数学物理学报, 2023, 43(1): 203-218

迁移率互异的可压电扩散模型的拟中性极限

姜利敏,1,*, 贺金满,1,2

1中原工学院理学院 郑州450007

2郑州大学数学与统计学院 郑州450001

Quasi-Neutral Limit of Compressible Electro-Diffusion System with the Different Mobilities

Jiang Limin,1,*, He Jinman,1,2

1College of Science, Zhongyuan University of Technology, Zhengzhou 450007

2College of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001

通讯作者: *姜利敏, E-mail: ajiang2001@126.com

收稿日期: 2022-03-21   修回日期: 2022-08-5  

基金资助: 国家自然科学基金(12102492)
河南省高等学科重点科研项目(22A110027)
河南省博士后基金

Received: 2022-03-21   Revised: 2022-08-5  

Fund supported: The NSFC(12102492)
Key Research Projects of Henan Higher Education Institutions(22A110027)
Henan Postdoctoral Foundation

作者简介 About authors

贺金满,E-mail:hejinman1026@163.com

摘要

该文应用 Sobolev 嵌入不等式、Green 公式以及加权的能量方法, 研究了电解液中迁移率互异的可压电扩散模型(Planck-Nernest-Poisson-Navier-Stokes) 的拟中性极限.

关键词： 能量方法; 拟中性极限; 索伯列夫不等式

Abstract

In this paper, by using the Sobolev inequality, the Green formula coupling and the elaborate energy method, we study the quasi-neutral limit of compressible Planck-Nernest-Poisson-Navier-Stokes(PNPNS) system with the general mobilities of two kinds of charges, which arises in the electro-hydrodynamics.

Keywords： Energy method; Quasi-neutral limit; Sobolev inequality

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本文引用格式

姜利敏, 贺金满. 迁移率互异的可压电扩散模型的拟中性极限[J]. 数学物理学报, 2023, 43(1): 203-218

Jiang Limin, He Jinman. Quasi-Neutral Limit of Compressible Electro-Diffusion System with the Different Mobilities[J]. Acta Mathematica Scientia, 2023, 43(1): 203-218

1 引言

该文研究了迁移率互异的可压电扩散模型PNPNS(Planck Nernest Poisson Navier Stokes) 的拟中性极限[1-2]. 该系统为

$\begin{equation} n_t^\lambda = {\rm div}(\mu_n (\nabla n^\lambda - n^\lambda \nabla \phi^\lambda ) - n^\lambda v^\lambda ), \label{eq:eqnp1} \end{equation}$
$\begin{equation} p_t^\lambda = {\rm div}(\mu_p (\nabla p^\lambda + p^\lambda \nabla \phi^\lambda ) - p^\lambda v^\lambda ),\label{eq:eqnp2} \end{equation}$
$\begin{equation} -\lambda^2 {\rm div}E^\lambda = n^\lambda - p^\lambda -D(x),\label{eq:eqnp3} \end{equation}$
$ \begin{equation} (\rho^\lambda v^\lambda)_t+{\rm div}(\rho^\lambda v^\lambda \otimes v^\lambda)+A \nabla(\rho^\lambda)^2=\mu \Delta v^\lambda+\nabla{\rm div}v^\lambda+(n^\lambda-p^\lambda)\nabla \phi^\lambda, \label{eq:eqnp4} \end{equation}$
$\begin{equation} \rho^\lambda_t+{\rm div}(\rho^\lambda v^\lambda)=0,\label{eq:eqnp5} \end{equation}$

初始条件

$\begin{equation} (n^\lambda,p^\lambda,v^\lambda,\rho^\lambda)(x,0)=(n_0^\lambda(x),p_0^\lambda(x),v_0^\lambda(x),\rho_0^\lambda(x)),\label{eq:eqnp7} \end{equation} $

其中$x\in {\Bbb T}^3, t>0$, ${\Bbb T}^3$是${\Bbb R}^3$中的周期区域; 函数$n^\lambda$, $p^\lambda$, $\phi^\lambda$, $v^\lambda$, $\rho^\lambda$ 分别为负电荷浓度, 正电荷浓度, 电势, 电解液的速度和压力. 令$E^\lambda = - \nabla\phi^\lambda$为电场. 参数$\lambda>0$ 表示标量化的Debye长度, 通常很小. 参数$\mu$$>$$0$为粘性系数. $D(x)$为已知的掺杂分布函数. 参数$\mu_n$, $\mu_p$为互异的迁移率, 且都为常数. $A$ 为任意的常数. 初始值$n_0^\lambda(x)$, $p_0^\lambda(x)$, $v_0^\lambda(x)$为光滑函数, 满足

$\begin{equation} \int_{{\Bbb T}^3} \left(n_0^\lambda-p_0^\lambda-D(x)\right) {\rm d}x=0.\label{eq:eqnp8} \end{equation} $

应用公式

$\begin{equation} {\rm div}(\vec{a}\otimes\bar{b})=({\rm div}\vec{a})\otimes\vec{b}+(a\cdot\bigtriangledown)\vec{b}, \label{eq:fab1} \end{equation} $

