha_{2}^{0}\|_{L^{2}(0, L)} \\&&+\|\psi_{1}^{0, 0}\|_{L^{\infty}(0, +\infty)}\|\partial_{x}u_{2}^{0}|_{z = 0}\|_{L^{2}(0, L)})\|\langle Z\rangle ^{l}{\rm e}^{-Z}\|_{L^{2}(0, +\infty)} \|\langle Z\rangle^{l}w_{3}\|_{L^{2}(\Omega_{\infty})}\\&& +(\|\partial_{x}u_{2}^{0}|_{z = 0}\|_{L^{2}(0, L)}\|\langle Z\rangle^{l}\varphi_{1}^{0, 0}\|_{L^{2}(0, +\infty)} +\|\partial_{x}b_{2}^{0}|_{z = 0}\|_{L^{2}(0, L)}\|\langle Z\rangle^{l}\psi_{1}^{0, 0}\|_{L^{2}(0, +\infty)})\\&& \times \|\langle Z\rangle ^{l}w_{2}\|_{L^{2}(\Omega_{\infty})} +(\|\partial_{x}b_{2}^{0}|_{z = 0}\|_{L^{2}(0, L)}\|\langle Z\rangle^{l}\varphi_{1}^{0, 0}\|_{L^{2}(0, +\infty)} \\&&+\|\partial_{x}u_{2}^{0}|_{z = 0}\|_{L^{2}(0, L)}\|\langle Z\rangle ^{l}\psi_{1}^{0, 0}\|_{L^{2}(0, +\infty)}) \|\langle Z\rangle ^{l}w_{3}\|_{L^{2}(\Omega_{\infty})} \\ && +l(2l-1)(\|\langle Z\rangle^{l}w_{2}\|^{2}_{L^{2}(\Omega_{\infty})} +\|\langle Z\rangle ^{l}w_{3}\|^{2}_{L^{2}(\Omega_{\infty})}), \end{eqnarray}$

$\begin{eqnarray} &&\|\langle Z\rangle^{l}w_{2}\|^{2}_{L^{\infty}(0, T;L^{2}(\Omega_{\infty}))}+\|\langle Z\rangle^{l}w_{3}\|^{2}_{L^{\infty}(0, T;L^{2}(\Omega_{\infty}))}\\ &&+ \|\langle Z\rangle^{l}\partial_{Z}w_{2}\|^{2}_{L^{2}((0, T)\times\Omega_{\infty})} +\|\langle Z\rangle^{l}\partial_{Z}w_{3}\|^{2}_{L^{2}((0, T)\times\Omega_{\infty})} \leq C_{7}, \end{eqnarray}$

$$$\left\{\begin{array}{ll} & {\partial_{t}u_{1}^{r}-\varepsilon_{1}\partial_{z}^{2}u_{1}^{r} = A+B+Y, }\\ & {u_{1}^{r}|_{z = 0} = u_{1}^{r}|_{z = h} = 0, }\\ & {u_{1}^{r}|_{t = 0} = 0, } \end{array} \right.$$$

$$$\left\{\begin{array}{ll} & {\partial_{t}b_{1}^{r}-\varepsilon_{2}\partial_{z}^{2}b_{1}^{r} = D+E+O, }\\ & {b_{1}^{r}|_{z = 0} = b_{1}^{r}|_{z = h} = 0, }\\ & {b_{1}^{r}|_{t = 0} = 0, } \end{array} \right.$$$

$$$\left\{\begin{array}{ll} & \partial_{t}u_{2}^{r}-\varepsilon_{1}(\partial_{x}^{2}u_{2}^{r}+\partial_{z}^{2}u_{2}^{r}) +u_{1}\partial_{x}u_{2}^{r}+u_{1}^{r}\partial_{x}u_{2}^{app}\\&-b_{1}\partial_{x}b_{2}^{r} -b_{1}^{r}\partial_{x}b_{2}^{app} = K+P+M+N+Q, \\ & \partial_{t}b_{2}^{r}-\varepsilon_{2}(\partial_{x}^{2}b_{2}^{r}+\partial_{z}^{2}b_{2}^{r}) +u_{1}\partial_{x}b_{2}^{r}+u_{1}^{r}\partial_{x}b_{2}^{app}\\&-b_{1}\partial_{x}u_{2}^{r} -b_{1}^{r}\partial_{x}u_{2}^{app} = R+T+U+V+W, \\ & {u_{2}^{r}|_{z = 0} = u_{2}^{r}|_{z = h} = 0, \ u_{2}^{r}|_{t = 0} = 0, }\\ & {b_{2}^{r}|_{z = 0} = b_{2}^{r}|_{z = h} = 0, \ b_{2}^{r}|_{t = 0} = 0, } \end{array} \right.$$$

