平面平行管道中的MHD方程组的边界层

平面平行管道中的MHD方程组的边界层

王娜^,¹, 王术²

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

Wang Na^,¹, Wang Shu²

通讯作者: 王娜, E-mail: wangna1989@hebtu.edu.cn

收稿日期: 2018-04-4

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

Received: 2018-04-4

Fund supported:

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

摘要

该文研究平面平行管道中不可压缩MHD方程组的边界层问题.利用多尺度分析和精细的能量方法，证明了当粘性系数与磁耗散系数趋近于0时，粘性与磁耗散MHD方程组的解收敛到理想MHD方程组的解.

关键词： 不可压缩MHD方程组 ; 边界层 ; 平面平行管道

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

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本文引用格式

王娜, 王术. 平面平行管道中的MHD方程组的边界层. 数学物理学报[J], 2019, 39(4): 738-760 doi:

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

磁流体力学(MHD)方程组是描述导电流体和电磁场之间相互作用的物理模型.在数学上,它是流体力学方程和磁场方程的强耦合组.在三维空间中,带有狄利克雷边界条件的不可压缩MHD方程组为

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

(1.1)

这里 $\Lambda$ 是 ${\mathbb{R} ^{3}}$ 中的有界区域, $u$ 表示速度, $b$ 表示磁场, $p$ 是压力, $\varepsilon_{1}$ 是粘性系数, $\varepsilon_{2}$ 是磁耗散系数. $F$ 和 $G$ 是给定的外力, $\alpha$ 和 $\beta$ 是给定的函数.对于此方程组的适定性和正则性问题可参看文献[2, 5, 7-8, 14, 18].当 $\varepsilon_{1}, \varepsilon_{2} = 0$ 时,上述方程组就变成了如下的理想MHD方程组

$\begin{equation} \left\{\begin{array}{ll} & {\partial_{t}u^{0}+u^{0} \cdot \nabla u^{0}-b^{0}\cdot \nabla b^{0}+\nabla p = F, }\\ & {\partial_{t}b^{0}+u^{0}\cdot\nabla b^{0}-b^{0}\cdot\nabla u^{0} = G, }\\ & {\nabla\cdot u^{0} = 0, \ \nabla\cdot b^{0} = 0, } \\ &{u^{0}|_{t = 0} = u_{0}, \ b^{0}|_{t = 0} = b_{0}. } \end{array} \right. \end{equation}$

(1.2)

带有滑移边界条件

$\begin{equation} u^{0}\cdot n = \alpha\cdot n, \ b^{0}\cdot n = \beta\cdot n, \end{equation}$

(1.3)

这里 $n$ 是区域边界 $\partial\Lambda$ 的单位外法向量.

可以观察到系统(1.1)的边界条件与系统(1.2)–(1.3)的边界条件是不同的,因此不能简单地断定在 $\varepsilon_{1}, \varepsilon_{2}\rightarrow 0$ 时这两个系统是否逼近.这就是所谓的边界层问题.它使得MHD方程组的极限问题变得同Navier-Stokes方程的极限问题一样具有困难性和挑战性.关于Navier-Stokes方程的边界层和零粘性消失极限问题,感兴趣的读者可参考文献[1, 3-4, 6, 9, 11-13, 16-17, 23].

对于不可压缩MHD方程组的边界层问题,已经有了一些结果.当速度满足特征狄利克雷边界条件(边界上无滑移无渗透),磁场满足完美传导墙边界条件, Liu等^[10]研究初始磁场的第一个分量在边界上非退化的情况,得到了二维空间中的 $L^{\infty}$ 收敛; Wang等^[21]证明了三维空间中的 $L^{2}$ 收敛以及二维空间中的 $L^{\infty}$ 收敛.当区域 $\Lambda = (0, L_{1})\times(0, L_{2})\times(0, h)$ ,系统(1.1)给定非特征狄利克雷边界条件(边界上无滑移可渗透),以及 $\varepsilon_{1} = \varepsilon_{2}$ , Xie等在文献[21]中考虑 $\alpha = (0, 0, -U), \ \beta = (0, 0, -V)$ , $|U|\neq|V|$ 的情况, Wang等在文献[20]中研究 $U = V>0$ 的情况.在这两种情况下,他们都得到了三维空间中误差函数的 $L^{2}$ 范数估计以及二维空间中误差函数的 $L^{\infty}$ 范数估计.

然而,当系统(1.1)关于速度和磁场都被赋予无渗透狄利克雷边界条件时,我们并不能得到像非特征狄利克雷边界条件那样令人满意的估计,这是因为在这两种边界条件下边界层的厚度是不同的.在无渗透狄利克雷边界条件下,更厚一点的边界层使得误差函数与边界层函数的混合项变得非常难处理. Wang等在文献[19]中研究了带有一类特殊初始值条件的粘性系数和磁耗散系数相等的情况,并且还研究了带有一般初始值条件的各向异性粘性和磁耗散系数的情况,在这两种情况下都得到了误差函数的 $L^{2}$ 范数估计.对于各向异性粘性和磁耗散MHD方程组的边界层问题, Wang等也在文献[22]中得到了同文献[19]类似的结果.

本文考虑无渗透狄利克雷边界条件的情形,将文献[1, 23-24]中关于Navier-Stokes方程的平面平行管道流推广到MHD方程组中去.这样误差函数与边界层函数的混合项就不再成为障碍了.在这种情形下,我们假定区域是无限长水平管道.并且假设在水平方向 $x$ 和 $y$ 上是周期的,因此定义

$\Lambda = [0, L]\times[0, L]\times[0, h],$

这里 $L$ 是水平方向的周期, $z = 0$ 和 $z = h$ 是垂直方向的两个边界.在本文中,我们假设所有函数在水平方向 $x$ 和 $y$ 上是周期的.令

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

(1.4)

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

(1.5)

寻找系统(1.1)具有与(1.4)–(1.5)式相同形式的解

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

(1.6)

将(1.5)和(1.6)式代入到方程组(1.1)中可以得到

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

(1.7)

初始和边界条件为

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

(1.8)

这里

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

(1.9)

其中 $\ F, \ G, \ m_{1}, \ m_{2}, \ n_{1}, \ n_{2}, \ \alpha^{i}, \ \beta^{i}$ 都不依赖于 $\varepsilon_{1}$ 和 $\varepsilon_{2}$ .定义 $\Omega\triangleq[0, L]\times[0, h]$ .并且假设初始值与边界值满足0阶兼容性条件.我们可以看到形为(1.6)式的解自动满足散度自由的条件.在(1.7)式中,令 $\varepsilon_{1}, \ \varepsilon_{2} = 0$ ,可以得到理想的平面平行管道流

$\begin{equation} \left\{\begin{array}{ll} & {\partial_{t}u_{1}^{0} = F_{1}, }\\ & {\partial_{t}u_{2}^{0}+u_{1}^{0}\partial_{x}u_{2}^{0}-b_{1}^{0}\partial_{x}b_{2}^{0} = F_{2}, }\\ & {\partial_{t}b_{1}^{0} = G_{1}, }\\ & {\partial_{t}b_{2}^{0}+u_{1}^{0}\partial_{x}b_{2}^{0}-b_{1}^{0}\partial_{x}u_{2}^{0} = G_{2}, } \end{array} \right. \end{equation}$

(1.10)

初始值为

$u_{1}^{0}|_{t = 0} = m_{1}(z), \ u_{2}^{0}|_{t = 0} = m_{2}(x, z),$

$b_{1}^{0}|_{t = 0} = n_{1}(z), \ b_{2}^{0}|_{t = 0} = n_{2}(x, z).$

我们可以很容易求解方程组(1.10).并且若假设 $F_{1}, F_{2}, G_{1}, G_{2}, m_{1}, m_{2}, n_{1}$ 和 $n_{2}$ 具有适当的正则性,我们就可以得到方程组解的正则性.在下面的部分中,我们将集中研究系统(1.7)和(1.10).本文的贡献是能够得到误差函数在空间和时间上的一致估计.

本文结构如下.在第2部分中,我们构造近似解并且给出本文的主要结果.第3部分致力于研究边界层函数的能量估计.第4部分是主要结果的证明.第5部分是总结.

2 近似解的构造

在引言中我们观察到若 $\varepsilon_{1}, \ \varepsilon_{2}\rightarrow 0$ ,则可以得到MHD方程组(1.7)的解在 $\Omega$ 内趋近于理想MHD方程组(1.10)的解.但是在边界 $\partial\Omega$ 上不成立.由此可推断出(1.7)的近似解应该包括内部函数与边界层函数两部分.由于有两个边界层,我们引进截断函数^[1]

$\rho(z)\in C^{\infty}([0, h]);\ 0\leq\rho(z)\leq1;\ \rho(z) = 1, z\in[0, \frac{h}{4}];\ \rho(z) = 0, z\in[\frac{h}{2}, h].$

从 $\rho(z)$ 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

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

(2.1)

这里 $\varphi^{0, 0}$ 和 $\psi^{0, 0}$ 分别是速度场和磁场在边界 $z = 0$ 附近的边界层函数, $\varphi^{0, h}$ 和 $\psi^{0, h}$ 分别是速度场和磁场在边界 $z = h$ 附近的边界层函数.令 $Z = \frac{z}{\sqrt{\varepsilon^{*}}}, \ Z^{h} = \frac{h-z}{\sqrt{\varepsilon^{*}}}$ .且边界层函数满足:当 $Z\rightarrow +\infty$ 时, $\varphi^{0, 0}, \psi^{0, 0}\rightarrow 0$ ;当 $Z^{h}\rightarrow +\infty$ 时, $\varphi^{0, h}, \psi^{0, h}\rightarrow 0$ .其中 $\sqrt{\varepsilon^{*}}$ 是边界层的厚度.

在整篇文章中, $C$ 表示不依赖于 $\varepsilon_{1}, \varepsilon_{2}$ 和 $\varepsilon^{*}$ 的常数,但是每行之间可能会不同.为了方便起见,我们不加以区分.现在给出本文的主要结果.

定理2.1 假设 $\alpha_{1}^{i}(t), \beta_{1}^{i}(t)\in H^{1}(0, T)$ , $\alpha_{2}^{i}(t, x), \beta_{2}^{i}(t, x)\in H^{1}(0, T;H^{4}(0, L))$ , $i = 0, h$ , $m_{1}(z), n_{1}(z)\in H^{2}(0, h)$ , $m_{2}(x, z), n_{2}(x, z)\in H^{4}(\Omega)$ , $F_{1}(t, z), G_{1}(t, z)\in L^{2}(0, T;H^{2}(0, h))$ , $F_{2}(t, x, z), G_{2}(t, x, z)\in L^{2}(0, T;H^{4}(\Omega))$ ,初始值和边界值满足零阶兼容性条件,并且假设当 $\varepsilon_{1}\rightarrow 0, \varepsilon_{2}\rightarrow 0, \varepsilon^{*}\rightarrow 0$ 时,

$\frac{\varepsilon_{1}}{\sqrt{\varepsilon^{*}}}\rightarrow 0, \frac{\varepsilon_{2}}{\sqrt{\varepsilon^{*}}}\rightarrow 0, \frac{|\varepsilon_{1}-\varepsilon^{*}|}{\varepsilon^{*}}\rightarrow 0, \frac{|\varepsilon_{2}-\varepsilon^{*}|}{\varepsilon^{*}}\rightarrow 0, \frac{|\varepsilon_{1}-\varepsilon^{*}|}{(\varepsilon^{*})^{\frac{1}{4}}(\varepsilon_{1})^{\frac{1}{2}}}\rightarrow 0, \frac{|\varepsilon_{2}-\varepsilon^{*}|}{(\varepsilon^{*})^{\frac{1}{4}}(\varepsilon_{2})^{\frac{1}{2}}}\rightarrow 0$

成立,则有

$\begin{eqnarray} &&\|u-u^{app}\|_{L^{\infty}(0, T;L^{2}(\Omega))}+\|b-b^{app}\|_{L^{\infty}(0, T;L^{2}(\Omega))}\\ &&+\sqrt{\varepsilon_{1}}\|u-u^{app}\|_{L^{2}(0, T;H^{1}(\Omega))} +\sqrt{\varepsilon_{2}}\|b-b^{app}\|_{L^{2}(0, T;H^{1}(\Omega))} \leq C\delta_{1}, \end{eqnarray}$

(2.2)

其中

$\delta_{1} = \max\bigg\{\varepsilon_{1}, \varepsilon_{2}, (\varepsilon^{*})^{\frac{3}{4}}, \frac{|\varepsilon_{1}-\varepsilon^{*}|}{(\varepsilon^{*})^{\frac{1}{4}}(\varepsilon_{1})^{\frac{1}{2}}} , \frac{|\varepsilon_{2}-\varepsilon^{*}|}{(\varepsilon^{*})^{\frac{1}{4}}(\varepsilon_{2})^{\frac{1}{2}}}\bigg\}.$

利用将在第3部分中获得的边界层函数的衰减性质,我们可以进一步推断出

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

(2.3)

定理2.2 假设 $\alpha_{1}^{i}(t), \beta_{1}^{i}(t)\in H^{1}(0, T)$ , $\alpha_{2}^{i}(t, x), \beta_{2}^{i}(t, x)\in H^{1}(0, T;H^{6}(0, L))$ , $i = 0, h$ , $m_{1}(z), n_{1}(z)\in H^{2}(0, h)$ , $m_{2}(x, z), n_{2}(x, z)\in H^{6}(\Omega)$ , $F_{1}(t, z), G_{1}(t, z)\in L^{2}(0, T;H^{2}(0, h))$ , $F_{2}(t, x, z), G_{2}(t, x, z)\in L^{2}(0, T;H^{6}(\Omega))$ ,初始值和边界值满足零阶兼容性条件,并且假设当 $\varepsilon_{1}\rightarrow 0, \varepsilon_{2}\rightarrow 0, \varepsilon^{*}\rightarrow 0$ 时,成立

$\frac{\varepsilon_{1}}{\sqrt{\varepsilon^{*}}}\rightarrow 0, \frac{\varepsilon_{2}}{\sqrt{\varepsilon^{*}}}\rightarrow 0, \frac{\varepsilon_{1}}{\sqrt{\varepsilon_{2}}}\rightarrow 0, \frac{\varepsilon_{2}}{\sqrt{\varepsilon_{1}}}\rightarrow 0, \frac{(\varepsilon^{*})^{\frac{3}{4}}}{\sqrt{\varepsilon_{1}}}\rightarrow 0, \frac{(\varepsilon^{*})^{\frac{3}{4}}}{\sqrt{\varepsilon_{2}}}\rightarrow 0,$