方程(1.4)可以简化为

$\begin{equation} \rho^\lambda v^\lambda_t+\rho^\lambda v^\lambda\cdot \bigtriangledown v^\lambda+A \nabla(\rho^\lambda)^2=\mu \Delta v^\lambda+\nabla{\rm div}v^\lambda+(n^\lambda-p^\lambda)\nabla \phi^\lambda.\label{eq:eqnp4t1} \end{equation}$

假设, 当$\lambda\rightarrow 0$时, 有$(n^\lambda, p^\lambda, E^\lambda, \rho^\lambda,v^\lambda )$$\rightarrow$$(n, p, \varepsilon, \rho,v)$ 成立, 其中 $\varepsilon=-\nabla\phi$, 那么形式上系统(1.1)-(1.3),(1.5),(1.9)式有极限系统为

$\begin{equation} n_t= {\rm div}(\mu_n (\nabla n + n \varepsilon ) - n v ),\label{eq:lim-eqnp1} \end{equation}$
$\begin{equation} p_t= {\rm div}(\mu_p (\nabla p - p \varepsilon ) - p v ),\label{eq:lim-eqnp2} \end{equation} $
$\begin{equation} 0 =n - p -D(x), \label{eq:lim-eqnp3} \end{equation} $
$\begin{equation} \rho v_t +\rho v\cdot \nabla v +A\nabla(\rho^2) -\mu \triangle v = \nabla{\rm div}v-(n - p ) \varepsilon, \label{eq:lim-eqnp4} \end{equation} $
$\begin{equation} \rho_t+{\rm div}(\rho v)=0,\label{eq:lim-eqnp51} \end{equation} $

初始条件为

$\begin{equation} (n,p,\rho,v)(x,0)=(n_0,p_0,\rho_0,v_0)(x),\label{eq:lim-eqnp7} \end{equation}$

其中$n_0(x)$, $p_0(x)$, $\rho_0(x)$,$v_0(x)$为光滑函数, 并且满足下面的条件

$\begin{equation} n_0-p_0-D(x)=0.\label{eq:lim-eqnp8} \end{equation} $

据作者所知, 关于此系统的一些结论. 在掺杂函数光滑的假设下, Li[3]证明了不可压电解液中电扩散方程的拟中性极限. Wang等[4] 研究了三维空间下带有不同迁移率电解液中不可压电扩散模型的拟中性极限和边界层问题. Wang等[5]研究了电解液中不可压电扩散模型的初始层问题. Yang等[6]研究了一般初值下量子 Navier-Stokes-Poisson 方程到不可压 Euler方程的收敛性问题. Liu等[7] 研究了三维空间中 Prandtl 非正定性. 拟中性问题被国内外专家学者广泛关注,并且有很多有意义的结论, 比如文献[8⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓-19] 以及其参考文献.

该文主要研究迁移率互异的不可压PNPNS(1.1)-(1.6)的拟中性极限,并假设两个迁移率$\mu_n$与$\mu_p$的差适当小. 该文区别于不可压PNPNS 系统主要在于方程(1.5), 由于方程(1.5)中的散度不再是0, 这给能量估计带来很多困难, 比如不等式(3.21)的项. 幸运的是,通过借助不等式(3.22)-(3.23), 这些困难可以被很好的解决. 另外, 随着能量估计变得更加复杂, 该文中引入的两个$\lambda$ -加权的Lyapunov 函数(1.19)-(1.20)也变得复杂.

该文通过奇异摄动理论中的渐近匹配展开和加权的能量估计证明该文的结论. 该文的一个技巧是应用$|\mu_n-\mu_p|$的小性, 因为这样能保证系统是严格的抛物- 椭圆型系统, 可以直接应用相关结论. 为证明该文结论, 引入Gronwall型熵积分不等式

$ \begin{matrix} \Gamma^\lambda(t)+\int_0^tG^\lambda(s){\rm d}s & \leq & M{\tilde{\Gamma}^\lambda}(t=0)+M\int_0^t(\Gamma^\lambda(s)+(\Gamma^\lambda(s))^2){\rm d}s\nonumber\\ && +M(\Gamma^\lambda(t))^2 +M\int_0^t\Gamma^\lambda(s)G^\lambda(s){\rm d}s+M\lambda,\label{eq:txg3}\end{matrix}$

其中

$\begin{matrix} \tilde{\Gamma^\lambda}(0)&=&\parallel(\bar{z}^\lambda,\bar{v}^\lambda,\bar{\rho}^\lambda,\bar{z}_t^\lambda,\bar{v}_t^\lambda,\bar{\rho}_t^\lambda,\nabla\bar{z}^\lambda,\nabla\bar{v}^\lambda,\nabla\bar{\rho}^\lambda,\nabla\bar{z}_t^\lambda,\nabla\bar{v}_t^\lambda,\nabla\bar{\rho}_t^\lambda)\parallel^2(t=0) \nonumber\\ &&+\lambda^2\parallel (\bar{E}^\lambda,\lambda{\rm div}\bar{E}^\lambda,\bar{E}_t^\lambda,\lambda{\rm div}\bar{E}_t^\lambda,\lambda\nabla{\rm div}\bar{E}^\lambda,\lambda\nabla{\rm div}\bar{E}_t^\lambda) \parallel^2(t=0).\label{eq:txg4} \end{matrix}$