$\begin{eqnarray} &&A = \varepsilon_{1}[\partial_{z}^{2}u_{1}^{0}+\rho''(z)\varphi_{1}^{0, 0}+\rho''(h-z)\varphi_{1}^{0, h}], \\&&B = \frac{2\varepsilon_{1}}{\sqrt{\varepsilon^{*}}}[\rho'(z)\partial_{Z}\varphi_{1}^{0, 0}+\rho'(h-z)\partial_{Z^{h}}\varphi_{1}^{0, h}], \\&& Y = \frac{\varepsilon_{1}-\varepsilon^{*}}{\varepsilon^{*}}[\rho(z)\partial_{Z}^{2}\varphi_{1}^{0, 0}+\rho(h-z)\partial_{Z^{h}}^{2}\varphi_{1}^{0, h}], \\&& D = \varepsilon_{2}[\partial_{z}^{2}b_{1}^{0}+\rho''(z)\psi_{1}^{0, 0}+\rho''(h-z)\psi_{1}^{0, h}], \\&& E = \frac{2\varepsilon_{2}}{\sqrt{\varepsilon^{*}}}[\rho'(z)\partial_{Z}\psi_{1}^{0, 0}+\rho'(h-z)\partial_{Z^{h}}\psi_{1}^{0, h}], \\&& O = \frac{\varepsilon_{2}-\varepsilon^{*}}{\varepsilon^{*}}[\rho(z)\partial_{Z}^{2}\psi_{1}^{0, 0}+\rho(h-z)\partial_{Z^{h}}^{2}\psi_{1}^{0, h}], \\ && K = \varepsilon_{1}[\partial_{x}^{2}u_{2}^{0}+\partial_{z}^{2}u_{2}^{0}+\rho(z)\partial_{x}^{2}\varphi_{2}^{0, 0} +\rho(h-z)\partial_{x}^{2}\varphi_{2}^{0, h}+\rho''(z)\varphi_{2}^{0, 0}+\rho''(h-z)\varphi_{2}^{0, h}], \\&& P = \frac{2\varepsilon_{1}}{\sqrt{\varepsilon^{*}}}[\rho'(z)\partial_{Z}\varphi_{2}^{0, 0}+\rho'(h-z)\partial_{Z^{h}}\varphi_{2}^{0, h}], \\&& M = \frac{\varepsilon_{1}-\varepsilon^{*}}{\varepsilon^{*}}[\rho(z)\partial_{Z}^{2}\varphi_{2}^{0, 0}+\rho(h-z)\partial_{Z^{h}}^{2}\varphi_{2}^{0, h}], \\&& N = -\sqrt{\varepsilon^{*}}[\rho(z)Z(\partial_{z}u_{1}^{0}|_{z = \xi_{1}}\partial_{x}\varphi_{2}^{0, 0} +\partial_{x}\partial_{z}u_{2}^{0}|_{z = \xi_{3}}\varphi_{1}^{0, 0}-\partial_{z}b_{1}^{0}|_{z = \xi_{5}}\partial_{x}\psi_{2}^{0, 0} \\&& \;\; -\partial_{x}\partial_{z}b_{2}^{0}|_{z = \xi_{7}}\psi_{1}^{0, 0}) -\rho(h-z)Z^{h}(\partial_{z}u_{1}^{0}|_{z = \xi_{2}}\partial_{x}\varphi_{2}^{0, h} +\partial_{x}\partial_{z}u_{2}^{0}|_{z = \xi_{4}}\varphi_{1}^{0, h}\\&& \;\; -\partial_{z}b_{1}^{0}|_{z = \xi_{6}}\partial_{x}\psi_{2}^{0, h} -\partial_{x}\partial_{z}b_{2}^{0}|_{z = \xi_{8}}\psi_{1}^{0, h})], \\&& Q = \rho(z)(\rho(z)-1)(-\varphi_{1}^{0, 0}\partial_{x}\varphi_{2}^{0, 0}+\psi_{1}^{0, 0}\partial_{x}\psi_{2}^{0, 0}) -\rho(h-z)(\rho(h-z)-1)\\&& \;\; \times(\varphi_{1}^{0, h}\partial_{x}\varphi_{2}^{0, h}-\psi_{1}^{0, h}\partial_{x}\psi_{2}^{0, h}), \\&& R = \varepsilon_{2}[\partial_{x}^{2}b_{2}^{0}+\partial_{z}^{2}b_{2}^{0}+\rho(z)\partial_{x}^{2}\psi_{2}^{0, 0} +\rho(h-z)\partial_{x}^{2}\psi_{2}^{0, h}+\rho''(z)\psi_{2}^{0, 0}+\rho''(h-z)\psi_{2}^{0, h}], \\&& T = \frac{2\varepsilon_{2}}{\sqrt{\varepsilon^{*}}}[\rho'(z)\partial_{Z}\psi_{2}^{0, 0}+\rho'(h-z)\partial_{Z^{h}}\psi_{2}^{0, h}], \\&& U = \frac{\varepsilon_{2}-\varepsilon^{*}}{\varepsilon^{*}}[\rho(z)\partial_{Z}^{2}\psi_{2}^{0, 0}+\rho(h-z)\partial_{Z^{h}}^{2}\psi_{2}^{0, h}], \\&& V = -\sqrt{\varepsilon^{*}}[\rho(z)Z(\partial_{z}u_{1}^{0}|_{z = \xi_{1}}\partial_{x}\psi_{2}^{0, 0} +\partial_{x}\partial_{z}b_{2}^{0}|_{z = \xi_{7}}\varphi_{1}^{0, 0}-\partial_{z}b_{1}^{0}|_{z = \xi_{5}}\partial_{x}\varphi_{2}^{0, 0} \\&& \;\; -\partial_{x}\partial_{z}u_{2}^{0}|_{z = \xi_{3}}\psi_{1}^{0, 0}) -\rho(h-z)Z^{h}(\partial_{z}u_{1}^{0}|_{z = \xi_{2}}\partial_{x}\psi_{2}^{0, h} +\partial_{x}\partial_{z}b_{2}^{0}|_{z = \xi_{8}}\varphi_{1}^{0, h}\\&& \;\; -\partial_{z}b_{1}^{0}|_{z = \xi_{6}}\partial_{x}\varphi_{2}^{0, h} -\partial_{x}\partial_{z}u_{2}^{0}|_{z = \xi_{4}}\psi_{1}^{0, h})], \\&& W = \rho(z)(\rho(z)-1)(-\varphi_{1}^{0, 0}\partial_{x}\psi_{2}^{0, 0}+\psi_{1}^{0, 0}\partial_{x}\varphi_{2}^{0, 0}) -\rho(h-z)(\rho(h-z)-1)\\&& \;\; \times(\varphi_{1}^{0, h}\partial_{x}\psi_{2}^{0, h}-\psi_{1}^{0, h}\partial_{x}\varphi_{2}^{0, h}), \\ && \xi_{i}\in(0, h), \ i = 1, 2\cdot\cdot\cdot8. \end{eqnarray}$

### 4.1 定理2.1的证明

$\begin{eqnarray} \frac{1}{2}\frac{\rm d}{{\rm d}t}\|u_{1}^{r}\|_{L^{2}(0, h)}^{2}+\varepsilon_{1}\|\partial_{z}u_{1}^{r}\|_{L^{2}(0, h)}^{2} = \int_{0}^{h}(A+B+Y)u_{1}^{r}{\rm d}z. \end{eqnarray}$