$\frac{|\varepsilon_{1}-\varepsilon^{*}|}{\varepsilon^{*}}\rightarrow 0, \frac{|\varepsilon_{2}-\varepsilon^{*}|}{\varepsilon^{*}}\rightarrow 0, \frac{|\varepsilon_{1}-\varepsilon^{*}|}{(\varepsilon^{*})^{\frac{1}{4}}\varepsilon_{1}}\rightarrow 0, \frac{|\varepsilon_{2}-\varepsilon^{*}|}{(\varepsilon^{*})^{\frac{1}{4}}\varepsilon_{2}}\rightarrow 0,$

$\frac{|\varepsilon_{1}-\varepsilon^{*}|}{(\varepsilon^{*})^{\frac{1}{4}}(\varepsilon_{1}\varepsilon_{2})^{\frac{1}{2}}}\rightarrow 0, \frac{|\varepsilon_{2}-\varepsilon^{*}|}{(\varepsilon^{*})^{\frac{1}{4}}(\varepsilon_{1}\varepsilon_{2})^{\frac{1}{2}}}\rightarrow 0, \frac{|\varepsilon_{1}-\varepsilon^{*}|}{(\varepsilon^{*})^{\frac{3}{4}}(\varepsilon_{1})^{\frac{1}{2}}}\rightarrow 0, \frac{|\varepsilon_{2}-\varepsilon^{*}|}{(\varepsilon^{*})^{\frac{3}{4}}(\varepsilon_{2})^{\frac{1}{2}}}\rightarrow 0,$

则有

$\begin{eqnarray} \|u-u^{app}\|_{L^{\infty}((0, T)\times\Omega)}+\|b-b^{app}\|_{L^{\infty}((0, T)\times\Omega)} \leq C(\delta_{1}\delta_{2})^{\frac{1}{2}}, \end{eqnarray}$

(2.4)

这里

$\begin{eqnarray*} \delta_{2}& = &\max\bigg\{\sqrt{\varepsilon_{1}}, \sqrt{\varepsilon_{2}}, \frac{\varepsilon_{1}}{\sqrt{\varepsilon_{2}}}, \frac{\varepsilon_{2}}{\sqrt{\varepsilon_{1}}}, \frac{(\varepsilon^{*})^{\frac{3}{4}}}{\sqrt{\varepsilon_{1}}}, \frac{(\varepsilon^{*})^{\frac{3}{4}}}{\sqrt{\varepsilon_{2}}}, \frac{|\varepsilon_{1}-\varepsilon^{*}|}{(\varepsilon^{*})^{\frac{1}{4}}\varepsilon_{1}} , \frac{|\varepsilon_{2}-\varepsilon^{*}|}{(\varepsilon^{*})^{\frac{1}{4}}\varepsilon_{2}}, \\ && \frac{|\varepsilon_{1}-\varepsilon^{*}|}{(\varepsilon^{*})^{\frac{1}{4}}(\varepsilon_{1}\varepsilon_{2})^{\frac{1}{2}}} , \frac{|\varepsilon_{2}-\varepsilon^{*}|}{(\varepsilon^{*})^{\frac{1}{4}}(\varepsilon_{1}\varepsilon_{2})^{\frac{1}{2}}} , \frac{|\varepsilon_{1}-\varepsilon^{*}|}{(\varepsilon^{*})^{\frac{3}{4}}(\varepsilon_{1})^{\frac{1}{2}}}, \frac{|\varepsilon_{2}-\varepsilon^{*}|}{(\varepsilon^{*})^{\frac{3}{4}}(\varepsilon_{2})^{\frac{1}{2}}}\bigg\}. \end{eqnarray*}$

注2.1 我们可以找到这样的 $\varepsilon_{1}, \ \varepsilon_{2}$ 和 $\varepsilon^{*}$ 使得它们满足定理2.1和2.2的假设条件.例如,取 $\varepsilon_{1} = \varepsilon^{\alpha}+\varepsilon^{\beta}, \ \varepsilon_{2} = \varepsilon^{\alpha}, \ \varepsilon^{*} = \varepsilon^{\alpha}, \ \beta>\frac{5}{4}\alpha>0;\$ 或者取 $\varepsilon_{1} = \varepsilon_{2} = \varepsilon^{*}.$

在下一部分中,我们将研究边界层函数,因为其范数估计将有助于定理的证明.

3 边界层函数

在本部分中,我们将在定理关于 $\varepsilon_{1}, \ \varepsilon_{2}$ 和 $\varepsilon^{*}$ 的假设条件下,利用能量方法得到边界层函数的范数估计.

我们发现(1.7)式可以改写为

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

将(2.1)式代入到上面的方程组中去,利用关于 $u_{1}^{0}, \ u_{2}^{0}, \ b_{1}^{0}$ 和 $b_{2}^{0}$ 的方程组(1.10),以及定理2.1中关于 $\varepsilon_{1}, \ \varepsilon_{2}$ 和 $\varepsilon^{*}$ 的假设条件:当 $\varepsilon_{1}, \ \varepsilon_{2}, \ \varepsilon^{*}\rightarrow 0$ 时,有 $\frac{|\varepsilon_{1}-\varepsilon^{*}|}{\varepsilon^{*}}, \ \frac{|\varepsilon_{2}-\varepsilon^{*}|}{\varepsilon^{*}}\rightarrow 0.$ 取零阶项,最终可以得到边界 $z = 0$ 附近的边界层方程

$\begin{equation} \left\{\begin{array}{ll} & {\partial_{t}\varphi_{1}^{0, 0}-\partial_{Z}^{2}\varphi_{1}^{0, 0} = 0, }\\ & {\partial_{t}\psi_{1}^{0, 0}-\partial_{Z}^{2}\psi_{1}^{0, 0} = 0, }\\ & \partial_{t}\varphi_{2}^{0, 0}-\partial_{Z}^{2}\varphi_{2}^{0, 0}+u_{1}^{0}|_{z = 0}\partial_{x}\varphi_{2}^{0, 0} +\varphi_{1}^{0, 0}\partial_{x}u_{2}^{0}|_{z = 0}+\varphi_{1}^{0, 0}\partial_{x}\varphi_{2}^{0, 0}\\& -b_{1}^{0}|_{z = 0}\partial_{x}\psi_{2}^{0, 0}-\psi_{1}^{0, 0}\partial_{x}b_{2}^{0}|_{z = 0}- \psi_{1}^{0, 0}\partial_{x}\psi_{2}^{0, 0} = 0, \\ & \partial_{t}\psi_{2}^{0, 0}-\partial_{Z}^{2}\psi_{2}^{0, 0}+u_{1}^{0}|_{z = 0}\partial_{x}\psi_{2}^{0, 0} +\varphi_{1}^{0, 0}\partial_{x}b_{2}^{0}|_{z = 0}+\varphi_{1}^{0, 0}\partial_{x}\psi_{2}^{0, 0}\\& -b_{1}^{0}|_{z = 0}\partial_{x}\varphi_{2}^{0, 0}-\psi_{1}^{0, 0}\partial_{x}u_{2}^{0}|_{z = 0}- \psi_{1}^{0, 0}\partial_{x}\varphi_{2}^{0, 0} = 0, \end{array} \right. \end{equation}$

(3.1)

初始和边界条件为

$\begin{eqnarray*} \left\{\begin{array}{ll} & {(\varphi_{1}^{0, 0}, \varphi_{2}^{0, 0})_{t = 0} = (0, 0), }\\ & {(\psi_{1}^{0, 0}, \psi_{2}^{0, 0})_{t = 0} = (0, 0), }\\ & {(\varphi_{1}^{0, 0}, \varphi_{2}^{0, 0})|_{Z = 0} = (\alpha_{1}^{0}(t)-u_{1}^{0}|_{z = 0}, \alpha_{2}^{0}(t, x)-u_{2}^{0}|_{z = 0}), }\\ & {(\psi_{1}^{0, 0}, \psi_{2}^{0, 0})|_{Z = 0} = (\beta_{1}^{0}(t)-b_{1}^{0}|_{z = 0}, \beta_{2}^{0}(t, x)-b_{2}^{0}|_{z = 0}), }\\ &{(\varphi_{1}^{0, 0}, \varphi_{2}^{0, 0})|_{Z = \infty} = (0, 0), }\\ &{(\psi_{1}^{0, 0}, \psi_{2}^{0, 0})|_{Z = \infty} = (0, 0), } \end{array} \right. \end{eqnarray*}$

以及 $z = h$ 附近的边界层方程

$\begin{equation} \left\{\begin{array}{ll} & {\partial_{t}\varphi_{1}^{0, h}-\partial_{Z^{h}}^{2}\varphi_{1}^{0, h} = 0, }\\ & {\partial_{t}\psi_{1}^{0, h}-\partial_{Z^{h}}^{2}\psi_{1}^{0, h} = 0, }\\ & \partial_{t}\varphi_{2}^{0, h}-\partial_{Z^{h}}^{2}\varphi_{2}^{0, h}+u_{1}^{0}|_{z = h}\partial_{x}\varphi_{2}^{0, h} +\varphi_{1}^{0, h}\partial_{x}u_{2}^{0}|_{z = h}+\varphi_{1}^{0, h}\partial_{x}\varphi_{2}^{0, h}\\& -b_{1}^{0}|_{z = h}\partial_{x}\psi_{2}^{0, h}-\psi_{1}^{0, h}\partial_{x}b_{2}^{0}|_{z = h}- \psi_{1}^{0, h}\partial_{x}\psi_{2}^{0, h} = 0, \\ & \partial_{t}\psi_{2}^{0, h}-\partial_{Z^{h}}^{2}\psi_{2}^{0, h}+u_{1}^{0}|_{z = h}\partial_{x}\psi_{2}^{0, h} +\varphi_{1}^{0, h}\partial_{x}b_{2}^{0}|_{z = h}+\varphi_{1}^{0, h}\partial_{x}\psi_{2}^{0, h}\\& -b_{1}^{0}|_{z = h}\partial_{x}\varphi_{2}^{0, h}-\psi_{1}^{0, h}\partial_{x}u_{2}^{0}|_{z = h}- \psi_{1}^{0, h}\partial_{x}\varphi_{2}^{0, h} = 0.\\ \end{array} \right. \end{equation}$

(3.2)

初始和边界条件为

$\begin{eqnarray*} \left\{\begin{array}{ll} & {(\varphi_{1}^{0, h}, \varphi_{2}^{0, h})_{t = 0} = (0, 0), }\\ & {(\psi_{1}^{0, h}, \psi_{2}^{0, h})_{t = 0} = (0, 0), }\\ & {(\varphi_{1}^{0, h}, \varphi_{2}^{0, h})|_{Z^{h} = 0} = (\alpha_{1}^{h}(t)-u_{1}^{0}|_{z = h}, \alpha_{2}^{h}(t, x)-u_{2}^{0}|_{z = h}), }\\ & {(\psi_{1}^{0, h}, \psi_{2}^{0, h})|_{Z^{h} = 0} = (\beta_{1}^{h}(t)-b_{1}^{0}|_{z = h}, \beta_{2}^{h}(t, x)-b_{2}^{0}|_{z = h}), }\\ &{(\varphi_{1}^{0, h}, \varphi_{2}^{0, h})|_{Z^{h} = \infty} = (0, 0), }\\ &{(\psi_{1}^{0, h}, \psi_{2}^{0, h})|_{Z^{h} = \infty} = (0, 0).}\\ \end{array} \right. \end{eqnarray*}$

很容易获得上述方程组的适定性.由于方程组(3.1)和方程组(3.2)的对称性,我们仅处理关于 $\varphi^{0, 0}$ 和 $\psi^{0, 0}$ 的方程组(3.1),继而可以得到关于 $\varphi^{0, h}$ 和 $\psi^{0, h}$ 的相应结果.

命题3.1 假设 $\alpha_{1}^{0}(t)\in H^{1}(0, T), \ m_{1}(z)\in H^{1}(0, h), \ F_{1}(t, z)\in L^{2}(0, T;H^{1}(0, h))$ ,满足零阶兼容性条件,则成立

$\begin{equation} \|\varphi_{1}^{0, 0}\|_{L^{\infty}(0, T;L^{2}(0, +\infty))}+\|\partial_{Z}\varphi_{1}^{0, 0}\|_{L^{2}((0, T)\times(0, +\infty))} \leq C, \end{equation}$

(3.3)

$\begin{equation} \|\partial_{Z}\varphi_{1}^{0, 0}\|_{L^{\infty}(0, T;L^{2}(0, +\infty))}+\|\partial_{Z}^{2}\varphi_{1}^{0, 0}\|_{L^{2}((0, T)\times(0, +\infty))} \leq C, \end{equation}$

(3.4)

$\begin{equation} \|\varphi_{1}^{0, 0}\|_{L^{\infty}((0, T)\times(0, +\infty))} \leq C, \end{equation}$

(3.5)

$\begin{equation} \|\langle Z\rangle^{l}\varphi_{1}^{0, 0}\|_{L^{\infty}(0, T;L^{2}(0, +\infty))}+\|\langle Z\rangle^{l}\partial_{Z}\varphi_{1}^{0, 0}\|_{L^{2}((0, T)\times(0, +\infty))} \leq C, \end{equation}$

(3.6)

其中 $\langle Z\rangle\triangleq\sqrt{Z^{2}+1}, \ l\in{{\Bbb Z}_{+}}$ ,以及 $C$ 是依赖于 $T, \ l, \ \|\alpha_{1}^{0}(t)\|_{H^{1}(0, T)}, \ \|m_{1}(z)\|_{H^{1}(0, h)}$ 和 $\|F_{1}(t, z)\|_{L^{2}(0, T;H^{1}(0, h))}$ 的正常数.

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

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

(3.7)

上述方程两端同时乘上 $w_{1}$ ,然后在 $(0, +\infty)$ 上积分,可以得到

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

(3.8)

利用Gronwall不等式,我们有

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

(3.9)

这里用到了Sobolev嵌入定理: $H^{1}(0, h)\circlearrowleft L^{\infty}(0, h)$ .