另外, 为给出关于$\lambda$的一致先验估计, 通过引入两个$\lambda$ -加权的Lyapunov型函数

$\begin{matrix} \Gamma^\lambda(t) &=&\parallel(\bar{z}^\lambda,\bar{v}^\lambda,\bar{E}^\lambda,\bar{\rho}^\lambda,\bar{z}_t^\lambda,\bar{v}_t^\lambda,\bar{\rho}_t^\lambda,\nabla\bar{z}^\lambda,\nabla\bar{v}^\lambda,{\rm div}\bar{v}^\lambda,\nabla\bar{\rho}^\lambda)\parallel^2\nonumber\\ &&+\parallel(\nabla\bar{z}_t^\lambda,\nabla\bar{v}_t^\lambda,\nabla\bar{\rho}_t^\lambda,\triangle\bar{z}^\lambda,\triangle\bar{v}^\lambda,\nabla{\rm div}\bar{v}^\lambda)\parallel^2\nonumber\\ &&+\lambda^2\parallel(\bar{E}^\lambda,\lambda{\rm div}\bar{E}^\lambda,\bar{E}_t^\lambda,\lambda{\rm div}\bar{E}_t^\lambda,\lambda\nabla{\rm div}\bar{E}^\lambda,\lambda\nabla{\rm div}\bar{E}_t^\lambda,\lambda\triangle{\rm div}\bar{E}^\lambda)\parallel^2\label{eq:tg1} \end{matrix} $

和

$\begin{matrix} G^\lambda(t)&=&\parallel(\bar{\rho}^\lambda,\bar{E}^\lambda,\bar{E}_t^\lambda,\nabla\bar{z}^\lambda,\nabla\bar{v}^\lambda,{\rm div}\bar{v}^\lambda,\nabla\bar{\rho}^\lambda,\nabla\bar{z}_t^\lambda,\nabla\bar{v}_t^\lambda,{\rm div}\bar{v}_t^\lambda,\nabla\bar{\rho}_t^\lambda)\parallel^2\nonumber\\ &&+\parallel(\triangle\bar{z}^\lambda,\triangle\bar{v}^\lambda,\nabla{\rm div }\bar{v}^\lambda,\triangle\bar{z}_t^\lambda,\triangle\bar{v}_t^\lambda,\nabla{\rm div}\bar{v}_t^\lambda)\parallel^2\nonumber\\ &&+\lambda^2\parallel(\lambda{\rm div}\bar{E}^\lambda,\lambda{\rm div}\bar{E}_t^\lambda,\lambda\nabla{\rm div}\bar{E}^\lambda,\lambda\nabla{\rm div}\bar{E}_t^\lambda,\lambda\triangle{\rm div}\bar{E}^\lambda,\lambda\triangle{\rm div}\bar{E}^\lambda_t)\parallel^2,\label{eq:tg2} \end{matrix} $

其中$\bar{E}^\lambda$, $\bar{v}^\lambda$和$\bar{\rho}^\lambda$由(2.1)式定义. 并且$\bar{z}^\lambda=\bar{n}^\lambda+\bar{p}^\lambda$. 另外,该文应用$\epsilon,M,M(\epsilon)$表示独立于$\lambda$的正常数, 可能行于行之间不同.

2 误差方程和定理

令

$\begin{equation} \bar{n}^\lambda=n^\lambda-n, \bar{p}^\lambda=p^\lambda-p, \bar{E}^\lambda=E^\lambda-\varepsilon, \bar{v}^\lambda=v^\lambda-v, \bar{\rho}^\lambda=\rho^\lambda-\rho, \label{eq:error-100} \end{equation}$