$\begin{eqnarray} \int_{0}^{h}Au_{1}^{r}{\rm d}z & = &\varepsilon_{1}\int_{0}^{h}\partial_{z}^{2}u_{1}^{0}u_{1}^{r}{\rm d}z +\varepsilon_{1}\int_{\frac{h}{4}}^{\frac{3h}{4}}[\rho''(z)\varphi_{1}^{0, 0}+ \rho''(h-z)\varphi_{1}^{0, h}]u_{1}^{r}{\rm d}z\\& \leq&\varepsilon_{1}\|\partial_{z}^{2}u_{1}^{0}\|_{L^{2}(0, h)}\|u_{1}^{r}\|_{L^{2}(0, h)} +C(h)\varepsilon_{1}(\varepsilon^{*})^{\frac{1}{4}}\|u_{1}^{r}\|_{L^{2}(0, h)} \bigg(\int_{\frac{h}{4\sqrt{\varepsilon^{*}}}}^{\frac{3h}{4\sqrt{\varepsilon^{*}}}}|\varphi_{1}^{0, 0}|^{2}{\rm d}Z\bigg)^{\frac{1}{2}} \\&& +C(h)\varepsilon_{1}(\varepsilon^{*})^{\frac{1}{4}}\|u_{1}^{r}\|_{L^{2}(0, h)} \bigg(\int_{\frac{h}{4\sqrt{\varepsilon^{*}}}}^{\frac{3h}{4\sqrt{\varepsilon^{*}}}}|\varphi_{1}^{0, h}|^{2}{\rm d}Z^{h}\bigg)^{\frac{1}{2}} \\ & \leq & \varepsilon_{1}\|\partial_{z}^{2}u_{1}^{0}\|_{L^{2}(0, h)}\|u_{1}^{r}\|_{L^{2}(0, h)}+ C(h)\varepsilon_{1}(\varepsilon^{*})^{\frac{1}{4}}(\|\varphi_{1}^{0, 0}\|_{L^{2}(0, +\infty)} +\|\varphi_{1}^{0, h}\|_{L^{2}(0, +\infty)})\\ &&\times\|u_{1}^{r}\|_{L^{2}(0, h)} \\ & \leq & \|u_{1}^{r}\|_{L^{2}(0, h)}^{2}+C(h)\varepsilon_{1}^{2}(\|u_{1}^{0}\|^{2}_{H^{2}(0, h)} +\|\varphi_{1}^{0, 0}\|^{2}_{L^{2}(0, +\infty)} +\|\varphi_{1}^{0, h}\|^{2}_{L^{2}(0, +\infty)}). \end{eqnarray}$

$\begin{eqnarray} \int_{0}^{h}Bu_{1}^{r}{\rm d}z & = &\frac{2\varepsilon_{1}}{\sqrt{\varepsilon^{*}}}\int_{\frac{h}{4}}^{\frac{3h}{4}}[\rho'(z)\partial_{Z}\varphi_{1}^{0, 0}+ \rho'(h-z)\partial_{Z^{h}}\varphi_{1}^{0, h}]u_{1}^{r}{\rm d}z\\& \leq & C(h)\varepsilon_{1}(\varepsilon^{*})^{\frac{1}{4}}\|u_{1}^{r}\|_{L^{2}(0, h)} \bigg[\bigg(\int_{\frac{h}{4\sqrt{\varepsilon^{*}}}}^{\frac{3h}{4\sqrt{\varepsilon^{*}}}}|\langle Z\rangle \partial_{Z}\varphi_{1}^{0, 0}|^{2}{\rm d}Z\bigg)^{\frac{1}{2}}\\&&+ \bigg(\int_{\frac{h}{4\sqrt{\varepsilon^{*}}}}^{\frac{3h}{4\sqrt{\varepsilon^{*}}}}|\langle Z^{h}\rangle \partial_{Z^{h}}\varphi_{1}^{0, h}|^{2}{\rm d}Z^{h}\bigg)^{\frac{1}{2}}\bigg]\\& \leq & C(h)\varepsilon_{1}(\varepsilon^{*})^{\frac{1}{4}}\|u_{1}^{r}\|_{L^{2}(0, h)}(\|\langle Z\rangle \partial_{Z}\varphi_{1}^{0, 0}\|_{L^{2}(0, +\infty)}+\|\langle Z^{h}\rangle \partial_{Z^{h}}\varphi_{1}^{0, h}\|_{L^{2}(0, +\infty)})\\& \leq & \|u_{1}^{r}\|_{L^{2}(0, h)}^{2}+C(h)\varepsilon_{1}^{2}(\varepsilon^{*})^{\frac{1}{2}}(\|\langle Z\rangle \partial_{Z}\varphi_{1}^{0, 0}\|^{2}_{L^{2}(0, +\infty)}\\ &&+\|\langle Z^{h}\rangle \partial_{Z^{h}}\varphi_{1}^{0, h}\|^{2}_{L^{2}(0, +\infty)}). \end{eqnarray}$

$\begin{eqnarray} \int_{0}^{h}Yu_{1}^{r}{\rm d}z & = &\frac{\varepsilon_{1}-\varepsilon^{*}}{\varepsilon^{*}}\int_{0}^{h} [\rho(z)\partial_{Z}^{2}\varphi_{1}^{0, 0}+ \rho(h-z)\partial_{Z^{h}}^{2}\varphi_{1}^{0, h}]u_{1}^{r}{\rm d}z\\& = &-\frac{\varepsilon_{1}-\varepsilon^{*}}{\sqrt{\varepsilon^{*}}}\int_{0}^{h} [\rho'(z)\partial_{Z}\varphi_{1}^{0, 0}+ \rho'(h-z)\partial_{Z^{h}}\varphi_{1}^{0, h}]u_{1}^{r}{\rm d}z\\&& -\frac{\varepsilon_{1}-\varepsilon^{*}}{\sqrt{\varepsilon^{*}}}\int_{0}^{h}[\rho(z)\partial_{Z}\varphi_{1}^{0, 0} -\rho(h-z)\partial_{Z^{h}}\varphi_{1}^{0, h}]\partial_{z}u_{1}^{r}{\rm d}z\\& \leq & C(h)\frac{|\varepsilon_{1}-\varepsilon^{*}|}{(\varepsilon^{*})^{\frac{1}{4}}}\|u_{1}^{r}\|_{L^{2}(0, h)} \bigg[\bigg(\int_{\frac{h}{4\sqrt{\varepsilon^{*}}}}^{\frac{3h}{4\sqrt{\varepsilon^{*}}}}| \partial_{Z}\varphi_{1}^{0, 0}|^{2}{\rm d}Z\bigg)^{\frac{1}{2}} \\ && +\bigg(\int_{\frac{h}{4\sqrt{\varepsilon^{*}}}}^{\frac{3h}{4\sqrt{\varepsilon^{*}}}} |\partial_{Z^{h}}\varphi_{1}^{0, h}|^{2}{\rm d}Z^{h}\bigg)^{\frac{1}{2}}\bigg]\\ && +\frac{|\varepsilon_{1}-\varepsilon^{*}|}{(\varepsilon^{*})^{\frac{1}{4}}}\|\partial_{z}u_{1}^{r}\|_{L^{2}(0, h)} \bigg[\bigg(\int_{0}^{\frac{h}{\sqrt{\varepsilon^{*}}}}| \partial_{Z}\varphi_{1}^{0, 0}|^{2}{\rm d}Z\bigg)^{\frac{1}{2}} +\bigg(\int_{0}^{\frac{h}{\sqrt{\varepsilon^{*}}}} |\partial_{Z^{h}}\varphi_{1}^{0, h}|^{2}{\rm d}Z^{h}\bigg)^{\frac{1}{2}}\bigg]\\& \leq & \|u_{1}^{r}\|_{L^{2}(0, h)}^{2}+\frac{\varepsilon_{1}}{2}\|\partial_{z}u_{1}^{r}\|^{2}_{L^{2}(0, h)} +C(h)\frac{(\varepsilon_{1}-\varepsilon^{*})^{2}}{\sqrt{\varepsilon^{*}}\varepsilon_{1}} (\|\partial_{Z}\varphi_{1}^{0, 0}\|^{2}_{L^{2}(0, +\infty)}\\ &&+\|\partial_{Z^{h}}\varphi_{1}^{0, h}\|^{2}_{L^{2}(0, +\infty)}).\\\end{eqnarray}$