在方程(3.7)两端同时乘上 $-\partial_{Z}^{2}w_{1}$ ,然后在 $(0, +\infty)$ 上积分,有

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

(3.10)

对上述不等式关于时间积分,可以得到

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

(3.11)

因此

$\begin{eqnarray*} \label{5-24}\nonumber \|w_{1}\|_{L^{\infty}((0, T)\times(0, +\infty))} &\leq& \sqrt{2} \|w_{1}\|^{\frac{1}{2}}_{L^{\infty}(0, T;L^{2}(0, +\infty))}\|\partial_{Z}w_{1}\|^{\frac{1}{2}}_{L^{\infty}(0, T;L^{2}(0, +\infty))} \\\nonumber &\leq & C(T)(\|\alpha_{1}^{0}\|_{H^{1}(0, T)}+\|m_{1}\|_{H^{1}(0, h)} +\|F_{1}\|_{L^{2}(0, T;H^{1}(0, h))}). \end{eqnarray*}$

事实上,由最大值原理也可以得到 $w_{1}$ 的 $L^{\infty}((0, T)\times(0, +\infty))$ 范数估计.

接下来,在(3.7)式两端同时乘上 $\langle Z\rangle ^{2l} w_{1}$ ,并且在 $(0, +\infty)$ 上积分,得到

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

(3.12)

由Gronwall不等式可以获得

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

(3.13)

最后,结合(3.9), (3.11)–(3.13)式以及 $w_{1}$ 的定义,可以立即得到(3.3)–(3.6)式.证毕.

因为关于 $\psi_{1}^{0, 0}$ 的方程同 $\varphi_{1}^{0, 0}$ 的类似,所以可以得到如下命题.

命题3.2 假设 $\beta_{1}^{0}(t)\in H^{1}(0, T), \ n_{1}(z)\in H^{1}(0, h), \ G_{1}(t, z)\in L^{2}(0, T;H^{1}(0, h))$ ,并且满足零阶兼容性条件,则有

$\begin{equation} \|\psi_{1}^{0, 0}\|_{L^{\infty}(0, T;L^{2}(0, +\infty))}+\|\partial_{Z}\psi_{1}^{0, 0}\|_{L^{2}((0, T)\times(0, +\infty))} \leq C, \end{equation}$

(3.14)

$\begin{equation} \|\partial_{Z}\psi_{1}^{0, 0}\|_{L^{\infty}(0, T;L^{2}(0, +\infty))}+\|\partial_{Z}^{2}\psi_{1}^{0, 0}\|_{L^{2}((0, T)\times(0, +\infty))} \leq C, \end{equation}$

(3.15)

$\begin{equation} \|\psi_{1}^{0, 0}\|_{L^{\infty}((0, T)\times(0, +\infty))} \leq C, \end{equation}$

(3.16)

$\begin{equation} \|\langle Z\rangle^{l}\psi_{1}^{0, 0}\|_{L^{\infty}(0, T;L^{2}(0, +\infty))}+\|\langle Z\rangle^{l}\partial_{Z}\psi_{1}^{0, 0}\|_{L^{2}((0, T)\times(0, +\infty))} \leq C, \end{equation}$

(3.17)

这里 $l\in{{\Bbb Z}_{+}}$ ,以及 $C$ 是依赖于 $T, l, \|\beta_{1}^{0}(t)\|_{H^{1}(0, T)},$ $\|n_{1}(z)\|_{H^{1}(0, h)}$ 和 $\|G_{1}(t, z)\|_{L^{2}(0, T;H^{1}(0, h))}$ 的正常数.

在下面的命题中,我们将得到 $\varphi_{2}^{0, 0}$ 和 $\psi_{2}^{0, 0}$ 的范数估计.

命题3.3 假设 $\alpha_{1}^{0}(t),$ $\beta_{1}^{0}(t)\in H^{1}(0, T)$ , $\alpha_{2}^{0}(t, x),$ $\beta_{2}^{0}(t, x)\in H^{1}(0, T;H^{3}(0, L))$ , $m_{1}(z),$ $n_{1}(z)\in H^{1}(0, h)$ , $m_{2}(x, z),$ $n_{2}(x, z)\in H^{3}(\Omega)$ , $F_{1}(t, z), G_{1}(t, z)\in L^{2}(0, T;H^{1}(0, h))$ , $F_{2}(t, x, z),$ $G_{2}(t, x, z)\in L^{2}(0, T;H^{3}(\Omega))$ ,初始值和边界值满足零阶兼容性条件,则有

$\begin{eqnarray} &&\|\varphi_{2}^{0, 0}\|_{L^{\infty}(0, T;L^{2}(\Omega_{\infty}))}+\|\psi_{2}^{0, 0}\|_{L^{\infty}(0, T;L^{2}(\Omega_{\infty}))} +\|\partial_{Z}\varphi_{2}^{0, 0}\|_{L^{2}((0, T)\times\Omega_{\infty})} \\ &&+\|\partial_{Z}\psi_{2}^{0, 0}\|_{L^{2}((0, T)\times\Omega_{\infty})} \leq C, \end{eqnarray}$

(3.18)

$\begin{eqnarray} &&\|\partial_{Z}\varphi_{2}^{0, 0}\|_{L^{\infty}(0, T;L^{2}(\Omega_{\infty}))}+\|\partial_{Z}\psi_{2}^{0, 0}\|_{L^{\infty}(0, T;L^{2}(\Omega_{\infty}))} +\|\partial_{Z}^{2}\varphi_{2}^{0, 0}\|_{L^{2}((0, T)\times\Omega_{\infty})} \\ &&+\|\partial_{Z}^{2}\psi_{2}^{0, 0}\|_{L^{2}((0, T)\times\Omega_{\infty})} \leq C, \end{eqnarray}$

(3.19)

$\begin{eqnarray} &&\|\varphi_{2}^{0, 0}\|_{L^{\infty}((0, T)\times\Omega_{\infty})} +\|\psi_{2}^{0, 0}\|_{L^{\infty}((0, T)\times\Omega_{\infty})} \leq C, \end{eqnarray}$

(3.20)

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

(3.21)

这里 $\Omega_{\infty}\triangleq [0, L]\times[0, +\infty),$ $l\in{{\Bbb Z}_{+}}$ 以及 $C$ 是依赖于 $T, l, \|\alpha_{1}^{0}(t)\|_{H^{1}(0, T)},$ $\|\beta_{1}^{0}(t)\|_{H^{1}(0, T)},$ $\|\alpha_{2}^{0}(t,$ $x)\|_{H^{1}(0, T;H^{3}(0, L))},$ $\|\beta_{2}^{0}(t, x)\|_{H^{1}(0, T;H^{3}(0, L))},$ $\|m_{1}(z)\|_{H^{1}(0, h)},$ $\|n_{1}(z)\|_{H^{1}(0, h)},$ $\|m_{2}(x, z)\|_{H^{3}(\Omega)},$ $\|n_{2}(x, z)\|_{H^{3}(\Omega)},$ $\|F_{1}(t, z)\|_{L^{2}(0, T;H^{1}(0, h))},$ $\|G_{1}(t, z)\|_{L^{2}(0, T;H^{1}(0, h))},$ $\|F_{2}(t, x, z)\|_{L^{2}(0, T;H^{3}(\Omega))}$ 和 $\|G_{2}(t, x, z)\|_{L^{2}(0, T;H^{3}(\Omega))}$ 的正常数.

证定义

$w_{2}(t, x, Z) = \varphi_{2}^{0, 0}(t, x, Z)-(\alpha_{2}^{0}(t, x)-u_{2}^{0}|_{z = 0}){\rm e}^{-Z},$

$w_{3}(t, x, Z) = \psi_{2}^{0, 0}(t, x, Z)-(\beta_{2}^{0}(t, x)-b_{2}^{0}|_{z = 0}){\rm e}^{-Z}.$

利用方程组(3.1)的第三个和第四个方程,可以得到

$\begin{eqnarray} \left\{\begin{array}{ll} &\partial_{t}w_{2}-\partial_{Z}^{2}w_{2}+(u_{1}^{0}|_{z = 0}+\varphi_{1}^{0, 0})\partial_{x}w_{2} -(b_{1}^{0}|_{z = 0}+\psi_{1}^{0, 0})\partial_{x}w_{3} \\& = \Gamma_{1}-\varphi_{1}^{0, 0}\partial_{x}u_{2}^{0}|_{z = 0}+\psi_{1}^{0, 0}\partial_{x}b_{2}^{0}|_{z = 0}, \\ &\partial_{t}w_{3}-\partial_{Z}^{2}w_{3}+(u_{1}^{0}|_{z = 0}+\varphi_{1}^{0, 0})\partial_{x}w_{3} -(b_{1}^{0}|_{z = 0}+\psi_{1}^{0, 0})\partial_{x}w_{2} \\& = \Gamma_{2}-\varphi_{1}^{0, 0}\partial_{x}b_{2}^{0}|_{z = 0}+\psi_{1}^{0, 0}\partial_{x}u_{2}^{0}|_{z = 0}, \\ &w_{2}|_{Z = 0} = w_{2}|_{Z = +\infty} = 0, \\ &w_{3}|_{Z = 0} = w_{3}|_{Z = +\infty} = 0, \\ & w_{2}|_{t = 0} = w_{3}|_{t = 0} = 0. \end{array} \right. \end{eqnarray}$

(3.22)

这里

$\begin{eqnarray*} \Gamma_{1}(t, x, Z)& = &(-\partial_{t}\alpha_{2}^{0}+F_{2}|_{z = 0}+\alpha_{2}^{0}-u_{2}^{0}|_{z = 0}- u_{1}^{0}|_{z = 0}\partial_{x}\alpha_{2}^{0}-\varphi_{1}^{0, 0}\partial_{x}\alpha_{2}^{0} \\&&+\varphi_{1}^{0, 0}\partial_{x}u_{2}^{0}|_{z = 0}+b_{1}^{0}|_{z = 0}\partial_{x}\beta_{2}^{0} +\psi_{1}^{0, 0}\partial_{x}\beta_{2}^{0}-\psi_{1}^{0, 0}\partial_{x}b_{2}^{0}|_{z = 0}){\rm e}^{-Z}, \\ \Gamma_{2}(t, x, Z)& = &(-\partial_{t}\beta_{2}^{0}+G_{2}|_{z = 0}+\beta_{2}^{0}-b_{2}^{0}|_{z = 0}- u_{1}^{0}|_{z = 0}\partial_{x}\beta_{2}^{0}-\varphi_{1}^{0, 0}\partial_{x}\beta_{2}^{0} \\&&+\varphi_{1}^{0, 0}\partial_{x}b_{2}^{0}|_{z = 0}+b_{1}^{0}|_{z = 0}\partial_{x}\alpha_{2}^{0} +\psi_{1}^{0, 0}\partial_{x}\alpha_{2}^{0}-\psi_{1}^{0, 0}\partial_{x}u_{2}^{0}|_{z = 0}){\rm e}^{-Z}. \end{eqnarray*}$

第1步在方程组(3.22)的第一个方程两端同时乘上 $w_{2}$ ,第二个方程两端同时乘上 $w_{3}$ ,然后在 $\Omega_{\infty}$ 上积分,可以得到

$\begin{eqnarray} && \frac{1}{2}\frac{\rm d}{{\rm d}t}(\|w_{2}\|_{L^{2}(\Omega_{\infty})}^{2}+\|w_{3}\|_{L^{2}(\Omega_{\infty})}^{2}) +\|\partial_{Z}w_{2}\|_{L^{2}(\Omega_{\infty})}^{2}+\|\partial_{Z}w_{3}\|_{L^{2}(\Omega_{\infty})}^{2}\\& = &\int_{\Omega_{\infty}}(\Gamma_{1}w_{2}+\Gamma_{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})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})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)}\\&&\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)})\|{\rm e}^{-Z}\|_{L^{2}(0, +\infty)} \|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)})\|{\rm e}^{-Z}\|_{L^{2}(0, +\infty)} \|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)}) \|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)}) \|w_{3}\|_{L^{2}(\Omega_{\infty})}\\& \leq & 6(\|w_{2}\|^{2}_{L^{2}(\Omega_{\infty})}+\|w_{3}\|^{2}_{L^{2}(\Omega_{\infty})}) +\frac{1}{2}(\|\alpha_{2}^{0}\|^{2}_{L^{2}(0, L)}+\|\partial_{t}\alpha_{2}^{0}\|^{2}_{L^{2}(0, L)} +\|\beta_{2}^{0}\|^{2}_{L^{2}(0, L)}\\&&+\|\partial_{t}\beta_{2}^{0}\|^{2}_{L^{2}(0, L)} +\|F_{2}\|^{2}_{H^{1}(\Omega)}+\|G_{2}\|^{2}_{H^{1}(\Omega)} +\|u_{2}^{0}\|^{2}_{H^{1}(\Omega)}+\|b_{2}^{0}\|^{2}_{H^{1}(\Omega)} \\&&+\|u_{1}^{0}\|^{2}_{H^{1}(0, h)}\|\alpha_{2}^{0}\|^{2}_{H^{1}(0, L)} +\|u_{1}^{0}\|^{2}_{H^{1}(0, h)}\|\beta_{2}^{0}\|^{2}_{H^{1}(0, L)} +\|\varphi_{1}^{0, 0}\|^{2}_{L^{\infty}(0, +\infty)}\|\alpha_{2}^{0}\|^{2}_{H^{1}(0, L)} \\&&+\|\varphi_{1}^{0, 0}\|^{2}_{L^{\infty}(0, +\infty)}\|\beta_{2}^{0}\|^{2}_{H^{1}(0, L)} +\|\varphi_{1}^{0, 0}\|^{2}_{L^{\infty}(0, +\infty)}\|u_{2}^{0}\|^{2}_{H^{1}(\Omega)} +\|\varphi_{1}^{0, 0}\|^{2}_{L^{\infty}(0, +\infty)}\|b_{2}^{0}\|^{2}_{H^{1}(\Omega)} \\&&+\|b_{1}^{0}\|^{2}_{H^{1}(0, h)}\|\beta_{2}^{0}\|^{2}_{H^{1}(0, L)} +\|b_{1}^{0}\|^{2}_{H^{1}(0, h)}\|\alpha_{2}^{0}\|^{2}_{H^{1}(0, L)} +\|\psi_{1}^{0, 0}\|^{2}_{L^{\infty}(0, +\infty)}\|\beta_{2}^{0}\|^{2}_{H^{1}(0, L)} \\&&+\|\psi_{1}^{0, 0}\|^{2}_{L^{\infty}(0, +\infty)}\|\alpha_{2}^{0}\|^{2}_{H^{1}(0, L)} +\|\psi_{1}^{0, 0}\|^{2}_{L^{\infty}(0, +\infty)}\|b_{2}^{0}\|^{2}_{H^{1}(\Omega)} +\|\psi_{1}^{0, 0}\|^{2}_{L^{\infty}(0, +\infty)}\|u_{2}^{0}\|^{2}_{H^{1}(\Omega)} \\&&+\|\varphi_{1}^{0, 0}\|^{2}_{L^{2}(0, +\infty)}\|u_{2}^{0}\|^{2}_{H^{1}(\Omega)} +\|\varphi_{1}^{0, 0}\|^{2}_{L^{2}(0, +\infty)}\|b_{2}^{0}\|^{2}_{H^{1}(\Omega)} +\|\psi_{1}^{0, 0}\|^{2}_{L^{2}(0, +\infty)}\|b_{2}^{0}\|^{2}_{H^{1}(\Omega)} \\ &&+\|\psi_{1}^{0, 0}\|^{2}_{L^{2}(0, +\infty)}\|u_{2}^{0}\|^{2}_{H^{1}(\Omega)}). \end{eqnarray}$