把(2.1)式代入系统(1.1)-(1.3),(1.5),(1.9)式. 并应用方程(1.10)-(1.14), 可得

$\begin{equation} \bar{n}^\lambda_t={\rm div}(\mu_n(\nabla\bar{n}^\lambda+n\bar{E}^\lambda)+\bar{n}^\lambda(\bar{E}^\lambda+\varepsilon)-\bar{n}^\lambda(\bar{v}^\lambda+v)-n\bar{v}^\lambda),\label{eq:eqn1} \end{equation}$
$\begin{equation} \bar{p}^\lambda_t={\rm div}(\mu_p(\nabla\bar{p}^\lambda-p\bar{E}^\lambda)-\bar{p}^\lambda(\bar{E}^\lambda+\varepsilon)-\bar{p}^\lambda(\bar{v}^\lambda+v)-p\bar{v}^\lambda),\label{eq:eqn2} \end{equation}$
$\begin{equation} -\lambda^2{\rm div}\bar{E}^\lambda=\bar{n}^\lambda-\bar{p}^\lambda+\lambda^2{\rm div}\varepsilon,\label{eq:eqn3} \end{equation}$
$\begin{matrix}\label{eqn4} &&(\bar{\rho}^\lambda+\rho)\bar{v}^\lambda_t+\bar{\rho} v_t+2A(\bar{\rho}^\lambda+\rho)\nabla\bar{\rho}^\lambda+2A\bar{\rho}^\lambda\nabla\rho+(\bar{\rho}^\lambda+\rho)(\bar{v}^\lambda+v)\nabla\bar{v}^\lambda\nonumber\\ &&+ (\bar{\rho}^\lambda+\rho)\bar{v}^\lambda\nabla v+\bar{\rho}^\lambda v \nabla v\nonumber\\ &=&\mu \Delta v+\nabla{\rm div} \bar{v}^\lambda+\lambda^2({\rm div}\bar{E}^\lambda+{\rm div}\varepsilon)(\bar{E}^\lambda+\varepsilon)-D \bar{E}^\lambda, \end{matrix}$
$\begin{equation} \bar{\rho}^\lambda_t+{\rm div}((\rho+\bar{\rho}^\lambda)\bar{v}^\lambda+\rho^\lambda v)=0,\label{eq:eqn5} \end{equation}$

初始条件

$\begin{equation} (n^\lambda,p^\lambda,v^\lambda,\rho^\lambda)(x,0)=(n_0^\lambda(x),p_0^\lambda(x),v_0^\lambda(x),\rho_0^\lambda(x)).\label{eq:eqn6} \end{equation}$

引入密度变换

$\begin{equation} \bar{z}^\lambda = \bar{n}^\lambda +\bar{p}^\lambda, ~~ \bar{n}^\lambda =\frac{\bar{z}^\lambda - \lambda^2( {\rm div}\bar{E}^\lambda+{\rm div}\varepsilon)}{2}, ~~ \bar{p}^\lambda =\frac{\bar{z}^\lambda + \lambda^2( {\rm div}\bar{E}^\lambda+{\rm div}\varepsilon)}{2}\label{eq:trans1}, \end{equation}$

应用公式

$\begin{equation} \mu_n c\pm\mu_p d =\frac{\mu_n\pm\mu_p}{2}(c+d)+\frac{\mu_n\mp\mu_p}{2}(c-d),\label{eq:fab2} \end{equation}$

为简化系统的写法, 引入记号

$\begin{equation} A=\nabla \bar{z}^\lambda + D \bar{E}^\lambda - \lambda^2 \varepsilon {\rm div}\bar{E}^\lambda-\lambda^2(\bar{E}^\lambda+\varepsilon){\rm div}\varepsilon,\label{eq:dena} \end{equation}$
$\begin{equation} B=-\lambda^2 \nabla({\rm div}\bar{E}^\lambda+{\rm div}\varepsilon)+ z\bar{E}^\lambda+\varepsilon \bar{z}^\lambda,\label{eq:denb} \end{equation}$

那么系统(2.2)-(2.7)可以等价的简化为

$\begin{matrix} \bar{z}_t^\lambda& =&\frac{\mu_n + \mu_p}{2} {\rm div}A+ \frac{\mu_n - \mu_p}{2} {\rm div}B-{\rm div}(\bar{z}^\lambda v+z\bar{v}^\lambda)-\frac{\mu_n + \mu_p}{2} \lambda^2{\rm div}(\bar{E}^\lambda{\rm div}\bar{E}^\lambda)\nonumber\\ &&+\frac{\mu_n - \mu_p}{2} {\rm div}(\bar{z}^\lambda \bar{E}^\lambda)-{\rm div}(\bar{z}^\lambda \vec{v}^\lambda),\label{eq:eqz1} \end{matrix}$
$\begin{matrix} -\lambda^2 {\rm div}E_t^\lambda &=&\frac{\mu_n + \mu_p}{2}{\rm div}B+\frac{\mu_n - \mu_p}{2}{\rm div}A+\lambda^2{\rm div}(v({\rm div}\bar{E}^\lambda+{\rm div}\varepsilon))\nonumber\\ &&+\lambda^2{\rm div}((\bar{v}^\lambda+v){\rm div}\varepsilon))-{\rm div}(D\bar{v}^\lambda)-\frac{\mu_n - \mu_p}{2} \lambda^2{\rm div}(\bar{E}^\lambda{\rm div}\bar{E}^\lambda)\nonumber\\ &&+\frac{\mu_n + \mu_p}{2} {\rm div}(\bar{z}^\lambda \bar{E}^\lambda)+\lambda^2{\rm div}(\bar{v}^\lambda {\rm div} \bar{E}^\lambda), \label{eq:eqz2} \end{matrix}$
$\begin{matrix} \bar{\rho}^\lambda \bar{v}_t^\lambda +2A\bar{\rho}^\lambda\nabla\bar{\rho}^\lambda &=&-\rho \bar{v}^\lambda_t-2A\rho\nabla\bar{\rho}^\lambda-2A\bar{\rho}^\lambda \nabla\rho-\bar{\rho}^\lambda v_t-\bar{\rho}^\lambda v \nabla v\nonumber\\ &&-\rho \bar{v}^\lambda \nabla v-\rho v\nabla \bar{v}^\lambda +\lambda^2\bar{E}^\lambda{\rm div} \varepsilon+\lambda^2\varepsilon{\rm div}\bar{E}^\lambda+\lambda^2\varepsilon^\lambda {\rm div}\varepsilon\nonumber\\ &&-D\bar{E}^\lambda+\mu\Delta \bar{v}^\lambda+\nabla{\rm div}\bar{v}^\lambda-\rho \bar{v}^\lambda \nabla \bar{v}^\lambda-\bar{\rho}^\lambda v \nabla \bar{v}^\lambda-\bar{\rho}^\lambda \bar{v}^\lambda \nabla v\nonumber\\ &&+\lambda^2\bar{E}^\lambda{\rm div}\bar{E}^\lambda-\bar{\rho}^\lambda \bar{v}^\lambda\nabla \bar{v}^\lambda,\label{eq:eqz3} \end{matrix}$
$\begin{equation} \bar{\rho}^\lambda_t+{\rm div}(\bar{\rho}^\lambda \bar{v}^\lambda)+{\rm div}(\rho \bar{v}^\lambda+v\bar{\rho}^\lambda)=0,\label{eq:eqz4} \end{equation}$