$\begin{eqnarray} &&\frac{1}{2}\frac{\rm d}{{\rm d}t}\|u_{1}^{r}\|_{L^{2}(0, h)}^{2}+\frac{\varepsilon_{1}}{2}\|\partial_{z}u_{1}^{r}\|_{L^{2}(0, h)}^{2}\\ &\leq& 3\|u_{1}^{r}\|_{L^{2}(0, h)}^{2} +C(h)\max\bigg \{\frac{(\varepsilon_{1}-\varepsilon^{*})^{2}}{\sqrt{\varepsilon^{*}}\varepsilon_{1}}, \varepsilon_{1}^{2}\bigg\} (\|u_{1}^{0}\|^{2}_{H^{2}(0, h)} +\|\varphi_{1}^{0, 0}\|^{2}_{L^{2}(0, +\infty)} \\&&+\|\varphi_{1}^{0, h}\|^{2}_{L^{2}(0, +\infty)}+\|\langle Z\rangle \partial_{Z}\varphi_{1}^{0, 0}\|^{2}_{L^{2}(0, +\infty)}\\ &&+\|\langle Z^{h}\rangle \partial_{Z^{h}}\varphi_{1}^{0, h}\|^{2}_{L^{2}(0, +\infty)}+\|\partial_{Z}\varphi_{1}^{0, 0}\|^{2}_{L^{2}(0, +\infty)} +\|\partial_{Z^{h}}\varphi_{1}^{0, h}\|^{2}_{L^{2}(0, +\infty)}), \end{eqnarray}$

$\begin{eqnarray} \|u_{1}^{r}\|_{L^{\infty}(0, T;L^{2}(0, h))}+\sqrt{\varepsilon_{1}}\|\partial_{z}u_{1}^{r}\|_{L^{2}((0, T)\times(0, h))} \leq C\max\bigg\{\frac{|\varepsilon_{1}-\varepsilon^{*}|}{(\varepsilon^{*})^{\frac{1}{4}}(\varepsilon_{1})^{\frac{1}{2}}}, \varepsilon_{1}\bigg\}. \end{eqnarray}$

$$$\|b_{1}^{r}\|_{L^{\infty}(0, T;L^{2}(0, h))}+\sqrt{\varepsilon_{2}}\|\partial_{z}b_{1}^{r}\|_{L^{2}((0, T)\times(0, h))} \leq C\max\bigg\{\frac{|\varepsilon_{2}-\varepsilon^{*}|}{(\varepsilon^{*})^{\frac{1}{4}}(\varepsilon_{2})^{\frac{1}{2}}}, \varepsilon_{2}\bigg\}.$$$

$\begin{eqnarray} &&\|\partial_{x}u_{2}^{r}\|_{L^{\infty}(0, T;L^{2}(\Omega))}+\|\partial_{x}b_{2}^{r}\|_{L^{\infty}(0, T;L^{2}(\Omega))} \\ &\leq & C\max\bigg\{\frac{|\varepsilon_{1}-\varepsilon^{*}|}{(\varepsilon^{*})^{\frac{1}{4}}(\varepsilon_{1})^{\frac{1}{2}}}, \frac{|\varepsilon_{2}-\varepsilon^{*}|}{(\varepsilon^{*})^{\frac{1}{4}}(\varepsilon_{2})^{\frac{1}{2}}}, \varepsilon_{1}, \varepsilon_{2}, (\varepsilon^{*})^{\frac{3}{4}}\bigg\}, \end{eqnarray}$

$\begin{eqnarray} &&\|\partial_{x}^{2}u_{2}^{r}\|_{L^{\infty}(0, T;L^{2}(\Omega))}+\|\partial_{x}^{2}b_{2}^{r}\|_{L^{\infty}(0, T;L^{2}(\Omega))} \\ &\leq & C\max\bigg\{\frac{|\varepsilon_{1}-\varepsilon^{*}|}{(\varepsilon^{*})^{\frac{1}{4}}(\varepsilon_{1})^{\frac{1}{2}}}, \frac{|\varepsilon_{2}-\varepsilon^{*}|}{(\varepsilon^{*})^{\frac{1}{4}}(\varepsilon_{2})^{\frac{1}{2}}}, \varepsilon_{1}, \varepsilon_{2}, (\varepsilon^{*})^{\frac{3}{4}}\bigg\}. \end{eqnarray}$