(3.23)

应用Gronwall不等式可以获得

$\begin{eqnarray} &&\|w_{2}\|^{2}_{L^{\infty}(0, T;L^{2}(\Omega_{\infty}))}+\|w_{3}\|^{2}_{L^{\infty}(0, T;L^{2}(\Omega_{\infty}))} \\ &&+\|\partial_{Z}w_{2}\|^{2}_{L^{2}((0, T)\times\Omega_{\infty})} +\|\partial_{Z}w_{3}\|^{2}_{L^{2}((0, T)\times\Omega_{\infty})} \leq C_{1}, \end{eqnarray}$

(3.24)

这里用到了命题3.1和3.2的结论,以及 $C_{1}$ 是依赖于 $T, \ \|\alpha_{1}^{0}\|_{H^{1}(0, T)},$ $\|\beta_{1}^{0}\|_{H^{1}(0, T)},$ $\|\alpha_{2}^{0}\|_{H^{1}((0, T)\times(0, L))},$ $\|\beta_{2}^{0}\|_{H^{1}((0, T)\times(0, L))},$ $\|m_{1}\|_{H^{1}(0, h)},$ $\|n_{1}\|_{H^{1}(0, h)}, \|m_{2}\|_{H^{1}(\Omega)},$ $\|n_{2}\|_{H^{1}(\Omega)},$ $\|F_{1}\|_{L^{2}(0, T;H^{1}(0, h))},$ $\|G_{1}\|_{L^{2}(0, T;H^{1}(0, h))},$ $\|F_{2}\|_{L^{2}(0, T;H^{1}(\Omega))}$ 和 $\|G_{2}\|_{L^{2}(0, T;H^{1}(\Omega))}$ 的正常数.

第2步对方程组(3.22)作用算子 $\partial_{x}$ ,我们会发现它与方程组(3.22)具有类似的结构.因此,利用同第1步类似的方法,可以得到

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

(3.25)

其中 $C_{2}$ 是依赖于 $T, \ \|\alpha_{1}^{0}\|_{H^{1}(0, T)},$ $\|\beta_{1}^{0}\|_{H^{1}(0, T)},$ $\|\alpha_{2}^{0}\|_{H^{1}(0, T;H^{2}(0, L))},$ $\|\beta_{2}^{0}\|_{H^{1}(0, T;H^{2}(0, L))},$ $\|m_{1}\|_{H^{1}(0, h)},$ $\|n_{1}\|_{H^{1}(0, h)}, \|m_{2}\|_{H^{2}(\Omega)},$ $\|n_{2}\|_{H^{2}(\Omega)},$ $\|F_{1}\|_{L^{2}(0, T;H^{1}(0, h))},$ $\|G_{1}\|_{L^{2}(0, T;H^{1}(0, h))},$ $\|F_{2}\|_{L^{2}(0, T;H^{2}(\Omega))}$ 和 $\|G_{2}\|_{L^{2}(0, T;H^{2}(\Omega))}$ 的正常数.

类似地,可以得到

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

(3.26)

这里 $C_{3}$ 是依赖于 $T, \ \|\alpha_{1}^{0}\|_{H^{1}(0, T)},$ $\|\beta_{1}^{0}\|_{H^{1}(0, T)},$ $\|\alpha_{2}^{0}\|_{H^{1}(0, T;H^{3}(0, L))},$ $\|\beta_{2}^{0}\|_{H^{1}(0, T;H^{3}(0, L))},$ $\|m_{1}\|_{H^{1}(0, h)},$ $\|n_{1}\|_{H^{1}(0, h)},$ $\|m_{2}\|_{H^{3}(\Omega)},$ $\|n_{2}\|_{H^{3}(\Omega)},$ $\|F_{1}\|_{L^{2}(0, T;H^{1}(0, h))},$ $\|G_{1}\|_{L^{2}(0, T;H^{1}(0, h))},$ $\|F_{2}\|_{L^{2}(0, T;H^{3}(\Omega))}$ 和 $\|G_{2}\|_{L^{2}(0, T;H^{3}(\Omega))}$ 的正常数.

第3步在方程组(3.22)的第一个方程两端同时乘上 $-\partial_{ZZ}w_{2}$ ,第二个方程两端同时乘上 $-\partial_{ZZ}w_{3}$ ,然后在 $\Omega_{\infty}$ 上积分,可以得到

$\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}$

(3.27)

然后应用柯西不等式以及关于时间积分,可得

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

(3.28)

这里 $C_{4}$ 是依赖于 $T,$ $\|\alpha_{1}^{0}\|_{H^{1}(0, T)},$ $\|\beta_{1}^{0}\|_{H^{1}(0, T)},$ $\|\alpha_{2}^{0}\|_{H^{1}(0, T;H^{2}(0, L))},$ $\|\beta_{2}^{0}\|_{H^{1}(0, T;H^{2}(0, L))},$ $\|m_{1}\|_{H^{1}(0, h)},$ $\|n_{1}\|_{H^{1}(0, h)},$ $\|m_{2}\|_{H^{2}(\Omega)},$ $\|n_{2}\|_{H^{2}(\Omega)},$ $\|F_{1}\|_{L^{2}(0, T;H^{1}(0, h))},$ $\|G_{1}\|_{L^{2}(0, T;H^{1}(0, h))},$ $\|F_{2}\|_{L^{2}(0, T;H^{2}(\Omega))}$ 和 $\|G_{2}\|_{L^{2}(0, T;H^{2}(\Omega))}$ 的正常数.

第4步对方程组(3.22)作用算子 $\partial_{x}$ ,将新得到的方程组的第一个方程两端同时乘上 $-\partial_{x}\partial_{Z}^{2}w_{2}$ ,第二个方程两端同时乘上 $-\partial_{x}\partial_{Z}^{2}w_{3}$ ,然后在 $\Omega_{\infty}$ 上积分,利用与第3步相同的方法可以获得

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

(3.29)

这里 $C_{5}$ 是依赖于 $T, \ \|\alpha_{1}^{0}\|_{H^{1}(0, T)},$ $\|\beta_{1}^{0}\|_{H^{1}(0, T)},$ $\|\alpha_{2}^{0}\|_{H^{1}(0, T;H^{3}(0, L))},$ $\|\beta_{2}^{0}\|_{H^{1}(0, T;H^{3}(0, L))},$ $\|m_{1}\|_{H^{1}(0, h)},$ $\|n_{1}\|_{H^{1}(0, h)},$ $\|m_{2}\|_{H^{3}(\Omega)},$ $\|n_{2}\|_{H^{3}(\Omega)},$ $\|F_{1}\|_{L^{2}(0, T;H^{1}(0, h))},$ $\|G_{1}\|_{L^{2}(0, T;H^{1}(0, h))},$ $\|F_{2}\|_{L^{2}(0, T;H^{3}(\Omega))}$ 和 $\|G_{2}\|_{L^{2}(0, T;H^{3}(\Omega))}$ 的正常数.

第5步结合(38), (39), (42)和(43)式,并且利用各向异性Sobolev不等式^{[15, 17]},我们可以得到

$\begin{eqnarray} \|w_{2}\|_{L^{\infty}((0, T)\times\Omega_{\infty})} &\leq & C(\|w_{2}\|^{\frac{1}{2}}_{L^{\infty}(0, T;L^{2}(\Omega_{\infty}))}\|\partial_{x}\partial_{Z}w_{2}\|^{\frac{1}{2}}_{L^{\infty}(0, T;L^{2}(\Omega_{\infty}))} \\ &&+\|\partial_{x}w_{2}\|^{\frac{1}{2}}_{L^{\infty}(0, T;L^{2}(\Omega_{\infty}))}\|\partial_{Z}w_{2}\|^{\frac{1}{2}}_{L^{\infty}(0, T;L^{2}(\Omega_{\infty}))}) \\&\leq& C_{6}, \end{eqnarray}$

(3.30)

$\begin{eqnarray} \|w_{3}\|_{L^{\infty}((0, T)\times\Omega_{\infty})}& \leq & C(\|w_{3}\|^{\frac{1}{2}}_{L^{\infty}(0, T;L^{2}(\Omega_{\infty}))}\|\partial_{x}\partial_{Z}w_{3}\|^{\frac{1}{2}}_{L^{\infty}(0, T;L^{2}(\Omega_{\infty}))} \\ &&+\|\partial_{x}w_{3}\|^{\frac{1}{2}}_{L^{\infty}(0, T;L^{2}(\Omega_{\infty}))}\|\partial_{Z}w_{3}\|^{\frac{1}{2}}_{L^{\infty}(0, T;L^{2}(\Omega_{\infty}))}) \\&\leq& C_{6}.\end{eqnarray}$

(3.31)

这里 $C_{6}$ 是依赖于 $T$ , ${{\left\| \alpha _{1}^{0} \right\|}_{{{H}^{1}}(0, T)}}$ , $\ \|\beta _{1}^{0}{{\|}_{{{H}^{1}}(0, T)}}$ , ${{\left\| \alpha _{2}^{0} \right\|}_{{{H}^{1}}(0, T;{{H}^{3}}(0, L))}}$ , $\|\beta _{2}^{0}{{\|}_{{{H}^{1}}(0, T;{{H}^{3}}(0, L))}}$ , $\|{{m}_{1}}{{\|}_{{{H}^{1}}(0, h)}}$ , ${{\left\| {{n}_{1}} \right\|}_{{{H}^{1}}(0, h)}}$ , $\|{{m}_{2}}{{\|}_{{{H}^{3}}(\Omega )}}$ , ${{\left\| {{n}_{2}} \right\|}_{{{H}^{3}}(\Omega )}}$ , $\|{{F}_{1}}{{\|}_{{{L}^{2}}(0, T;{{H}^{1}}(0, h))}}$ , ${{\left\| {{G}_{1}} \right\|}_{{{L}^{2}}(0, T;{{H}^{1}}(0, h))}}$ , ${{\left\| {{F}_{2}} \right\|}_{{{L}^{2}}(0, T;{{H}^{3}}(\Omega ))}}$ 和 $\|G_{2}\|_{L^{2}(0, T;H^{3}(\Omega))}$ 的正常数.

第6步在方程组(3.22)的第一个方程两端同时乘上 $\langle Z\rangle ^{2l} w_{2}$ ,第二个方程两端同时乘上 $\langle Z\rangle ^{2l} w_{3}$ ,然后在 $\Omega_{\infty}$ 上积分,利用函数在 $x$ 方向的周期性,最终可以得到

$\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}$

(3.32)

利用柯西不等式和Gronwall不等式,可以获得

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

(3.33)

这里 $C_{7}$ 是依赖于 $T, \ l, \ \|\alpha_{1}^{0}\|_{H^{1}(0, T)},$ $\|\beta_{1}^{0}\|_{H^{1}(0, T)},$ $\|\alpha_{2}^{0}\|_{H^{1}((0, T)\times(0, L))},$ $\|\beta_{2}^{0}\|_{H^{1}((0, T)\times(0, L))},$ $\|m_{1}\|_{H^{1}(0, h)},$ $\|n_{1}\|_{H^{1}(0, h)}, \|m_{2}\|_{H^{1}(\Omega)},$ $\|n_{2}\|_{H^{1}(\Omega)},$ $\|F_{1}\|_{L^{2}(0, T;H^{1}(0, h))},$ $\|G_{1}\|_{L^{2}(0, T;H^{1}(0, h))},$ $\|F_{2}\|_{L^{2}(0, T;H^{1}(\Omega))}$ 和 $\|G_{2}\|_{L^{2}(0, T;H^{1}(\Omega))}$ 的正常数.

最后,结合(3.24), (3.28), (3.30), (3.31)和(3.33)式以及 $w_{2}, \ w_{3}$ 的定义,能够很容易得到(3.18)–(3.21)式.证毕.

由于 $\partial^{k}_{x}\varphi_{2}^{0, 0}, \ \partial^{k}_{x}\psi_{2}^{0, 0}, \ k\in{{\Bbb Z}_{+}}$ 同 $\varphi_{2}^{0, 0}, \psi_{2}^{0, 0}$ 满足一样的方程,所以当 $\alpha_{2}^{0}, \ \beta_{2}^{0}, \ m_{2}, \ n_{2}, \ F_{2}$ 和 $G_{2}$ 在空间上提高 $k$ 阶正则性时,我们可以得到关于 $\partial^{k}_{x}\varphi_{2}^{0, 0}$ 和 $\partial^{k}_{x}\psi_{2}^{0, 0}$ 的与命题3.3相同的结论.

4 定理的证明

现在我们开始证明定理2.1和2.2.定义误差函数为

$u_{1}^{r} = u_{1}-u_{1}^{app}, \ \ \ \ b_{1}^{r} = b_{1}-b_{1}^{app},$

$u_{2}^{r} = u_{2}-u_{2}^{app}, \ \ \ \ b_{2}^{r} = b_{2}-b_{2}^{app}.$

利用方程组(1.7), (1.10), (3.1)和(3.2),可以获得关于误差函数 $u_{1}^{r}, \ b_{1}^{r}, \ u_{2}^{r}$ 和 $b_{2}^{r}$ 的方程:

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

(4.1)

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

(4.2)

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

(4.3)

这里

$\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}$

(4.4)

4.1 定理2.1的证明

在(4.1)式两端同时乘上 $u_{1}^{r}$ ,然后在 $[0, h]$ 上积分,得

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

(4.5)

我们估计右端的每一项,

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

(4.6)

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

(4.7)

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

(4.8)

因此

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

(4.9)

然后应用Gronwall不等式以及命题3.1,可以得到

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

(4.10)

由于关于 $b_{1}^{r}$ 的方程(4.2)同关于 $u_{1}^{r}$ 的方程类似,故可以利用相同的方法得到

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

(4.11)

接下来,我们对 $u_{2}^{r}$ 和 $b_{2}^{r}$ 进行能量估计.