初始条件

$\begin{equation} (z^\lambda,v^\lambda,\rho^\lambda)(x,0)=(z_0^\lambda(x),v_0^\lambda(x),\rho_0^\lambda(x)). \label{eq:eqz6} \end{equation}$

形式上当$\lambda\rightarrow0$时, 有$(\bar{z}^\lambda, \bar{\rho}^\lambda, \bar{E}^\lambda, \bar{v}^\lambda )$$\rightarrow$$(z, \rho, \varepsilon, v)$, 因此有极限系统

$\begin{equation} z_t = \frac{\mu_n + \mu_p}{2} {\rm div}(\nabla z + D \varepsilon ) + \frac{\mu_n - \mu_p}{2} {\rm div} (3z \varepsilon ) -{\rm div}(3z v),\label{eq:lim-eqz1} \end{equation}$
$\begin{equation} 0 = \frac{\mu_n + \mu_p}{2}{\rm div}(3z \varepsilon) + \frac{\mu_n - \mu_p}{2}{\rm div}(\nabla z + D \varepsilon )-{\rm div}(Dv), \label{eq:lim-eqz2} \end{equation}$
$\begin{equation} \rho v_t +\rho v\cdot \nabla v +A\nabla(\rho^2) -\mu \triangle v = \nabla{\rm div}v-(n - p ) \varepsilon, \label{eq:lim-eqz3} \end{equation}$
$\begin{equation} \rho_t+{\rm div}(\rho v)=0,\label{eq:lim-eqz4} \end{equation}$

初始条件

$\begin{equation} (n,p,\rho,v)(x,0)=(n_0,p_0,\rho_0,v_0)(x).\label{eq:lim-eqz5} \end{equation}$

易证系统(1.1)-(1.6)和系统(2.12)-(2.16), 以及极限系统(1.10)-(1.15)和(2.17)-(2.21)式的等价性,此处省略证明过程.

下面给出该文的主要结论.

定理2.1 假设函数$(n^\lambda,p^\lambda,E^\lambda,v^\lambda,\rho^\lambda)$是系统(1.1)-(1.6)的局部解, 它们定义在 ${\Bbb T}^2\times[0,T^*)$上, 其中$T^*: 0<T^*<\infty$是极限系统(1.10)-(1.15)的局部光滑解$(n,p,\varepsilon,v,\rho)$ 的最大时间存在区间, 并且$n+p>k_0,0<\underline{\rho}<\rho<\overline{\rho}$, 其中$k_0,\underline{\rho}<\rho<\overline{\rho}$是正常数, 同时假设存在正常数$M_1$使得

$\begin{equation} \sum_{i=0}^3\parallel\partial_t^i(\bar{z}^\lambda,\lambda\bar{E}^\lambda,\lambda^2{\rm div}\bar{E}^\lambda,\bar{v}^\lambda,\bar{\rho}^\lambda)\parallel^2(t=0)\leq M_1\lambda \label{eq:ass1} \end{equation}$

和

$\begin{equation} \sum_{i=0}^3\parallel\partial_t^i(\nabla\bar{z}^\lambda,\lambda{\rm div}\bar{E}^\lambda,\lambda^2\nabla{\rm div}\bar{E}^\lambda,\nabla\bar{v}^\lambda,\nabla\bar{\rho}^\lambda)\parallel^2(t=0)\leq M_1\lambda \label{eq:ass2} \end{equation}$

成立. 那么系统(2.12)-(2.16)存在唯一光滑解 $(\bar{z}^\lambda,\bar{p}^\lambda,\bar{E}^\lambda,\bar{v}^\lambda,\bar{\rho}^\lambda)$ 满足对任意的$T\in(0,T^*)$ 有正常数$M,\delta<1$ 和 $\lambda_0\ll1$, 使得对任意的 $\lambda\in(0,\lambda_0]$有

$\begin{equation} \sup_{0\leq t\leq T}(\parallel(\bar{z}^\lambda,\bar{v}^\lambda,\lambda\bar{E}^\lambda_t)\parallel^2_{H^2}+\parallel(\bar{\rho}^\lambda,\bar{z}^\lambda_t,\bar{v}^\lambda_t,\bar{\rho}^\lambda_t)\parallel^2_{H^1}+\parallel\bar{E}^\lambda\parallel^2_{H^3})\leq M \lambda^{1-\delta}. \label{eq:reus1} \end{equation}$

成立.