$\begin{eqnarray} && \frac{1}{2}\frac{\rm d}{{\rm d}t}(\|\partial_{z}u_{2}^{r}\|_{L^{2}(\Omega)}^{2} +\|\partial_{z}b_{2}^{r}\|_{L^{2}(\Omega)}^{2})+\varepsilon_{1}(\|\partial_{x}\partial_{z}u_{2}^{r}\|_{L^{2}(\Omega)}^{2} +\|\partial_{z}^{2}u_{2}^{r}\|_{L^{2}(\Omega)}^{2}) \\ &&+\varepsilon_{2}(\|\partial_{x}\partial_{z}b_{2}^{r}\|_{L^{2}(\Omega)}^{2}+\|\partial_{z}^{2}b_{2}^{r}\|_{L^{2}(\Omega)}^{2})\\ & = & \int_{\Omega}(u_{1}^{r}\partial_{x}u_{2}^{app}-b_{1}^{r}\partial_{x}b_{2}^{app}+u_{1}\partial_{x}u_{2}^{r} -b_{1}\partial_{x}b_{2}^{r}-K-P-M-N-Q)\partial_{z}^{2}u_{2}^{r}{\rm d}x{\rm d}z\\&& +\int_{\Omega}(u_{1}^{r}\partial_{x}b_{2}^{app}-b_{1}^{r}\partial_{x}u_{2}^{app} +u_{1}\partial_{x}b_{2}^{r}-b_{1}\partial_{x}u_{2}^{r}-R-T-U-V-W)\partial_{z}^{2}b_{2}^{r}{\rm d}x{\rm d}z\\& \leq& \|\partial_{z}^{2}u_{2}^{r}\|_{L^{2}(\Omega)} [L^{\frac{1}{2}}\|u_{1}^{r}\|_{L^{2}(0, h)}(\|\partial_{x}u_{2}^{0}\|_{L^{\infty}(\Omega)}+\|\partial_{x}\varphi_{2}^{0, 0}\|_{L^{\infty}(\Omega_{\infty})} +\|\partial_{x}\varphi_{2}^{0, h}\|_{L^{\infty}(\Omega_{\infty})})\\&& +L^{\frac{1}{2}}\|b_{1}^{r}\|_{L^{2}(0, h)}(\|\partial_{x}b_{2}^{0}\|_{L^{\infty}(\Omega)}+\|\partial_{x}\psi_{2}^{0, 0}\|_{L^{\infty}(\Omega_{\infty})} +\|\partial_{x}\psi_{2}^{0, h}\|_{L^{\infty}(\Omega_{\infty})}) \\&&+\|u_{1}\|_{L^{\infty}(0, h)}\|\partial_{x}u_{2}^{r}\|_{L^{2}(\Omega)} +\|b_{1}\|_{L^{\infty}(0, h)}\|\partial_{x}b_{2}^{r}\|_{L^{2}(\Omega)} +\|K\|_{L^{2}(\Omega)}+\|P\|_{L^{2}(\Omega)}\\&&+\|M\|_{L^{2}(\Omega)}+ \|N\|_{L^{2}(\Omega)}+\|Q\|_{L^{2}(\Omega)}] +\|\partial_{z}^{2}b_{2}^{r}\|_{L^{2}(\Omega)}[L^{\frac{1}{2}}\|u_{1}^{r}\|_{L^{2}(0, h)}(\|\partial_{x}b_{2}^{0}\|_{L^{\infty}(\Omega)} \\&&+\|\partial_{x}\psi_{2}^{0, 0}\|_{L^{\infty}(\Omega_{\infty})} +\|\partial_{x}\psi_{2}^{0, h}\|_{L^{\infty}(\Omega_{\infty})}) +L^{\frac{1}{2}}\|b_{1}^{r}\|_{L^{2}(0, h)}(\|\partial_{x}u_{2}^{0}\|_{L^{\infty}(\Omega)}+\|\partial_{x}\varphi_{2}^{0, 0}\|_{L^{\infty}(\Omega_{\infty})} \\&&+\|\partial_{x}\varphi_{2}^{0, h}\|_{L^{\infty}(\Omega_{\infty})}) +\|u_{1}\|_{L^{\infty}(0, h)}\|\partial_{x}b_{2}^{r}\|_{L^{2}(\Omega)} +\|b_{1}\|_{L^{\infty}(0, h)}\|\partial_{x}u_{2}^{r}\|_{L^{2}(\Omega)} +\|R\|_{L^{2}(\Omega)} \\ && +\|T\|_{L^{2}(\Omega)}+\|U\|_{L^{2}(\Omega)}+ \|V\|_{L^{2}(\Omega)}+\|W\|_{L^{2}(\Omega)}] \\ & \leq & \frac{\varepsilon_{1}}{2}\|\partial_{z}^{2}u_{2}^{r}\|^{2}_{L^{2}(\Omega)} +\frac{\varepsilon_{2}}{2}\|\partial_{z}^{2}b_{2}^{r}\|^{2}_{L^{2}(\Omega)} +\frac{C}{\varepsilon_{1}}[\|u_{1}^{r}\|^{2}_{L^{2}(0, h)}(\|u_{2}^{0}\|^{2}_{H^{2+s}(\Omega)}+\|\partial_{x}\varphi_{2}^{0, 0}\|^{2}_{L^{\infty}(\Omega_{\infty})} \\&&+\|\partial_{x}\varphi_{2}^{0, h}\|^{2}_{L^{\infty}(\Omega_{\infty})}) +\|b_{1}^{r}\|^{2}_{L^{2}(0, h)}(\|b_{2}^{0}\|^{2}_{H^{2+s}(\Omega)}+\|\partial_{x}\psi_{2}^{0, 0}\|^{2}_{L^{\infty}(\Omega_{\infty})} \\&&+\|\partial_{x}\psi_{2}^{0, h}\|^{2}_{L^{\infty}(\Omega_{\infty})}) +\|u_{1}\|^{2}_{L^{\infty}(0, h)}\|\partial_{x}u_{2}^{r}\|^{2}_{L^{2}(\Omega)} +\|b_{1}\|^{2}_{L^{\infty}(0, h)}\|\partial_{x}b_{2}^{r}\|^{2}_{L^{2}(\Omega)}]\\&& +\frac{C}{\varepsilon_{2}}[\|u_{1}^{r}\|^{2}_{L^{2}(0, h)}(\|b_{2}^{0}\|^{2}_{H^{2+s}(\Omega)}+\|\partial_{x}\psi_{2}^{0, 0}\|^{2}_{L^{\infty}(\Omega_{\infty})} +\|\partial_{x}\psi_{2}^{0, h}\|^{2}_{L^{\infty}(\Omega_{\infty})}) \\&&+\|b_{1}^{r}\|^{2}_{L^{2}(0, h)}(\|u_{2}^{0}\|^{2}_{H^{2+s}(\Omega)}+\|\partial_{x}\varphi_{2}^{0, 0}\|^{2}_{L^{\infty}(\Omega_{\infty})} +\|\partial_{x}\varphi_{2}^{0, h}\|^{2}_{L^{\infty}(\Omega_{\infty})}) \\&&+\|u_{1}\|^{2}_{L^{\infty}(0, h)}\|\partial_{x}b_{2}^{r}\|^{2}_{L^{2}(\Omega)} +\|b_{1}\|^{2}_{L^{\infty}(0, h)}\|\partial_{x}u_{2}^{r}\|^{2}_{L^{2}(\Omega)}]\\&&+C\varepsilon_{1} (\|u_{2}^{0}\|^{2}_{H^{2}(\Omega)}+\|\partial_{x}^{2}\varphi_{2}^{0, 0}\|^{2}_{L^{2}(\Omega_{\infty})} +\|\partial_{x}^{2}\varphi_{2}^{0, h}\|^{2}_{L^{2}(\Omega_{\infty})} +\|\varphi_{2}^{0, 0}\|^{2}_{L^{2}(\Omega_{\infty})}+\|\varphi_{2}^{0, h}\|^{2}_{L^{2}(\Omega_{\infty})} \\&&+\|\langle Z\rangle\partial_{Z}\varphi_{2}^{0, 0}\|^{2}_{L^{2}(\Omega_{\infty})} +\|\langle Z^{h}\rangle\partial_{Z^{h}}\varphi_{2}^{0, h}\|^{2}_{L^{2}(\Omega_{\infty})}) +C\varepsilon_{2} (\|b_{2}^{0}\|^{2}_{H^{2}(\Omega)}+\|\partial_{x}^{2}\psi_{2}^{0, 0}\|^{2}_{L^{2}(\Omega_{\infty})}\\&& +\|\partial_{x}^{2}\psi_{2}^{0, h}\|^{2}_{L^{2}(\Omega_{\infty})} +\|\psi_{2}^{0, 0}\|^{2}_{L^{2}(\Omega_{\infty})}+\|\psi_{2}^{0, h}\|^{2}_{L^{2}(\Omega_{\infty})} +\|\langle Z\rangle\partial_{Z}\psi_{2}^{0, 0}\|^{2}_{L^{2}(\Omega_{\infty})} \\&&+\|\langle Z^{h}\rangle\partial_{Z^{h}}\psi_{2}^{0, h}\|^{2}_{L^{2}(\Omega_{\infty})}) +\frac{C(\varepsilon_{1}-\varepsilon^{*})^{2}}{(\varepsilon^{*})^{\frac{3}{2}}\varepsilon_{1}} (\|\partial_{Z}^{2}\varphi_{2}^{0, 0}\|^{2}_{L^{2}(\Omega_{\infty})} +\|\partial_{Z^{h}}^{2}\varphi_{2}^{0, h}\|^{2}_{L^{2}(\Omega_{\infty})}) \\&&+\frac{C(\varepsilon_{2}-\varepsilon^{*})^{2}}{(\varepsilon^{*})^{\frac{3}{2}}\varepsilon_{2}} (\|\partial_{Z}^{2}\psi_{2}^{0, 0}\|^{2}_{L^{2}(\Omega_{\infty})} +\|\partial_{Z^{h}}^{2}\psi_{2}^{0, h}\|^{2}_{L^{2}(\Omega_{\infty})}) \\&&+\frac{C(\varepsilon^{*})^{\frac{3}{2}}}{\varepsilon_{1}}[\|u_{1}^{0}\|^{2}_{H^{2}(0, h)} (\|\langle Z\rangle\partial_{x}\varphi_{2}^{0, 0}\|^{2}_{L^{2}(\Omega_{\infty})}+\|\langle Z^{h}\rangle\partial_{x}\varphi_{2}^{0, h}\|^{2}_{L^{2}(\Omega_{\infty})}) \\&&+\|b_{1}^{0}\|^{2}_{H^{2}(0, h)} (\|\langle Z\rangle\partial_{x}\psi_{2}^{0, 0}\|^{2}_{L^{2}(\Omega_{\infty})}+\|\langle Z^{h}\rangle\partial_{x}\psi_{2}^{0, h}\|^{2}_{L^{2}(\Omega_{\infty})})\\&& +\|u_{2}^{0}\|^{2}_{H^{3+s}(\Omega)} (\|\langle Z\rangle\varphi_{1}^{0, 0}\|^{2}_{L^{2}(0, +\infty)}+\|\langle Z^{h}\rangle\varphi_{1}^{0, h}\|^{2}_{L^{2}(0, +\infty)})\\&& +\|b_{2}^{0}\|^{2}_{H^{3+s}(\Omega)} (\|\langle Z\rangle\psi_{1}^{0, 0}\|^{2}_{L^{2}(0, +\infty)}+\|\langle Z^{h}\rangle\psi_{1}^{0, h}\|^{2}_{L^{2}(0, +\infty)})\\&&+\|\varphi_{1}^{0, 0}\|^{2}_{L^{\infty}(0, +\infty)} \|\langle Z\rangle\partial_{x}\varphi_{2}^{0, 0}\|^{2}_{L^{2}(\Omega_{\infty})} +\|\psi_{1}^{0, 0}\|^{2}_{L^{\infty}(0, +\infty)} \|\langle Z\rangle\partial_{x}\psi_{2}^{0, 0}\|^{2}_{L^{2}(\Omega_{\infty})} \\&&+\|\varphi_{1}^{0, h}\|^{2}_{L^{\infty}(0, +\infty)} \|\langle Z^{h}\rangle\partial_{x}\varphi_{2}^{0, h}\|^{2}_{L^{2}(\Omega_{\infty})} +\|\psi_{1}^{0, h}\|^{2}_{L^{\infty}(0, +\infty)} \|\langle Z^{h}\rangle\partial_{x}\psi_{2}^{0, h}\|^{2}_{L^{2}(\Omega_{\infty})}] \\&&+\frac{C(\varepsilon^{*})^{\frac{3}{2}}}{\varepsilon_{2}}[\|u_{1}^{0}\|^{2}_{H^{2}(0, h)} (\|\langle Z\rangle\partial_{x}\psi_{2}^{0, 0}\|^{2}_{L^{2}(\Omega_{\infty})}+\|\langle Z^{h}\rangle\partial_{x}\psi_{2}^{0, h}\|^{2}_{L^{2}(\Omega_{\infty})}) \\&&+\|b_{1}^{0}\|^{2}_{H^{2}(0, h)} (\|\langle Z\rangle\partial_{x}\varphi_{2}^{0, 0}\|^{2}_{L^{2}(\Omega_{\infty})}+\|\langle Z^{h}\rangle\partial_{x}\varphi_{2}^{0, h}\|^{2}_{L^{2}(\Omega_{\infty})}) \\&&+\|u_{2}^{0}\|^{2}_{H^{3+s}(\Omega)} (\|\langle Z\rangle\psi_{1}^{0, 0}\|^{2}_{L^{2}(0, +\infty)}+\|\langle Z^{h}\rangle\psi_{1}^{0, h}\|^{2}_{L^{2}(0, +\infty)}) \\&&+\|b_{2}^{0}\|^{2}_{H^{3+s}(\Omega)} (\|\langle Z\rangle\varphi_{1}^{0, 0}\|^{2}_{L^{2}(0, +\infty)}+\|\langle Z^{h}\rangle\varphi_{1}^{0, h}\|^{2}_{L^{2}(0, +\infty)})\\&&+\|\varphi_{1}^{0, 0}\|^{2}_{L^{\infty}(0, +\infty)} \|\langle Z\rangle\partial_{x}\psi_{2}^{0, 0}\|^{2}_{L^{2}(\Omega_{\infty})} +\|\psi_{1}^{0, 0}\|^{2}_{L^{\infty}(0, +\infty)} \|\langle Z\rangle\partial_{x}\varphi_{2}^{0, 0}\|^{2}_{L^{2}(\Omega_{\infty})} \\ &&+\|\varphi_{1}^{0, h}\|^{2}_{L^{\infty}(0, +\infty)} \|\langle Z^{h}\rangle\partial_{x}\psi_{2}^{0, h}\|^{2}_{L^{2}(\Omega_{\infty})} +\|\psi_{1}^{0, h}\|^{2}_{L^{\infty}(0, +\infty)} \|\langle Z^{h}\rangle\partial_{x}\varphi_{2}^{0, h}\|^{2}_{L^{2}(\Omega_{\infty})}]. \end{eqnarray}$