在方程组(4.3)的第一个方程两端同时乘上 $u_{2}^{r}$ 以及第二个方程两端同时乘上 $b_{2}^{r}$ ,在 $\Omega$ 上积分

$\begin{eqnarray} && \frac{1}{2}\frac{\rm d}{{\rm d}t}(\|u_{2}^{r}\|_{L^{2}(\Omega)}^{2} +\|b_{2}^{r}\|_{L^{2}(\Omega)}^{2})+\varepsilon_{1}(\|\partial_{x}u_{2}^{r}\|_{L^{2}(\Omega)}^{2}+\|\partial_{z}u_{2}^{r}\|_{L^{2}(\Omega)}^{2}) \\&&+\varepsilon_{2}(\|\partial_{x}b_{2}^{r}\|_{L^{2}(\Omega)}^{2}+\|\partial_{z}b_{2}^{r}\|_{L^{2}(\Omega)}^{2}) \\& = &-\int_{\Omega}u_{1}^{r}\partial_{x}u_{2}^{app}u_{2}^{r}{\rm d}x{\rm d}z+\int_{\Omega}b_{1}^{r}\partial_{x}b_{2}^{app}u_{2}^{r}{\rm d}x{\rm d}z -\int_{\Omega}u_{1}^{r}\partial_{x}b_{2}^{app}b_{2}^{r}{\rm d}x{\rm d}z \\&&+\int_{\Omega}b_{1}^{r}\partial_{x}u_{2}^{app}b_{2}^{r}{\rm d}x{\rm d}z +\int_{\Omega}Ku_{2}^{r}{\rm d}x{\rm d}z+\int_{\Omega}Rb_{2}^{r}{\rm d}x{\rm d}z \\&&+\int_{\Omega}Pu_{2}^{r}{\rm d}x{\rm d}z+\int_{\Omega}Tb_{2}^{r}{\rm d}x{\rm d}z +\int_{\Omega}Mu_{2}^{r}{\rm d}x{\rm d}z+\int_{\Omega}Ub_{2}^{r}{\rm d}x{\rm d}z+\int_{\Omega}Nu_{2}^{r}{\rm d}x{\rm d}z \\ &&+\int_{\Omega}Vb_{2}^{r}{\rm d}x{\rm d}z +\int_{\Omega}Qu_{2}^{r}{\rm d}x{\rm d}z +\int_{\Omega}Wb_{2}^{r}{\rm d}x{\rm d}z = \sum\limits_{j = 1}^{14}I_{j}. \end{eqnarray}$

(4.12)

然后开始估计上面等式右端的每一项.

$\begin{eqnarray} I_{1}& = & -\int_{\Omega}u_{1}^{r}[\partial_{x}u_{2}^{0}+\rho(z)\partial_{x}\varphi_{2}^{0, 0}+\rho(h-z)\partial_{x}\varphi_{2}^{0, h}]u_{2}^{r}{\rm d}x{\rm d}z\\ &\leq & L^{\frac{1}{2}}\|u_{1}^{r}\|_{L^{2}(0, h)}(\|\partial_{x}u_{2}^{0}\|_{L^{\infty}(\Omega)} +\|\partial_{x}\varphi_{2}^{0, 0}\|_{L^{\infty}(\Omega_{\infty})} +\|\partial_{x}\varphi_{2}^{0, h}\|_{L^{\infty}(\Omega_{\infty})})\|u_{2}^{r}\|_{L^{2}(\Omega)}\\ &\leq & \|u_{2}^{r}\|^{2}_{L^{2}(\Omega)}+C\|u_{1}^{r}\|^{2}_{L^{2}(0, h)}(\|u_{2}^{0}\|^{2}_{H^{2+s}(\Omega)} +\|\partial_{x}\varphi_{2}^{0, 0}\|^{2}_{L^{\infty}(\Omega_{\infty})} +\|\partial_{x}\varphi_{2}^{0, h}\|^{2}_{L^{\infty}(\Omega_{\infty})}), \end{eqnarray}$

(4.13)

这里用到了Sobolev嵌入定理: $H^{2+s}(\Omega)\circlearrowleft C^{1}(\Omega), s>0$ .

同理

$\begin{eqnarray} &&I_{2}\leq \|u_{2}^{r}\|^{2}_{L^{2}(\Omega)}+C\|b_{1}^{r}\|^{2}_{L^{2}(0, h)}(\|b_{2}^{0}\|^{2}_{H^{2+s}(\Omega)} +\|\partial_{x}\psi_{2}^{0, 0}\|^{2}_{L^{\infty}(\Omega_{\infty})} +\|\partial_{x}\psi_{2}^{0, h}\|^{2}_{L^{\infty}(\Omega_{\infty})}), \end{eqnarray}$

(4.14)

$\begin{eqnarray} &&I_{3}\leq \|b_{2}^{r}\|^{2}_{L^{2}(\Omega)}+C\|u_{1}^{r}\|^{2}_{L^{2}(0, h)}(\|b_{2}^{0}\|^{2}_{H^{2+s}(\Omega)} +\|\partial_{x}\psi_{2}^{0, 0}\|^{2}_{L^{\infty}(\Omega_{\infty})} +\|\partial_{x}\psi_{2}^{0, h}\|^{2}_{L^{\infty}(\Omega_{\infty})}), \end{eqnarray}$

(4.15)

$\begin{eqnarray} &&I_{4}\leq \|b_{2}^{r}\|^{2}_{L^{2}(\Omega)}+C\|b_{1}^{r}\|^{2}_{L^{2}(0, h)}(\|u_{2}^{0}\|^{2}_{H^{2+s}(\Omega)} +\|\partial_{x}\varphi_{2}^{0, 0}\|^{2}_{L^{\infty}(\Omega_{\infty})} +\|\partial_{x}\varphi_{2}^{0, h}\|^{2}_{L^{\infty}(\Omega_{\infty})}). \end{eqnarray}$

(4.16)

接下来

$\begin{eqnarray} I_{5}& = &\int_{\Omega}\varepsilon_{1}[\partial_{x}^{2}u_{2}^{0}+\partial_{z}^{2}u_{2}^{0}+\rho(z)\partial_{x}^{2}\varphi_{2}^{0, 0} +\rho(h-z)\partial_{x}^{2}\varphi_{2}^{0, h}+\rho''(z)\varphi_{2}^{0, 0} +\rho''(h-z)\varphi_{2}^{0, h}]u_{2}^{r}{\rm d}x{\rm d}z\\& \leq & \varepsilon_{1}(\|\partial^{2}_{x}u_{2}^{0}\|_{L^{2}(\Omega)}+\|\partial^{2}_{z}u_{2}^{0}\|_{L^{2}(\Omega)})\|u_{2}^{r}\|_{L^{2}(\Omega)} +C\varepsilon_{1}(\varepsilon^{*})^{\frac{1}{4}} \\&&\times(\|\partial^{2}_{x}\varphi_{2}^{0, 0}\|_{L^{2}(\Omega_{\infty})} +\|\partial^{2}_{x}\varphi_{2}^{0, h}\|_{L^{2}(\Omega_{\infty})} +\|\varphi_{2}^{0, 0}\|_{L^{2}(\Omega_{\infty})}+\|\varphi_{2}^{0, h}\|_{L^{2}(\Omega_{\infty})})\|u_{2}^{r}\|_{L^{2}(\Omega)} \\&\leq & \|u_{2}^{r}\|^{2}_{L^{2}(\Omega)}+C\varepsilon_{1}^{2}(\|u_{2}^{0}\|^{2}_{H^{2}(\Omega)}+\|\partial^{2}_{x}\varphi_{2}^{0, 0}\|^{2}_{L^{2}(\Omega_{\infty})} +\|\partial^{2}_{x}\varphi_{2}^{0, h}\|^{2}_{L^{2}(\Omega_{\infty})}+\|\varphi_{2}^{0, 0}\|^{2}_{L^{2}(\Omega_{\infty})}\\ &&+\|\varphi_{2}^{0, h}\|^{2}_{L^{2}(\Omega_{\infty})}). \end{eqnarray}$

(4.17)

类似地

$\begin{eqnarray} I_{6}& = &\int_{\Omega}\varepsilon_{2}[\partial_{x}^{2}b_{2}^{0}+\partial_{z}^{2}b_{2}^{0}+\rho(z)\partial_{x}^{2}\psi_{2}^{0, 0} +\rho(h-z)\partial_{x}^{2}\psi_{2}^{0, h}+\rho''(z)\psi_{2}^{0, 0} +\rho''(h-z)\psi_{2}^{0, h}]b_{2}^{r}{\rm d}x{\rm d}z\\& \leq & \|b_{2}^{r}\|^{2}_{L^{2}(\Omega)}+C\varepsilon_{2}^{2}(\|b_{2}^{0}\|^{2}_{H^{2}(\Omega)}+\|\partial^{2}_{x}\psi_{2}^{0, 0}\|^{2}_{L^{2}(\Omega_{\infty})} +\|\partial^{2}_{x}\psi_{2}^{0, h}\|^{2}_{L^{2}(\Omega_{\infty})}\\ &&+\|\psi_{2}^{0, 0}\|^{2}_{L^{2}(\Omega_{\infty})}+\|\psi_{2}^{0, h}\|^{2}_{L^{2}(\Omega_{\infty})}). \end{eqnarray}$

(4.18)

对于 $I_{7}$ ,有

$\begin{eqnarray} I_{7}& = &\frac{2\varepsilon_{1}}{\sqrt{\varepsilon^{*}}}\int_{\Omega}[\rho'(z)\partial_{Z}\varphi_{2}^{0, 0} +\rho'(h-z)\partial_{Z^{h}}\varphi_{2}^{0, h}]u_{2}^{r}{\rm d}x{\rm d}z\\& \leq & C(h)\varepsilon_{1}(\varepsilon^{*})^{\frac{1}{4}}\|u_{2}^{r}\|_{L^{2}(\Omega)} \bigg[\bigg(\int_{0}^{L}\int_{\frac{h}{4\sqrt{\varepsilon^{*}}}}^{\frac{3h}{4\sqrt{\varepsilon^{*}}}} |\langle Z\rangle\partial_{Z}\varphi_{2}^{0, 0}|^{2}{\rm d}Z{\rm d}x\bigg)^{\frac{1}{2}} \\&&+\bigg(\int_{0}^{L}\int_{\frac{h}{4\sqrt{\varepsilon^{*}}}}^{\frac{3h}{4\sqrt{\varepsilon^{*}}}} |\langle Z^{h}\rangle\partial_{Z^{h}}\varphi_{2}^{0, h}|^{2}{\rm d}Z^{h}{\rm d}x\bigg)^{\frac{1}{2}}\bigg] \\ &\leq & C(h)\varepsilon_{1}(\varepsilon^{*})^{\frac{1}{4}}\|u_{2}^{r}\|_{L^{2}(\Omega)} (\|\langle Z\rangle\partial_{Z}\varphi_{2}^{0, 0}\|_{L^{2}(\Omega_{\infty})}+\|\langle Z^{h}\rangle\partial_{Z^{h}}\varphi_{2}^{0, h}\|_{L^{2}(\Omega_{\infty})})\\ &\leq & \|u_{2}^{r}\|^{2}_{L^{2}(\Omega)}+C(h)\varepsilon_{1}^{2}(\varepsilon^{*})^{\frac{1}{2}}(\|\langle Z\rangle\partial_{Z}\varphi_{2}^{0, 0}\|^{2}_{L^{2}(\Omega_{\infty})} +\|\langle Z^{h}\rangle\partial_{Z^{h}}\varphi_{2}^{0, h}\|^{2}_{L^{2}(\Omega_{\infty})}). \end{eqnarray}$

(4.19)

同理

$\begin{eqnarray} I_{8}& = &\frac{2\varepsilon_{2}}{\sqrt{\varepsilon^{*}}}\int_{\Omega}[\rho'(z)\partial_{Z}\psi_{2}^{0, 0} +\rho'(h-z)\partial_{Z^{h}}\psi_{2}^{0, h}]b_{2}^{r}{\rm d}x{\rm d}z\\ & \leq & \|b_{2}^{r}\|^{2}_{L^{2}(\Omega)}+C(h)\varepsilon_{2}^{2}(\varepsilon^{*})^{\frac{1}{2}}(\|\langle Z\rangle\partial_{Z}\psi_{2}^{0, 0}\|^{2}_{L^{2}(\Omega_{\infty})} +\|\langle Z^{h}\rangle\partial_{Z^{h}}\psi_{2}^{0, h}\|^{2}_{L^{2}(\Omega_{\infty})}). \end{eqnarray}$

(4.20)

接下来估计 $I_{9},$ 有

$\begin{eqnarray} I_{9}& = &\frac{\varepsilon_{1}-\varepsilon^{*}}{\varepsilon^{*}}\int_{\Omega}[\rho(z)\partial_{Z}^{2}\varphi_{2}^{0, 0} +\rho(h-z)\partial_{Z^{h}}^{2}\varphi_{2}^{0, h}]u_{2}^{r}{\rm d}x{\rm d}z\\&& = -\frac{\varepsilon_{1}-\varepsilon^{*}}{\sqrt{\varepsilon^{*}}}\int_{0}^{L}\int_{0}^{h}[\rho'(z)\partial_{Z}\varphi_{2}^{0, 0} +\rho'(h-z)\partial_{Z^{h}}\varphi_{2}^{0, h}]u_{2}^{r}{\rm d}z{\rm d}x\\ && -\frac{\varepsilon_{1}-\varepsilon^{*}}{\sqrt{\varepsilon^{*}}}\int_{0}^{L}\int_{0}^{h}[\rho(z)\partial_{Z}\varphi_{2}^{0, 0} -\rho(h-z)\partial_{Z^{h}}\varphi_{2}^{0, h}]\partial_{z}u_{2}^{r}{\rm d}z{\rm d}x \\ & \leq & C(h)\frac{|\varepsilon_{1}-\varepsilon^{*}|}{(\varepsilon^{*})^{\frac{1}{4}}} (\|\partial_{Z}\varphi_{2}^{0, 0}\|_{L^{2}(\Omega_{\infty})}+\|\partial_{Z^{h}}\varphi_{2}^{0, h}\|_{L^{2}(\Omega_{\infty})}) \|u_{2}^{r}\|_{L^{2}(\Omega)}\\ && +\frac{|\varepsilon_{1}-\varepsilon^{*}|}{(\varepsilon^{*})^{\frac{1}{4}}} (\|\partial_{Z}\varphi_{2}^{0, 0}\|_{L^{2}(\Omega_{\infty})}+\|\partial_{Z^{h}}\varphi_{2}^{0, h}\|_{L^{2}(\Omega_{\infty})}) \|\partial_{z}u_{2}^{r}\|_{L^{2}(\Omega)}\\ & \leq & \|u_{2}^{r}\|^{2}_{L^{2}(\Omega)}+\frac{\varepsilon_{1}}{2}\|\partial_{z}u_{2}^{r}\|^{2}_{L^{2}(\Omega)} +C(h)\frac{(\varepsilon_{1}-\varepsilon^{*})^{2}}{\sqrt{\varepsilon^{*}}\varepsilon_{1}} (\|\partial_{Z}\varphi_{2}^{0, 0}\|^{2}_{L^{2}(\Omega_{\infty})}\\ &&+\|\partial_{Z^{h}}\varphi_{2}^{0, h}\|^{2}_{L^{2}(\Omega_{\infty})}). \end{eqnarray}$