3 能量估计

下面应用能量估计证明定理2.1.

为了简化符号, 引入$\lambda$ -加权Sobolev模如下

$\begin{equation} \mid\mid\mid W\mid\mid\mid^2=\parallel(\bar{z}^\lambda,\bar{v}^\lambda,\lambda\bar{E}_t^\lambda)\parallel^2_{H^2}+\parallel (\bar{\rho}^\lambda,\bar{z}_t^\lambda,\bar{v}_t^\lambda,\bar{\rho}_t^\lambda) \parallel^2_{H^1}+\parallel\bar{E}^\lambda\parallel^2_{H^3}.~~~~\label{eq:tg3} \end{equation}$

定理3.1 在定理 2.1的假设下, 有

$\begin{matrix} &&\parallel(\bar{z}^\lambda,\lambda \bar{E}^\lambda,\lambda^2{\rm div}\bar{E}^\lambda,\bar{v}^\lambda,\bar{\rho}^\lambda)\parallel^2\nonumber\\ &&+M \int_0^t\parallel (\nabla \bar{z}^\lambda, \bar{E}^\lambda,\lambda {\rm div}\bar{E}^\lambda, \lambda^2\nabla {\rm div}\bar{E}^\lambda,\nabla \bar{v}^\lambda,{\rm div}\bar{v}^\lambda,\bar{\rho}^\lambda) \parallel^2{\rm d}t\nonumber\\ &\leq&\parallel(\bar{z}^\lambda,\lambda \bar{E}^\lambda,\lambda^2{\rm div}\bar{E}^\lambda,\bar{v}^\lambda,\bar{\rho}^\lambda)\parallel^2(t=0)\nonumber\\ &&+M\int_0^t\parallel ( \bar{z}^\lambda,\bar{v}^\lambda,\bar{\rho}^\lambda) \parallel^2{\rm d}t+M\int_0^t \mid\mid\mid W\mid\mid\mid^4(s){\rm d}s+M\lambda.\label{eq:eq-zevest1} \end{matrix}$

证 在系统(2.12)-(2.14)两端分别乘以$\bar{z}^\lambda,-\bar{\phi}^\lambda,\bar{v}^\lambda$, 另外再在方程(2.13)两端乘以$\lambda^2 {\rm div}\bar{E}^\lambda$, 再关于$x$在$ {\Bbb T}^3$上积分, 可得

$\begin{matrix} &&\frac{\rm d}{{\rm d}t}\parallel(\bar{z}^\lambda,\lambda \bar{E}^\lambda,\lambda^2{\rm div}\bar{E}^\lambda,\bar{v}^\lambda,\bar{\rho}^\lambda)\parallel^2\nonumber \\ & &+k\parallel(\nabla \bar{z}^\lambda, \bar{E}^\lambda,\lambda {\rm div}\bar{E}^\lambda, \lambda^2\nabla {\rm div}\bar{E}^\lambda,\nabla \bar{v}^\lambda,{\rm div}\bar{v}^\lambda,\bar{\rho}^\lambda)\parallel^2\nonumber\\ & \leq & M\parallel(\bar{z}^\lambda,\bar{v}^\lambda,\bar{\rho}^\lambda)\parallel^2+M\mid\mid\mid W\mid\mid\mid^4+M\lambda, \label{eq:eq-zxy1} \end{matrix}$

对不等式(3.3)关于$t$在$[t]$上积分,可以得到不等式(3.2). 证毕

注3.1 定理3.1与定理3.2的证明类似, 此处省略该定理的证明.