$\begin{eqnarray} && \|\partial_{z}u_{2}^{r}\|_{L^{\infty}(0, T;L^{2}(\Omega))} +\|\partial_{z}b_{2}^{r}\|_{L^{\infty}(0, T;L^{2}(\Omega))} \\&&+\sqrt{\varepsilon_{1}}(\|\partial_{x}\partial_{z}u_{2}^{r}\|_{L^{2}((0, T)\times\Omega)} +\|\partial_{z}^{2}u_{2}^{r}\|_{L^{2}((0, T)\times\Omega)})\\&& +\sqrt{\varepsilon_{2}}(\|\partial_{x}\partial_{z}b_{2}^{r}\|_{L^{2}((0, T)\times\Omega)} +\|\partial_{z}^{2}b_{2}^{r}\|_{L^{2}((0, T)\times\Omega)}) \\ & \leq & C\max\bigg\{\frac{|\varepsilon_{1}-\varepsilon^{*}|} {(\varepsilon^{*})^{\frac{3}{4}}(\varepsilon_{1})^{\frac{1}{2}}}, \frac{|\varepsilon_{2}-\varepsilon^{*}|} {(\varepsilon^{*})^{\frac{3}{4}}(\varepsilon_{2})^{\frac{1}{2}}}, \frac{|\varepsilon_{1}-\varepsilon^{*}|} {(\varepsilon^{*})^{\frac{1}{4}}\varepsilon_{1}}, \frac{|\varepsilon_{2}-\varepsilon^{*}|} {(\varepsilon^{*})^{\frac{1}{4}}\varepsilon_{2}}, \frac{|\varepsilon_{1}-\varepsilon^{*}|} {(\varepsilon^{*})^{\frac{1}{4}}(\varepsilon_{1}\varepsilon_{2})^{\frac{1}{2}}}, \\ &&\frac{|\varepsilon_{2}-\varepsilon^{*}|} {(\varepsilon^{*})^{\frac{1}{4}}(\varepsilon_{1}\varepsilon_{2})^{\frac{1}{2}}}, \frac{(\varepsilon^{*})^{\frac{3}{4}}}{(\varepsilon_{1})^{\frac{1}{2}}}, \frac{(\varepsilon^{*})^{\frac{3}{4}}}{(\varepsilon_{2})^{\frac{1}{2}}}, \frac{\varepsilon_{1}}{(\varepsilon_{2})^{\frac{1}{2}}}, \frac{\varepsilon_{2}}{(\varepsilon_{1})^{\frac{1}{2}}}, (\varepsilon_{1})^{\frac{1}{2}}, (\varepsilon_{2})^{\frac{1}{2}}\bigg\}. \end{eqnarray}$