(4.21)

同理

$\begin{eqnarray} I_{10}& = &\frac{\varepsilon_{2}-\varepsilon^{*}}{\varepsilon^{*}}\int_{\Omega}[\rho(z)\partial_{Z}^{2}\psi_{2}^{0, 0} +\rho(h-z)\partial_{Z^{h}}^{2}\psi_{2}^{0, h}]b_{2}^{r}{\rm d}x{\rm d}z\\ &\leq & \|b_{2}^{r}\|^{2}_{L^{2}(\Omega)}+\frac{\varepsilon_{2}}{2}\|\partial_{z}b_{2}^{r}\|^{2}_{L^{2}(\Omega)} +C(h)\frac{(\varepsilon_{2}-\varepsilon^{*})^{2}}{\sqrt{\varepsilon^{*}}\varepsilon_{2}} (\|\partial_{Z}\psi_{2}^{0, 0}\|^{2}_{L^{2}(\Omega_{\infty})}、、&&+\|\partial_{Z^{h}}\psi_{2}^{0, h}\|^{2}_{L^{2}(\Omega_{\infty})}). \end{eqnarray}$

(4.22)

对于 $I_{11}$ ,有

$\begin{eqnarray} I_{11}& = &-\sqrt{\varepsilon^{*}}\int_{\Omega}[\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})]u_{2}^{r}{\rm d}x{\rm d}z\\& \leq & C(\varepsilon^{*})^{\frac{3}{4}}\|u_{2}^{r}\|_{L^{2}(\Omega)}(|\partial_{z}u_{1}^{0}|_{z = \xi_{1}}| \|\langle Z\rangle\partial_{x}\varphi_{2}^{0, 0}\|_{L^{2}(\Omega_{\infty})} \\&&+\|\partial_{x}\partial_{z}u_{2}^{0}|_{z = \xi_{3}}\|_{L^{\infty}(0, L)} \|\langle Z\rangle\varphi_{1}^{0, 0}\|_{L^{2}(0, +\infty)} +|\partial_{z}b_{1}^{0}|_{z = \xi_{5}}| \|\langle Z\rangle\partial_{x}\psi_{2}^{0, 0}\|_{L^{2}(\Omega_{\infty})} \\&&+\|\partial_{x}\partial_{z}b_{2}^{0}|_{z = \xi_{7}}\|_{L^{\infty}(0, L)} \|\langle Z\rangle\psi_{1}^{0, 0}\|_{L^{2}(0, +\infty)} +|\partial_{z}u_{1}^{0}|_{z = \xi_{2}}| \|\langle Z^{h}\rangle\partial_{x}\varphi_{2}^{0, h}\|_{L^{2}(\Omega_{\infty})} \\&&+\|\partial_{x}\partial_{z}u_{2}^{0}|_{z = \xi_{4}}\|_{L^{\infty}(0, L)} \|\langle Z^{h}\rangle\varphi_{1}^{0, h}\|_{L^{2}(0, +\infty)} +|\partial_{z}b_{1}^{0}|_{z = \xi_{6}}| \|\langle Z^{h}\rangle\partial_{x}\psi_{2}^{0, h}\|_{L^{2}(\Omega_{\infty})} \\&&+\|\partial_{x}\partial_{z}b_{2}^{0}|_{z = \xi_{8}}\|_{L^{\infty}(0, L)} \|\langle Z^{h}\rangle\psi_{1}^{0, h}\|_{L^{2}(0, +\infty)})\\ & \leq & \|u_{2}^{r}\|^{2}_{L^{2}(\Omega)}+C(\varepsilon^{*})^{\frac{3}{2}}(\|u_{1}^{0}\|^{2}_{H^{2}(0, h)} \|\langle Z\rangle\partial_{x}\varphi_{2}^{0, 0}\|^{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)} +\|b_{1}^{0}\|^{2}_{H^{2}(0, h)} \|\langle Z\rangle\partial_{x}\psi_{2}^{0, 0}\|^{2}_{L^{2}(\Omega_{\infty})} \\&&+\|b_{2}^{0}\|^{2}_{H^{3+s}(\Omega)} \|\langle Z\rangle\psi_{1}^{0, 0}\|^{2}_{L^{2}(0, +\infty)}+\|u_{1}^{0}\|^{2}_{H^{2}(0, h)} \|\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^{h}\rangle\varphi_{1}^{0, h}\|^{2}_{L^{2}(0, +\infty)} +\|b_{1}^{0}\|^{2}_{H^{2}(0, h)} \|\langle Z^{h}\rangle\partial_{x}\psi_{2}^{0, h}\|^{2}_{L^{2}(\Omega_{\infty})} \\ &&+\|b_{2}^{0}\|^{2}_{H^{3+s}(\Omega)} \|\langle Z^{h}\rangle\psi_{1}^{0, h}\|^{2}_{L^{2}(0, +\infty)}), \end{eqnarray}$

(4.23)

这里用到了Sobolev嵌入定理.

同理对于 $I_{12}$ 有

$\begin{eqnarray} I_{12}& = &-\sqrt{\varepsilon^{*}}\int_{\Omega}[\rho(z)Z(\partial_{z}u_{1}^{0}|_{z = \xi_{1}}\partial_{x}\psi_{2}^{0, 0} +\partial_{x}\partial_{z}b_{2}^{0}|_{z = \xi_{7}}\varphi_{1}^{0, 0}-\partial_{z}b_{1}^{0}|_{z = \xi_{5}}\partial_{x}\varphi_{2}^{0, 0} \\&&-\partial_{x}\partial_{z}u_{2}^{0}|_{z = \xi_{3}}\psi_{1}^{0, 0}) -\rho(h-z)Z^{h}(\partial_{z}u_{1}^{0}|_{z = \xi_{2}}\partial_{x}\psi_{2}^{0, h} +\partial_{x}\partial_{z}b_{2}^{0}|_{z = \xi_{8}}\varphi_{1}^{0, h}\\&&-\partial_{z}b_{1}^{0}|_{z = \xi_{6}}\partial_{x}\varphi_{2}^{0, h} -\partial_{x}\partial_{z}u_{2}^{0}|_{z = \xi_{4}}\psi_{1}^{0, h})]b_{2}^{r}{\rm d}x{\rm d}z \\ & \leq & \|b_{2}^{r}\|^{2}_{L^{2}(\Omega)}+C(\varepsilon^{*})^{\frac{3}{2}}(\|u_{1}^{0}\|^{2}_{H^{2}(0, h)} \|\langle Z\rangle\partial_{x}\psi_{2}^{0, 0}\|^{2}_{L^{2}(\Omega_{\infty})} \\&&+\|b_{2}^{0}\|^{2}_{H^{3+s}(\Omega)} \|\langle Z\rangle\varphi_{1}^{0, 0}\|^{2}_{L^{2}(0, +\infty)} +\|b_{1}^{0}\|^{2}_{H^{2}(0, h)} \|\langle Z\rangle\partial_{x}\varphi_{2}^{0, 0}\|^{2}_{L^{2}(\Omega_{\infty})} \\&&+\|u_{2}^{0}\|^{2}_{H^{3+s}(\Omega)} \|\langle Z\rangle\psi_{1}^{0, 0}\|^{2}_{L^{2}(0, +\infty)}+\|u_{1}^{0}\|^{2}_{H^{2}(0, h)} \|\langle Z^{h}\rangle\partial_{x}\psi_{2}^{0, h}\|^{2}_{L^{2}(\Omega_{\infty})} \\&&+\|b_{2}^{0}\|^{2}_{H^{3+s}(\Omega)} \|\langle Z^{h}\rangle\varphi_{1}^{0, h}\|^{2}_{L^{2}(0, +\infty)} +\|b_{1}^{0}\|^{2}_{H^{2}(0, h)} \|\langle Z^{h}\rangle\partial_{x}\varphi_{2}^{0, h}\|^{2}_{L^{2}(\Omega_{\infty})} \\ &&+\|u_{2}^{0}\|^{2}_{H^{3+s}(\Omega)} \|\langle Z^{h}\rangle\psi_{1}^{0, h}\|^{2}_{L^{2}(0, +\infty)}). \end{eqnarray}$

(4.24)

接下来对于 $I_{13},$ 有

$\begin{eqnarray} I_{13}& = &\int_{0}^{L}\int_{\frac{h}{4}}^{\frac{3h}{4}}\rho(z)(\rho(z)-1)(-\varphi_{1}^{0, 0}\partial_{x}\varphi_{2}^{0, 0} +\psi_{1}^{0, 0}\partial_{x}\psi_{2}^{0, 0})u_{2}^{r}{\rm d}z{\rm d}x\\&& -\int_{0}^{L}\int_{\frac{h}{4}}^{\frac{3h}{4}}\rho(h-z)(\rho(h-z)-1)(\varphi_{1}^{0, h}\partial_{x}\varphi_{2}^{0, h} -\psi_{1}^{0, h}\partial_{x}\psi_{2}^{0, h})u_{2}^{r}{\rm d}z{\rm d}x\\ & \leq & C(h)(\varepsilon^{*})^{\frac{3}{4}}\|u_{2}^{r}\|_{L^{2}(\Omega)} (\|\varphi_{1}^{0, 0}\|_{L^{\infty}(0, +\infty)}\|\langle Z\rangle \partial_{x}\varphi_{2}^{0, 0}\|_{L^{2}(\Omega_{\infty})} \\&&+\|\psi_{1}^{0, 0}\|_{L^{\infty}(0, +\infty)}\|\langle Z\rangle \partial_{x}\psi_{2}^{0, 0}\|_{L^{2}(\Omega_{\infty})} +\|\varphi_{1}^{0, h}\|_{L^{\infty}(0, +\infty)}\|\langle Z^{h}\rangle \partial_{x}\varphi_{2}^{0, h}\|_{L^{2}(\Omega_{\infty})} \\&&+\|\psi_{1}^{0, h}\|_{L^{\infty}(0, +\infty)}\|\langle Z^{h}\rangle \partial_{x}\psi_{2}^{0, h}\|_{L^{2}(\Omega_{\infty})})\\& \leq & \|u_{2}^{r}\|^{2}_{L^{2}(\Omega)}+C(h)(\varepsilon^{*})^{\frac{3}{2}}(\|\varphi_{1}^{0, 0}\|^{2}_{L^{\infty}(0, +\infty)}\|\langle Z\rangle \partial_{x}\varphi_{2}^{0, 0}\|^{2}_{L^{2}(\Omega_{\infty})} \\&&+\|\psi_{1}^{0, 0}\|^{2}_{L^{\infty}(0, +\infty)}\|\langle Z\rangle \partial_{x}\psi_{2}^{0, 0}\|^{2}_{L^{2}(\Omega_{\infty})} +\|\varphi_{1}^{0, h}\|^{2}_{L^{\infty}(0, +\infty)}\|\langle Z^{h}\rangle \partial_{x}\varphi_{2}^{0, h}\|^{2}_{L^{2}(\Omega_{\infty})} \\ &&+\|\psi_{1}^{0, h}\|^{2}_{L^{\infty}(0, +\infty)}\|\langle Z^{h}\rangle \partial_{x}\psi_{2}^{0, h}\|^{2}_{L^{2}(\Omega_{\infty})}). \end{eqnarray}$

(4.25)

同理对于最后一项可以得到

$\begin{eqnarray} I_{14}& = &\int_{0}^{L}\int_{\frac{h}{4}}^{\frac{3h}{4}}\rho(z)(\rho(z)-1) (-\varphi_{1}^{0, 0}\partial_{x}\psi_{2}^{0, 0}+\psi_{1}^{0, 0}\partial_{x}\varphi_{2}^{0, 0})b_{2}^{r}{\rm d}z{\rm d}x\\&& -\int_{0}^{L}\int_{\frac{h}{4}}^{\frac{3h}{4}}\rho(h-z)(\rho(h-z)-1) (\varphi_{1}^{0, h}\partial_{x}\psi_{2}^{0, h}-\psi_{1}^{0, h}\partial_{x}\varphi_{2}^{0, h})b_{2}^{r}{\rm d}z{\rm d}x\\& \leq & \|b_{2}^{r}\|^{2}_{L^{2}(\Omega)}+C(h)(\varepsilon^{*})^{\frac{3}{2}}(\|\varphi_{1}^{0, 0}\|^{2}_{L^{\infty}(0, +\infty)}\|\langle Z\rangle \partial_{x}\psi_{2}^{0, 0}\|^{2}_{L^{2}(\Omega_{\infty})} \\&&+\|\psi_{1}^{0, 0}\|^{2}_{L^{\infty}(0, +\infty)}\|\langle Z\rangle \partial_{x}\varphi_{2}^{0, 0}\|^{2}_{L^{2}(\Omega_{\infty})} +\|\varphi_{1}^{0, h}\|^{2}_{L^{\infty}(0, +\infty)}\|\langle Z^{h}\rangle \partial_{x}\psi_{2}^{0, h}\|^{2}_{L^{2}(\Omega_{\infty})} \\ &&+\|\psi_{1}^{0, h}\|^{2}_{L^{\infty}(0, +\infty)}\|\langle Z^{h}\rangle \partial_{x}\varphi_{2}^{0, h}\|^{2}_{L^{2}(\Omega_{\infty})}). \end{eqnarray}$

(4.26)

将(4.13)–(4.26)式代入到(4.12)式中,应用Gronwall不等式,命题3.1–3.3,以及(4.10)–(4.11)式,最终可以获得

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

(4.27)

这样就完成了定理2.1的证明.