定理3.2 在定理 2.1的假设下, 有

$\begin{matrix} &&\parallel(\bar{z}_t^\lambda,\lambda \bar{E}_t^\lambda,\lambda^2{\rm div}\bar{E}_t^\lambda,\bar{v}_t^\lambda,\bar{\rho}_t^\lambda)\parallel^2\nonumber \\ &&+M \int_0^t\parallel (\nabla \bar{z}_t^\lambda, \bar{E}_t^\lambda,\lambda {\rm div}\bar{E}_t^\lambda, \lambda^2\nabla {\rm div}\bar{E}_t^\lambda,\nabla \bar{v}_t^\lambda,{\rm div}\bar{v}_t^\lambda) \parallel^2{\rm d}t\nonumber \\ &\leq&\parallel(\bar{z}_t^\lambda,\lambda \bar{E}_t^\lambda,\lambda^2{\rm div}\bar{E}_t^\lambda,\bar{v}_t^\lambda,\bar{\rho}_t^\lambda)\parallel^2(t=0)\nonumber \\ &&+M\int_0^t\parallel ( \bar{z}^\lambda,\bar{z}_t^\lambda,\bar{v}^\lambda,\nabla\bar{v}^\lambda,{\rm div}\bar{v}^\lambda,\bar{v}_t^\lambda,\bar{\rho}^\lambda,\nabla\bar{\rho}^\lambda,\bar{\rho}_t^\lambda,\bar{E}^\lambda,{\rm div}\bar{E}^\lambda) \parallel^2{\rm d}t\nonumber \\ &&+M\int_0^t (\mid\mid\mid W\mid\mid\mid^4(s)+\mid\mid\mid W\mid\mid\mid^2G^\lambda(t)){\rm d}s+M\lambda.\label{eq:eq-zevest2} \end{matrix}$

证 对方程(2.12)关于$t$求导, 在方程两端乘以$\bar{z}^\lambda_t$, 然后关于$x$在区间${\Bbb T}^3$上积分, 应用Green公式, 有

$\begin{matrix} &&\frac{1}{2}\frac{\rm d}{{\rm d}t}\parallel \bar{z}_t^\lambda \parallel^2+\frac{\mu_n+\mu_p}{2}\parallel \nabla \bar{z}_t^\lambda\parallel^2\nonumber \\ &=&-\frac{\mu_n+\mu_p}{2}\int_{{\Bbb T}^3} (A-\nabla \bar{z}^\lambda)_t\nabla \bar{z}_t^\lambda {\rm d}x-\frac{\mu_n-\mu_p}{2}\int_{{\Bbb T}^3} B_t\nabla \bar{z}^\lambda_{t}{\rm d}x+\int_{{\Bbb T}^3} (z\bar{v}^\lambda+\bar{z}^\lambda v)_t\nabla \bar{z}^\lambda_t{\rm d}x\nonumber \\ &&-\frac{\mu_n-\mu_p}{2}\int_{{\Bbb T}^3}(\bar{z}^\lambda \bar{E}^\lambda)_t\nabla \bar{z}^\lambda_t{\rm d}x+\lambda^2\frac{\mu_n+\mu_p}{2}\int_{{\Bbb T}^3}(\bar{E}^\lambda{\rm div}\bar{E}^\lambda)_t\nabla \bar{z}^\lambda_t{\rm d}x+\int_{{\Bbb T}^3}(\bar{z}^\lambda \bar{v}^\lambda)_t\nabla \bar{z}^\lambda_t {\rm d}x. \label{eq:eq-zest8} \end{matrix}$

估计方程(3.5)右端各项. 对于线性项, 应用Cauchy-Schwarz 不等式, Sobolev嵌入定理, 线性项可以被(3.6)式控制

$\begin{matrix} && \epsilon\frac{\mu_n+\mu_p}{2}\parallel\nabla \bar{z}^\lambda_{t}\parallel^2+\epsilon\parallel (\bar{E}^\lambda,\bar{E}^\lambda_{t},{\rm div}\bar{E}^\lambda,\bar{z}^\lambda,\bar{z}^\lambda_{t},\bar{v}^\lambda,\bar{v}^\lambda_{t}) \parallel^2\nonumber\\ && +M(\epsilon)\lambda^4\parallel ({\rm div}\bar{E}^\lambda_{t},\nabla{\rm div}\bar{E}^\lambda_{t}) \parallel^2+M\lambda.\label{eq:eq-zest9} \end{matrix}$