$\begin{eqnarray} && \|\partial_{x}\partial_{z}u_{2}^{r}\|_{L^{\infty}(0, T;L^{2}(\Omega))} +\|\partial_{x}\partial_{z}b_{2}^{r}\|_{L^{\infty}(0, T;L^{2}(\Omega))} \\&&+\sqrt{\varepsilon_{1}}(\|\partial_{x}^{2}\partial_{z}u_{2}^{r}\|_{L^{2}((0, T)\times\Omega)} +\|\partial_{x}\partial_{z}^{2}u_{2}^{r}\|_{L^{2}((0, T)\times\Omega)})\\&& +\sqrt{\varepsilon_{2}}(\|\partial_{x}^{2}\partial_{z}b_{2}^{r}\|_{L^{2}((0, T)\times\Omega)} +\|\partial_{x}\partial_{z}^{2}b_{2}^{r}\|_{L^{2}((0, T)\times\Omega)})\\& \leq &C\max\bigg\{\frac{|\varepsilon_{1}-\varepsilon^{*}|} {(\varepsilon^{*})^{\frac{3}{4}}(\varepsilon_{1})^{\frac{1}{2}}}, \frac{|\varepsilon_{2}-\varepsilon^{*}|} {(\varepsilon^{*})^{\frac{3}{4}}(\varepsilon_{2})^{\frac{1}{2}}}, \frac{|\varepsilon_{1}-\varepsilon^{*}|} {(\varepsilon^{*})^{\frac{1}{4}}\varepsilon_{1}}, \frac{|\varepsilon_{2}-\varepsilon^{*}|} {(\varepsilon^{*})^{\frac{1}{4}}\varepsilon_{2}}, \frac{|\varepsilon_{1}-\varepsilon^{*}|} {(\varepsilon^{*})^{\frac{1}{4}}(\varepsilon_{1}\varepsilon_{2})^{\frac{1}{2}}}, \\ && \frac{|\varepsilon_{2}-\varepsilon^{*}|} {(\varepsilon^{*})^{\frac{1}{4}}(\varepsilon_{1}\varepsilon_{2})^{\frac{1}{2}}}, \frac{(\varepsilon^{*})^{\frac{3}{4}}}{(\varepsilon_{1})^{\frac{1}{2}}}, \frac{(\varepsilon^{*})^{\frac{3}{4}}}{(\varepsilon_{2})^{\frac{1}{2}}}, \frac{\varepsilon_{1}}{(\varepsilon_{2})^{\frac{1}{2}}}, \frac{\varepsilon_{2}}{(\varepsilon_{1})^{\frac{1}{2}}}, (\varepsilon_{1})^{\frac{1}{2}}, (\varepsilon_{2})^{\frac{1}{2}}\bigg\}. \end{eqnarray}$

