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

(1.1) $ \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} $

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

(1.2) $ \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} $

带有滑移边界条件

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

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

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

(1.5) $ \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} $

寻找系统(1.1)具有与(1.4)–(1.5)式相同形式的解

(1.6) $ \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} $

将(1.5)和(1.6)式代入到方程组(1.1)中可以得到

(1.7) $ \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} $

初始和边界条件为

(1.8) $ \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} $

这里

(1.9) $ \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} $

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

(1.10) $ \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} $

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.

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

(2.1) $ \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} $

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

(2.2) $ \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} $

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

(2.3) $ \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} $

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

(2.4) $ \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} $

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.

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

(3.1) $ \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} $

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

(3.2) $ \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} $

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

(3.3) $ \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} $

(3.4) $ \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} $

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

(3.6) $ \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} $

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

(3.7) $ \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} $

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

(3.8) $ \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} $

利用Gronwall不等式,我们有

(3.9) $ \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} $

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

(3.10) $ \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} $

对上述不等式关于时间积分,可以得到

(3.11) $ \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} $

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

(3.12) $ \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} $

由Gronwall不等式可以获得

(3.13) $ \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} $

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

(3.14) $ \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} $

(3.15) $ \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} $

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

(3.17) $ \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} $

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

(3.18) $ \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} $

(3.19) $ \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} $

(3.20) $ \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} $

(3.21) $ \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} $

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

(3.22) $ \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} $

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

(3.23) $ \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} $

应用Gronwall不等式可以获得

(3.24) $ \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} $

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

(3.25) $ \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} $

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

(3.26) $ \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} $

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

(3.27) $ \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} $

然后应用柯西不等式以及关于时间积分,可得

(3.28) $ \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} $

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

(3.29) $ \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} $

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

(3.30) $ \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} $

(3.31) $ \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} $

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

(3.32) $ \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} $

利用柯西不等式和Gronwall不等式,可以获得

(3.33) $ \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} $

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.

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:

(4.1) $ \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} $

(4.2) $ \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} $

(4.3) $ \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} $

这里

(4.4) $ \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} $

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

(4.5) $ \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} $

我们估计右端的每一项,

(4.6) $ \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} $

(4.7) $ \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} $

(4.8) $ \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} $

因此

(4.9) $ \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} $

然后应用Gronwall不等式以及命题3.1,可以得到

(4.10) $ \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} $

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

(4.11) $ \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} $

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

(4.12) $ \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} $

然后开始估计上面等式右端的每一项.

(4.13) $ \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} $

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

同理

(4.14) $ \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} $

(4.15) $ \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} $

(4.16) $ \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} $

接下来

(4.17) $ \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} $

类似地

(4.18) $ \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} $

对于$ I_{7} $,有

(4.19) $ \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} $

同理

(4.20) $ \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} $

接下来估计$ I_{9}, $有

(4.21) $ \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} $

同理

(4.22) $ \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} $

对于$ I_{11} $,有

(4.23) $ \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} $

这里用到了Sobolev嵌入定理.

同理对于$ I_{12} $有

(4.24) $ \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} $

接下来对于$ I_{13}, $有

(4.25) $ \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} $

同理对于最后一项可以得到

(4.26) $ \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} $

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

(4.27) $ \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} $

这样就完成了定理2.1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

(4.28) $ \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} $

开始估计右端的每一项.

(4.29) $ \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} $

(4.30) $ \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} $

(4.31) $ \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} $

将(4.29)–(4.31)式代入到(4.28)式中,可以获得

(4.32) $ \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} $

关于时间积分,有

(4.33) $ \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} $

结合(4.10)式,可以得到

(4.34) $ \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} $

对于$ b_{1}^{r} $,利用类似的方法可以得到

(4.35) $ \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} $

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

(4.36) $ \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} $

(4.37) $ \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} $

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

(4.38) $ \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} $

关于时间积分,可以得到

(4.39) $ \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} $

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

(4.40) $ \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} $

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

(4.41) $ \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} $

这就完成了证明.

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.