4.2 定理2.2的证明

我们将证明分为4步.

第1步方程(4.1)两端同时乘上 $-\partial_{z}^{2}u_{1}^{r}$ ,然后在 $[0, h]$ 上积分,可以得到

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

(4.28)

开始估计右端的每一项.

$\begin{eqnarray} \int_{0}^{h}A(-\partial_{z}^{2}u_{1}^{r}){\rm d}z & = &-\varepsilon_{1}\int_{0}^{h}\partial_{z}^{2}u_{1}^{0}\partial_{z}^{2}u_{1}^{r}{\rm d}z -\varepsilon_{1}\int_{\frac{h}{4}}^{\frac{3h}{4}}[\rho''(z)\varphi_{1}^{0, 0}+ \rho''(h-z)\varphi_{1}^{0, h}]\partial_{z}^{2}u_{1}^{r}{\rm d}z\\& \leq & \varepsilon_{1}\|\partial_{z}^{2}u_{1}^{0}\|_{L^{2}(0, h)}\|\partial_{z}^{2}u_{1}^{r}\|_{L^{2}(0, h)} \\&& + C(h)\varepsilon_{1}(\varepsilon^{*})^{\frac{1}{4}}(\|\varphi_{1}^{0, 0}\|_{L^{2}(0, +\infty)} +\|\varphi_{1}^{0, h}\|_{L^{2}(0, +\infty)})\|\partial_{z}^{2}u_{1}^{r}\|_{L^{2}(0, h)}\\ & \leq & \frac{\varepsilon_{1}}{6}\|\partial_{z}^{2}u_{1}^{r}\|_{L^{2}(0, h)}^{2}+C(h)\varepsilon_{1}(\|u_{1}^{0}\|^{2}_{H^{2}(0, h)} +\|\varphi_{1}^{0, 0}\|^{2}_{L^{2}(0, +\infty)} \\&& +\|\varphi_{1}^{0, h}\|^{2}_{L^{2}(0, +\infty)}). \end{eqnarray}$

(4.29)

$\begin{eqnarray} \int_{0}^{h}B(-\partial_{z}^{2}u_{1}^{r}){\rm d}z & = &-\frac{2\varepsilon_{1}}{\sqrt{\varepsilon^{*}}}\int_{\frac{h}{4}}^{\frac{3h}{4}}[\rho'(z)\partial_{Z}\varphi_{1}^{0, 0}+ \rho'(h-z)\partial_{Z^{h}}\varphi_{1}^{0, h}]\partial_{z}^{2}u_{1}^{r}{\rm d}z\\ & \leq & C(h)\varepsilon_{1}(\varepsilon^{*})^{\frac{1}{4}}\|\partial_{z}^{2}u_{1}^{r}\|_{L^{2}(0, h)}(\|\langle Z\rangle \partial_{Z}\varphi_{1}^{0, 0}\|_{L^{2}(0, +\infty)}\\ &&+\|\langle Z^{h}\rangle \partial_{Z^{h}}\varphi_{1}^{0, h}\|_{L^{2}(0, +\infty)})\\& \leq & \frac{\varepsilon_{1}}{6}\|\partial_{z}^{2}u_{1}^{r}\|_{L^{2}(0, h)}^{2}+C(h)\varepsilon_{1}(\varepsilon^{*})^{\frac{1}{2}}(\|\langle Z\rangle \partial_{Z}\varphi_{1}^{0, 0}\|^{2}_{L^{2}(0, +\infty)}\\ &&+\|\langle Z^{h}\rangle \partial_{Z^{h}}\varphi_{1}^{0, h}\|^{2}_{L^{2}(0, +\infty)}). \end{eqnarray}$

(4.30)

$\begin{eqnarray} \int_{0}^{h}Y(-\partial_{z}^{2}u_{1}^{r}){\rm d}z & = &-\frac{\varepsilon_{1}-\varepsilon^{*}}{\varepsilon^{*}}\int_{0}^{h}[\rho(z)\partial_{Z}^{2}\varphi_{1}^{0, 0}+ \rho(h-z)\partial_{Z^{h}}^{2}\varphi_{1}^{0, h}]\partial_{z}^{2}u_{1}^{r}{\rm d}z\\& \leq & \frac{|\varepsilon_{1}-\varepsilon^{*}|}{(\varepsilon^{*})^{\frac{3}{4}}}\|\partial_{z}^{2}u_{1}^{r}\|_{L^{2}(0, h)} (\|\partial_{Z}^{2}\varphi_{1}^{0, 0}\|_{L^{2}(0, +\infty)}+\|\partial_{Z^{h}}^{2}\varphi_{1}^{0, h}\|_{L^{2}(0, +\infty)})\\& \leq & \frac{\varepsilon_{1}}{6}\|\partial_{z}^{2}u_{1}^{r}\|_{L^{2}(0, h)}^{2}+C\frac{(\varepsilon_{1}-\varepsilon^{*})^{2}} {(\varepsilon^{*})^{\frac{3}{2}}\varepsilon_{1}}(\|\partial_{Z}^{2}\varphi_{1}^{0, 0}\|^{2}_{L^{2}(0, +\infty)}\\ &&+\|\partial_{Z^{h}}^{2}\varphi_{1}^{0, h}\|^{2}_{L^{2}(0, +\infty)}). \end{eqnarray}$

(4.31)

将(4.29)–(4.31)式代入到(4.28)式中,可以获得

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

(4.32)

关于时间积分,有

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

(4.33)

结合(4.10)式,可以得到

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

(4.34)

对于 $b_{1}^{r}$ ,利用类似的方法可以得到

$\begin{equation} \|b_{1}^{r}\|_{L^{\infty}((0, T)\times(0, h))} \leq C\max\bigg\{\frac{|\varepsilon_{2}-\varepsilon^{*}|^{\frac{1}{2}}}{(\varepsilon^{*})^{\frac{1}{8}}(\varepsilon_{2})^{\frac{1}{4}}}, (\varepsilon_{2})^{\frac{1}{2}}\bigg\} \max\bigg\{\frac{|\varepsilon_{2}-\varepsilon^{*}|^{\frac{1}{2}}} {(\varepsilon^{*})^{\frac{3}{8}}(\varepsilon_{2})^{\frac{1}{4}}}, (\varepsilon_{2})^{\frac{1}{4}}\bigg\}. \end{equation}$

(4.35)

第2步对方程组(4.3)分别作用算子 $\partial_{x}$ 和 $\partial_{x}^{2}$ ,我们会发现得到的两个新方程组同方程组(4.3)具有类似的结构.重复在证明定理2.1中关于 $u_{2}^{r}$ 和 $b_{2}^{r}$ 的 $L^{\infty}(0, T;L^{2}(\Omega))$ 范数估计的过程,我们可以得到

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

(4.36)

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

(4.37)

第3步在方程组(4.3)的第一个方程两端同时乘上 $-\partial_{z}^{2}u_{2}^{r}$ 以及第二个方程两端同时乘上 $-\partial_{z}^{2}b_{2}^{r}$ ,然后在 $\Omega$ 上积分,有

$\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}$

(4.38)

关于时间积分,可以得到

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

(4.39)

这里用到了(4.10), (4.11)和(4.36)式以及

$\|u_{1}\|_{L^{\infty}((0, T)\times(0, h))}\leq \|u_{1}^{r}\|_{L^{\infty}((0, T)\times(0, h))}+\|u_{1}^{app}\|_{L^{\infty}((0, T)\times(0, h))}\leq C,$

$\|b_{1}\|_{L^{\infty}((0, T)\times(0, h))}\leq \|b_{1}^{r}\|_{L^{\infty}((0, T)\times(0, h))}+\|b_{1}^{app}\|_{L^{\infty}((0, T)\times(0, h))}\leq C.$

第4步对方程组(4.3)作用算子 $\partial_{x},$ 然后将新得到的方程组的第一个方程两端同乘 $-\partial_{x}\partial_{z}^{2}u_{2}^{r}$ 以及第二个方程两端同乘 $-\partial_{x}\partial_{z}^{2}b_{2}^{r}$ ,在 $\Omega$ 上积分,重复第3步中的过程可得

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

(4.40)

这里用到了(4.37)式.

最后,利用(4.27), (4.36), (4.39)和(4.40)式以及各向异性Sobolev不等式,可以得到

$\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}$

(4.41)

这就完成了证明.

5 总结

本文通过寻找带有无渗透狄利克雷边界条件的不可压缩MHD方程组的平面平行管道流解,构造出了方程组的近似解.在粘性系数、磁耗散系数以及边界层的厚度满足一定的关系时,我们得到了误差函数的 $L^{2}$ 收敛和 $L^{\infty}$ 收敛,从而说明了近似解的正确性.

参考文献

原文顺序

文献年度倒序

文中引用次数倒序

被引期刊影响因子

[1]

Mazzucato

A L

, Niu

D J

, Wang

X M

Boundary layer associated with a class of 3D nonlinear plane parallel channel flows

Indiana Univ Math J, 2011, 60: 1113- 1136

DOI:10.1512/iumj.2011.60.4479 [本文引用: 3]

[2]

Biskamp

Nonlinear Magnetohydrodynamics. Cambridge: Cambridge University Press, 1993

[本文引用: 1]

[3]

W N

Boundary layer theory and the zero viscosity limit of the Navier-Stokes equations

Acta Math Sin (English Series), 2000, 16: 207- 218

DOI:10.1007/s101140000034 [本文引用: 1]

[4]

Han

D Z

, Mazzucato

A L

, Niu

D J

, Wang

X M

Boundary layer for a class of nonlinear pipe flow

J Differential Equations, 2012, 252: 6387- 6413

DOI:10.1016/j.jde.2012.02.012 [本文引用: 1]

[5]

Duvaut

, Lions

J L

Inéquation en themoélasticite et magnétohydrodynamique

Arch Ration Mech Anal, 1972, 46: 241- 279

DOI:10.1007/BF00250512 [本文引用: 1]

[6]

Gie G M, Kelliher J P, Lopes Filho M C, et al. The vanishing viscosity limit for some symmetric flows. 2017, arXiv: 1706.06039

[本文引用: 1]

[7]

, Xin

Z P

On the regularity of weak solutions to the magnetohydrodynamic equations

J Differential Equations, 2005, 213: 235- 254

DOI:10.1016/j.jde.2004.07.002 [本文引用: 1]

[8]

, Xin

Z P

Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations

J Funct Anal, 2005, 227: 113- 152

DOI:10.1016/j.jfa.2005.06.009 [本文引用: 1]

[9]

Masmoudi

The Euler limit of the Navier-Stokes equations, and rotating fluids with boundary

Arch Ration Mech Anal, 1998, 142: 375- 394

DOI:10.1007/s002050050097 [本文引用: 1]

[10]

Liu C J, Xie F, Yang T. MHD boundary layers in Sobolev spaces without monotonicity Ⅱ: convergence theory. 2017, arXiv: 1704.00523[math.AP]

[本文引用: 1]

[11]

Masmoudi

Remarks about the inviscid limit of the Navier-Stokes system

Comm Math Phys, 2007, 270 (3): 777- 788

DOI:10.1007/s00220-006-0171-5 [本文引用: 1]

[12]

Sammartino

, Caflisch

R E

Zero viscosity limit for analytic solutions of the Navier-Stokes equations on a half space I:Existence for Euler and Prandtl equations

Comm Math Phys, 1998, 192: 433- 461

DOI:10.1007/s002200050304

[13]

Sammartino

, Caflisch

R E

Zero viscosity limit for analytic solutions of the Navier-Stokes equations on a half space Ⅱ:Construction of Navier-Stokes solution

Comm Math Phys, 1998, 192: 463- 491

DOI:10.1007/s002200050305 [本文引用: 1]

[14]

Sermange

, Temam

Some mathematical questions related to the MHD equations

Comm Pure Appl Math, 1983, 36: 635- 664

DOI:10.1002/cpa.3160360506 [本文引用: 1]

[15]

Temam

, Wang

X M

Asymptotic analysis of Oseen type equations in a channel at small viscosity

Indiana Univ Math J, 1996, 45: 863- 916

URL [本文引用: 1]

[16]

Temam

, Wang

X M

Remarks on the Prandtl equation for a permeable wall

Z Angew Math Mech, 2000, 80: 835- 843

DOI:10.1002/1521-4001(200011)80:11/12<835::AID-ZAMM835>3.0.CO;2-9 [本文引用: 1]

[17]

Temam

, Wang

X M

Boundary layer associated with incompressible Navier-Stokes equations:The noncharacteristic boundary case

J Differential Equations, 2002, 179: 647- 686

DOI:10.1006/jdeq.2001.4038 [本文引用: 2]

[18]

J H

Regularity criteria for the generalized MHD equations

Comm Partial Differential Equations, 2008, 33: 285- 306

DOI:10.1080/03605300701382530 [本文引用: 1]

[19]

Wang

, Wang

B Y

, Liu

C D

, Wang

Boundary layer problem and zero viscosity-diffusion vanishing limit of the incompressible Magnetohydrodynamic system with no-slip boundary conditions

J Differential Equations, 2017, 263: 4723- 4749

DOI:10.1016/j.jde.2017.05.025 [本文引用: 2]

[20]

王术, 王娜.

不可压缩MHD方程组的边界层问题

北京工业大学学报, 2017, 43 (10): 1596- 1603

URL [本文引用: 1]

Wang

, Wang

The boundary layer problem for the incompressible MHD equations

J Beijing Univ Technol, 2017, 43 (10): 1596- 1603

URL [本文引用: 1]

[21]

Wang

, Wang

Boundary layer problem of MHD system with noncharacteristic perfect conducting wall

Applicable Analysis, 2019, 98 (3): 516- 535

DOI:10.1080/00036811.2017.1395867 [本文引用: 2]

[22]

Wang

, Wang

Vanishing vertical limit of the incompressible combined viscosity and magnetic diffusion magnetohydrodynamic system

Math Meth Appl Sci, 2018, 41: 5015- 5049

DOI:10.1002/mma.4950 [本文引用: 1]

[23]

Nguyen

T T

, Sueur

Boundary-layer interactions in the plane-parallel incompressible flows

Nonlinearity, 2012, 25 (12): 3327- 3342

DOI:10.1088/0951-7715/25/12/3327 [本文引用: 2]

[24]

Wang

X M

A kato type theorem on zero viscosity limit of Navier-Stokes flows

Indiana Univ Math J, 2001, 50: 223- 241

DOI:10.1512/iumj.2001.50.2098 [本文引用: 1]

[25]

谢晓强, 罗琳, 李常敏.