$\begin{eqnarray} \|u_{2}^{r}\|_{L^{\infty}((0, T)\times\Omega)} &\leq & C(\|u_{2}^{r}\|^{\frac{1}{2}}_{L^{\infty}(0, T;L^{2}(\Omega))}\|\partial_{x}\partial_{z}u_{2}^{r}\|^{\frac{1}{2}}_{L^{\infty}(0, T;L^{2}(\Omega))} \\ &&+\|\partial_{x}u_{2}^{r}\|^{\frac{1}{2}}_{L^{\infty}(0, T;L^{2}(\Omega))}\|\partial_{z}u_{2}^{r}\|^{\frac{1}{2}}_{L^{\infty}(0, T;L^{2})}) \\ &\leq & C\max\bigg\{\frac{|\varepsilon_{1}-\varepsilon^{*}|^{\frac{1}{2}}}{(\varepsilon^{*})^{\frac{1}{8}}(\varepsilon_{1})^{\frac{1}{4}}}, \frac{|\varepsilon_{2}-\varepsilon^{*}|^{\frac{1}{2}}}{(\varepsilon^{*})^{\frac{1}{8}}(\varepsilon_{2})^{\frac{1}{4}}}, (\varepsilon_{1})^{\frac{1}{2}}, (\varepsilon_{2})^{\frac{1}{2}}, (\varepsilon^{*})^{\frac{3}{8}}\bigg\}\\ &&\max\bigg\{\frac{|\varepsilon_{1}-\varepsilon^{*}|^{\frac{1}{2}}} {(\varepsilon^{*})^{\frac{3}{8}}(\varepsilon_{1})^{\frac{1}{4}}}, \frac{|\varepsilon_{2}-\varepsilon^{*}|^{\frac{1}{2}}} {(\varepsilon^{*})^{\frac{3}{8}}(\varepsilon_{2})^{\frac{1}{4}}}, \frac{|\varepsilon_{1}-\varepsilon^{*}|^{\frac{1}{2}}} {(\varepsilon^{*})^{\frac{1}{8}}(\varepsilon_{1})^{\frac{1}{2}}}, \frac{|\varepsilon_{2}-\varepsilon^{*}|^{\frac{1}{2}}} {(\varepsilon^{*})^{\frac{1}{8}}(\varepsilon_{2})^{\frac{1}{2}}}, \frac{|\varepsilon_{1}-\varepsilon^{*}|^{\frac{1}{2}}} {(\varepsilon^{*})^{\frac{1}{8}}(\varepsilon_{1}\varepsilon_{2})^{\frac{1}{4}}}, \\&&\frac{|\varepsilon_{2}-\varepsilon^{*}|^{\frac{1}{2}}} {(\varepsilon^{*})^{\frac{1}{8}}(\varepsilon_{1}\varepsilon_{2})^{\frac{1}{4}}}, \frac{(\varepsilon^{*})^{\frac{3}{8}}}{(\varepsilon_{1})^{\frac{1}{4}}}, \frac{(\varepsilon^{*})^{\frac{3}{8}}}{(\varepsilon_{2})^{\frac{1}{4}}}, \frac{(\varepsilon_{1})^{\frac{1}{2}}}{(\varepsilon_{2})^{\frac{1}{4}}}, \frac{(\varepsilon_{2})^{\frac{1}{2}}}{(\varepsilon_{1})^{\frac{1}{4}}}, (\varepsilon_{1})^{\frac{1}{4}}, (\varepsilon_{2})^{\frac{1}{4}}\bigg\}.\\\|b_{2}^{r}\|_{L^{\infty}((0, T)\times\Omega)} &\leq & C\max\bigg\{\frac{|\varepsilon_{1}-\varepsilon^{*}|^{\frac{1}{2}}}{(\varepsilon^{*})^{\frac{1}{8}}(\varepsilon_{1})^{\frac{1}{4}}}, \frac{|\varepsilon_{2}-\varepsilon^{*}|^{\frac{1}{2}}}{(\varepsilon^{*})^{\frac{1}{8}}(\varepsilon_{2})^{\frac{1}{4}}}, (\varepsilon_{1})^{\frac{1}{2}}, (\varepsilon_{2})^{\frac{1}{2}}, (\varepsilon^{*})^{\frac{3}{8}}\bigg\}\\ &&\max\bigg\{\frac{|\varepsilon_{1}-\varepsilon^{*}|^{\frac{1}{2}}} {(\varepsilon^{*})^{\frac{3}{8}}(\varepsilon_{1})^{\frac{1}{4}}}, \frac{|\varepsilon_{2}-\varepsilon^{*}|^{\frac{1}{2}}} {(\varepsilon^{*})^{\frac{3}{8}}(\varepsilon_{2})^{\frac{1}{4}}}, \frac{|\varepsilon_{1}-\varepsilon^{*}|^{\frac{1}{2}}} {(\varepsilon^{*})^{\frac{1}{8}}(\varepsilon_{1})^{\frac{1}{2}}}, \frac{|\varepsilon_{2}-\varepsilon^{*}|^{\frac{1}{2}}} {(\varepsilon^{*})^{\frac{1}{8}}(\varepsilon_{2})^{\frac{1}{2}}}, \frac{|\varepsilon_{1}-\varepsilon^{*}|^{\frac{1}{2}}} {(\varepsilon^{*})^{\frac{1}{8}}(\varepsilon_{1}\varepsilon_{2})^{\frac{1}{4}}}, \\&&\frac{|\varepsilon_{2}-\varepsilon^{*}|^{\frac{1}{2}}} {(\varepsilon^{*})^{\frac{1}{8}}(\varepsilon_{1}\varepsilon_{2})^{\frac{1}{4}}}, \frac{(\varepsilon^{*})^{\frac{3}{8}}}{(\varepsilon_{1})^{\frac{1}{4}}}, \frac{(\varepsilon^{*})^{\frac{3}{8}}}{(\varepsilon_{2})^{\frac{1}{4}}}, \frac{(\varepsilon_{1})^{\frac{1}{2}}}{(\varepsilon_{2})^{\frac{1}{4}}}, \frac{(\varepsilon_{2})^{\frac{1}{2}}}{(\varepsilon_{1})^{\frac{1}{4}}}, (\varepsilon_{1})^{\frac{1}{4}}, (\varepsilon_{2})^{\frac{1}{4}}\bigg\}. \end{eqnarray}$

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