非特征边界的MHD方程的边界层

数学年刊, 2014, 35A (2): 171- 192

URL

Xie

X Q

, Luo

, Li

C M

Boundary layer for MHD equations with the noncharacteristic boundary conditions

Chin Ann Math, 2014, 35A (2): 171- 192

URL

Boundary layer associated with a class of 3D nonlinear plane parallel channel flows

2011

... 可以观察到系统(1.1)的边界条件与系统(1.2)–(1.3)的边界条件是不同的,因此不能简单地断定在

$\varepsilon_{1}, \varepsilon_{2}\rightarrow 0$

时这两个系统是否逼近.这就是所谓的边界层问题.它使得MHD方程组的极限问题变得同Navier-Stokes方程的极限问题一样具有困难性和挑战性.关于Navier-Stokes方程的边界层和零粘性消失极限问题,感兴趣的读者可参考文献[1, 3-4, 6, 9, 11-13, 16-17, 23]. ...

... 本文考虑无渗透狄利克雷边界条件的情形,将文献[1, 23-24]中关于Navier-Stokes方程的平面平行管道流推广到MHD方程组中去.这样误差函数与边界层函数的混合项就不再成为障碍了.在这种情形下,我们假定区域是无限长水平管道.并且假设在水平方向

$x$

和

$y$

上是周期的,因此定义 ...

... 在引言中我们观察到若

$\varepsilon_{1}, \ \varepsilon_{2}\rightarrow 0$

,则可以得到MHD方程组(1.7)的解在

$\Omega$

内趋近于理想MHD方程组(1.10)的解.但是在边界

$\partial\Omega$

上不成立.由此可推断出(1.7)的近似解应该包括内部函数与边界层函数两部分.由于有两个边界层,我们引进截断函数^[1] ...

1993

... 这里

$\Lambda$

是

${\mathbb{R} ^{3}}$

中的有界区域,

$u$

表示速度,

$b$

表示磁场,

$p$

是压力,

$\varepsilon_{1}$

是粘性系数,

$\varepsilon_{2}$

是磁耗散系数.

$F$

和

$G$

是给定的外力,

$\alpha$

和

$\beta$

是给定的函数.对于此方程组的适定性和正则性问题可参看文献[2, 5, 7-8, 14, 18].当

$\varepsilon_{1}, \varepsilon_{2} = 0$

时,上述方程组就变成了如下的理想MHD方程组 ...

Boundary layer theory and the zero viscosity limit of the Navier-Stokes equations

2000

... 可以观察到系统(1.1)的边界条件与系统(1.2)–(1.3)的边界条件是不同的,因此不能简单地断定在

$\varepsilon_{1}, \varepsilon_{2}\rightarrow 0$

Boundary layer for a class of nonlinear pipe flow

2012

... 可以观察到系统(1.1)的边界条件与系统(1.2)–(1.3)的边界条件是不同的,因此不能简单地断定在

$\varepsilon_{1}, \varepsilon_{2}\rightarrow 0$

Inéquation en themoélasticite et magnétohydrodynamique

1972

... 这里

$\Lambda$

是

${\mathbb{R} ^{3}}$

中的有界区域,

$u$

表示速度,

$b$

表示磁场,

$p$

是压力,

$\varepsilon_{1}$

是粘性系数,

$\varepsilon_{2}$

是磁耗散系数.

$F$

和

$G$

是给定的外力,

$\alpha$

和

$\beta$

是给定的函数.对于此方程组的适定性和正则性问题可参看文献[2, 5, 7-8, 14, 18].当

$\varepsilon_{1}, \varepsilon_{2} = 0$

时,上述方程组就变成了如下的理想MHD方程组 ...

... 可以观察到系统(1.1)的边界条件与系统(1.2)–(1.3)的边界条件是不同的,因此不能简单地断定在

$\varepsilon_{1}, \varepsilon_{2}\rightarrow 0$

On the regularity of weak solutions to the magnetohydrodynamic equations

2005

... 这里

$\Lambda$

是

${\mathbb{R} ^{3}}$

中的有界区域,

$u$

表示速度,

$b$

表示磁场,

$p$

是压力,

$\varepsilon_{1}$

是粘性系数,

$\varepsilon_{2}$

是磁耗散系数.

$F$

和

$G$

是给定的外力,

$\alpha$

和

$\beta$

是给定的函数.对于此方程组的适定性和正则性问题可参看文献[2, 5, 7-8, 14, 18].当

$\varepsilon_{1}, \varepsilon_{2} = 0$

时,上述方程组就变成了如下的理想MHD方程组 ...

Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations

2005

... 这里

$\Lambda$

是

${\mathbb{R} ^{3}}$

中的有界区域,

$u$

表示速度,

$b$

表示磁场,

$p$

是压力,

$\varepsilon_{1}$

是粘性系数,

$\varepsilon_{2}$

是磁耗散系数.

$F$

和

$G$

是给定的外力,

$\alpha$

和

$\beta$

是给定的函数.对于此方程组的适定性和正则性问题可参看文献[2, 5, 7-8, 14, 18].当

$\varepsilon_{1}, \varepsilon_{2} = 0$

时,上述方程组就变成了如下的理想MHD方程组 ...

The Euler limit of the Navier-Stokes equations, and rotating fluids with boundary

1998

... 可以观察到系统(1.1)的边界条件与系统(1.2)–(1.3)的边界条件是不同的,因此不能简单地断定在

$\varepsilon_{1}, \varepsilon_{2}\rightarrow 0$

... 对于不可压缩MHD方程组的边界层问题,已经有了一些结果.当速度满足特征狄利克雷边界条件(边界上无滑移无渗透),磁场满足完美传导墙边界条件, Liu等^[10]研究初始磁场的第一个分量在边界上非退化的情况,得到了二维空间中的

$L^{\infty}$

收敛; Wang等^[21]证明了三维空间中的

$L^{2}$

收敛以及二维空间中的

$L^{\infty}$

收敛.当区域

$\Lambda = (0, L_{1})\times(0, L_{2})\times(0, h)$

,系统(1.1)给定非特征狄利克雷边界条件(边界上无滑移可渗透),以及

$\varepsilon_{1} = \varepsilon_{2}$

, Xie等在文献[21]中考虑

$\alpha = (0, 0, -U), \ \beta = (0, 0, -V)$

$|U|\neq|V|$

的情况, Wang等在文献[20]中研究

$U = V>0$

的情况.在这两种情况下,他们都得到了三维空间中误差函数的

$L^{2}$

范数估计以及二维空间中误差函数的

$L^{\infty}$

范数估计. ...

Remarks about the inviscid limit of the Navier-Stokes system

2007

... 可以观察到系统(1.1)的边界条件与系统(1.2)–(1.3)的边界条件是不同的,因此不能简单地断定在

$\varepsilon_{1}, \varepsilon_{2}\rightarrow 0$

Zero viscosity limit for analytic solutions of the Navier-Stokes equations on a half space I:Existence for Euler and Prandtl equations

1998

Zero viscosity limit for analytic solutions of the Navier-Stokes equations on a half space Ⅱ:Construction of Navier-Stokes solution

1998

... 可以观察到系统(1.1)的边界条件与系统(1.2)–(1.3)的边界条件是不同的,因此不能简单地断定在

$\varepsilon_{1}, \varepsilon_{2}\rightarrow 0$

Some mathematical questions related to the MHD equations

1983

... 这里

$\Lambda$

是

${\mathbb{R} ^{3}}$

中的有界区域,

$u$

表示速度,

$b$

表示磁场,

$p$

是压力,

$\varepsilon_{1}$

是粘性系数,

$\varepsilon_{2}$

是磁耗散系数.

$F$

和

$G$

是给定的外力,

$\alpha$

和

$\beta$

是给定的函数.对于此方程组的适定性和正则性问题可参看文献[2, 5, 7-8, 14, 18].当

$\varepsilon_{1}, \varepsilon_{2} = 0$

时,上述方程组就变成了如下的理想MHD方程组 ...

Asymptotic analysis of Oseen type equations in a channel at small viscosity

1996

... 第5步结合(38), (39), (42)和(43)式,并且利用各向异性Sobolev不等式^{[15, 17]},我们可以得到 ...

Remarks on the Prandtl equation for a permeable wall

2000

... 可以观察到系统(1.1)的边界条件与系统(1.2)–(1.3)的边界条件是不同的,因此不能简单地断定在

$\varepsilon_{1}, \varepsilon_{2}\rightarrow 0$

Boundary layer associated with incompressible Navier-Stokes equations:The noncharacteristic boundary case

2002

... 可以观察到系统(1.1)的边界条件与系统(1.2)–(1.3)的边界条件是不同的,因此不能简单地断定在

$\varepsilon_{1}, \varepsilon_{2}\rightarrow 0$

... 第5步结合(38), (39), (42)和(43)式,并且利用各向异性Sobolev不等式^{[15, 17]},我们可以得到 ...

Regularity criteria for the generalized MHD equations

2008

... 这里

$\Lambda$

是

${\mathbb{R} ^{3}}$

中的有界区域,

$u$

表示速度,

$b$

表示磁场,

$p$

是压力,

$\varepsilon_{1}$

是粘性系数,

$\varepsilon_{2}$

是磁耗散系数.

$F$

和

$G$

是给定的外力,

$\alpha$

和

$\beta$

是给定的函数.对于此方程组的适定性和正则性问题可参看文献[2, 5, 7-8, 14, 18].当

$\varepsilon_{1}, \varepsilon_{2} = 0$

时,上述方程组就变成了如下的理想MHD方程组 ...

Boundary layer problem and zero viscosity-diffusion vanishing limit of the incompressible Magnetohydrodynamic system with no-slip boundary conditions

2017

... 然而,当系统(1.1)关于速度和磁场都被赋予无渗透狄利克雷边界条件时,我们并不能得到像非特征狄利克雷边界条件那样令人满意的估计,这是因为在这两种边界条件下边界层的厚度是不同的.在无渗透狄利克雷边界条件下,更厚一点的边界层使得误差函数与边界层函数的混合项变得非常难处理. Wang等在文献[19]中研究了带有一类特殊初始值条件的粘性系数和磁耗散系数相等的情况,并且还研究了带有一般初始值条件的各向异性粘性和磁耗散系数的情况,在这两种情况下都得到了误差函数的

$L^{2}$

范数估计.对于各向异性粘性和磁耗散MHD方程组的边界层问题, Wang等也在文献[22]中得到了同文献[19]类似的结果. ...

... ]中得到了同文献[19]类似的结果. ...

不可压缩MHD方程组的边界层问题

2017

$L^{\infty}$

收敛; Wang等^[21]证明了三维空间中的

$L^{2}$

收敛以及二维空间中的

$L^{\infty}$

收敛.当区域

$\Lambda = (0, L_{1})\times(0, L_{2})\times(0, h)$

,系统(1.1)给定非特征狄利克雷边界条件(边界上无滑移可渗透),以及

$\varepsilon_{1} = \varepsilon_{2}$

, Xie等在文献[21]中考虑

$\alpha = (0, 0, -U), \ \beta = (0, 0, -V)$

$|U|\neq|V|$

的情况, Wang等在文献[20]中研究

$U = V>0$

的情况.在这两种情况下,他们都得到了三维空间中误差函数的

$L^{2}$

范数估计以及二维空间中误差函数的

$L^{\infty}$

范数估计. ...

不可压缩MHD方程组的边界层问题

2017

$L^{\infty}$

收敛; Wang等^[21]证明了三维空间中的

$L^{2}$

收敛以及二维空间中的

$L^{\infty}$

收敛.当区域

$\Lambda = (0, L_{1})\times(0, L_{2})\times(0, h)$

,系统(1.1)给定非特征狄利克雷边界条件(边界上无滑移可渗透),以及

$\varepsilon_{1} = \varepsilon_{2}$

, Xie等在文献[21]中考虑

$\alpha = (0, 0, -U), \ \beta = (0, 0, -V)$

$|U|\neq|V|$

的情况, Wang等在文献[20]中研究

$U = V>0$

的情况.在这两种情况下,他们都得到了三维空间中误差函数的

$L^{2}$

范数估计以及二维空间中误差函数的

$L^{\infty}$

范数估计. ...

Boundary layer problem of MHD system with noncharacteristic perfect conducting wall

2019

$L^{\infty}$

收敛; Wang等^[21]证明了三维空间中的

$L^{2}$

收敛以及二维空间中的

$L^{\infty}$

收敛.当区域

$\Lambda = (0, L_{1})\times(0, L_{2})\times(0, h)$

,系统(1.1)给定非特征狄利克雷边界条件(边界上无滑移可渗透),以及

$\varepsilon_{1} = \varepsilon_{2}$

, Xie等在文献[21]中考虑

$\alpha = (0, 0, -U), \ \beta = (0, 0, -V)$

$|U|\neq|V|$

的情况, Wang等在文献[20]中研究

$U = V>0$

的情况.在这两种情况下,他们都得到了三维空间中误差函数的

$L^{2}$

范数估计以及二维空间中误差函数的

$L^{\infty}$

范数估计. ...

... , Xie等在文献[21]中考虑

$\alpha = (0, 0, -U), \ \beta = (0, 0, -V)$

$|U|\neq|V|$

的情况, Wang等在文献[20]中研究

$U = V>0$

的情况.在这两种情况下,他们都得到了三维空间中误差函数的

$L^{2}$

范数估计以及二维空间中误差函数的

$L^{\infty}$

范数估计. ...

Vanishing vertical limit of the incompressible combined viscosity and magnetic diffusion magnetohydrodynamic system

2018

$L^{2}$

范数估计.对于各向异性粘性和磁耗散MHD方程组的边界层问题, Wang等也在文献[22]中得到了同文献[19]类似的结果. ...

Boundary-layer interactions in the plane-parallel incompressible flows

2012

... 可以观察到系统(1.1)的边界条件与系统(1.2)–(1.3)的边界条件是不同的,因此不能简单地断定在

$\varepsilon_{1}, \varepsilon_{2}\rightarrow 0$

$x$

和

$y$

上是周期的,因此定义 ...

A kato type theorem on zero viscosity limit of Navier-Stokes flows

2001

$x$

和

$y$

上是周期的,因此定义 ...

非特征边界的MHD方程的边界层

2014

非特征边界的MHD方程的边界层

2014

〈

〉