对于非线性项, 有

$ \begin{matrix} &&\int_{{\Bbb T}^3} ({\bar{z}^\lambda \bar{v}^\lambda})_t\nabla \bar{z}^\lambda_{t} {\rm d}x\nonumber \\ & \leq & M(\epsilon)\parallel \bar{v}^\lambda_{t}\bar{z}^\lambda \parallel^2+M(\epsilon)\parallel \bar{v}^\lambda\bar{z}_t^\lambda \parallel^2+\epsilon\parallel \nabla \bar{z}^\lambda_{t} \parallel^2\nonumber \\ & \leq & M(\epsilon) \parallel \bar{v}^\lambda_{t} \parallel^2\parallel \bar{z}^\lambda\parallel_{H^2}^2+M(\epsilon) \parallel \bar{z}^\lambda_{t} \parallel^2\parallel \bar{v}^\lambda\parallel_{H^2}^2+\epsilon\parallel \nabla \bar{z}^\lambda_{t} \parallel^2\nonumber \\ &\leq& \epsilon\parallel \nabla \bar{z}^\lambda_{t} \parallel^2+M\mid\mid\mid W\mid\mid\mid^4,\label{eq:eq-zest10} \end{matrix}$
$\begin{matrix} &&-\frac{\mu_n-\mu_p}{2}\int_{{\Bbb T}^3} ({\bar{z}^\lambda\bar{E}^\lambda})_t\nabla \bar{z}^\lambda_t {\rm d}x\nonumber \\ &=&-\frac{\mu_n-\mu_p}{2}\int_{{\Bbb T}^3} ({\bar{z}^\lambda_t\bar{E}^\lambda+\bar{z}^\lambda\bar{E}_t^\lambda})\nabla \bar{z}^\lambda_t {\rm d}x\nonumber \\ &\leq & M(\epsilon)(\parallel\bar{E}^\lambda_{t}\bar{z}^\lambda\parallel^2+\parallel \bar{E}^\lambda\bar{z}^\lambda_{t} \parallel^2)+\epsilon\frac{\mu_n-\mu_p}{2}\parallel \nabla \bar{z}^\lambda_{t} \parallel^2\nonumber\\ & \leq& M(\epsilon)( \parallel \bar{z}^\lambda\parallel_{H^2}^2\parallel \bar{E}^\lambda_{t} \parallel^2+\parallel \bar{z}^\lambda_{t} \parallel_{H^1}^2\parallel \bar{E}^\lambda\parallel_{H^1}^2)+\epsilon\frac{\mu_n-\mu_p}{2}\parallel \nabla \bar{z}^\lambda_{t} \parallel^2\nonumber \\ &\leq&\epsilon\frac{\mu_n-\mu_p}{2}\parallel \nabla \bar{z}^\lambda_{t} \parallel^2+M\mid\mid\mid W\mid\mid\mid^4,\label{eq:eq-zest11} \end{matrix}$
$ \begin{matrix} &&\lambda^2\frac{\mu_n+\mu_p}{2}\int_{{\Bbb T}^3} ({\bar{E}^\lambda{\rm div}\bar{E}^\lambda})_t\nabla \bar{z}^\lambda_{t} {\rm d}x\nonumber \\ &=&\lambda^2\frac{\mu_n+\mu_p}{2}\int_{{\Bbb T}^3}({\bar{E}^\lambda_{t}{\rm div}\bar{E}^\lambda}+{\bar{E}^\lambda{\rm div}\bar{E}^\lambda_{t}})\nabla \bar{z}^\lambda_{t} {\rm d}x \nonumber \\ & \leq &\lambda^4M(\epsilon)( \parallel \bar{E}^\lambda_{t}{\rm div}\bar{E}^\lambda \parallel^2+\parallel \bar{E}^\lambda{\rm div}\bar{E}^\lambda_{t} \parallel^2)+\epsilon\frac{\mu_n+\mu_p}{2}\parallel \nabla \bar{z}^\lambda_{t} \parallel^2\nonumber \\ & \leq &\lambda^4M(\epsilon)(\parallel \bar{E}^\lambda \parallel_{H^2}^2\parallel {\rm div}\bar{E}^\lambda_{t} \parallel^2+\parallel \bar{E}^\lambda_{t} \parallel_{H^1}^2\parallel {\rm div}\bar{E}^\lambda \parallel_{H^1}^2)+\epsilon\frac{\mu_n+\mu_p}{2}\parallel \nabla \bar{z}^\lambda_{t} \parallel^2\nonumber \\ &\leq&\epsilon\frac{\mu_n+\mu_p}{2}\parallel \nabla \bar{z}^\lambda_{t} \parallel^2+M\mid\mid\mid W\mid\mid\mid^4.\label{eq:eq-zest12} \end{matrix}$

应用不等式(3.5)-(3.9), 可得

$\begin{matrix} && \frac{\rm d}{{\rm d}t}\parallel \bar{z}^\lambda_{t} \parallel^2+k\parallel \nabla \bar{z}^\lambda_{t}\parallel^2\nonumber \\ & \leq &M\parallel(\bar{z}^\lambda,\bar{z}^\lambda_{t},\bar{E}^\lambda,\bar{E}^\lambda_{t},{\rm div}\bar{E}^\lambda,\bar{v}^\lambda,\bar{v}^\lambda_{t})\parallel^2 +M\lambda^4\parallel({\rm div}\bar{E}^\lambda_{t},\nabla {\rm div}\bar{E}^\lambda_{t})\parallel^2\nonumber \\ && +M\mid\mid\mid W\mid\mid\mid^4+M\lambda.\label{eq:eq-zest17} \end{matrix}$

对方程(2.13)关于$t$求导, 两端乘以$\bar{\phi}^\lambda_{t}$, 再关于$x$在区间${\Bbb T}^3$ 积分, 可得

$\begin{matrix} &&\frac{\lambda^2}{2}\frac{\rm d}{{\rm d}t}\parallel \bar{E}^\lambda_{t} \parallel^2+\lambda^2\parallel {\rm div} \bar{E}^\lambda_{t}\parallel^2+\frac{\mu_n+\mu_p}{2}\int_{{\Bbb T}^3} z |\bar{E}^\lambda_{t}|^2{\rm d}x\nonumber \\ & =&\frac{\mu_n+\mu_p}{2}\int_{{\Bbb T}^3}B_t\bar{E}^\lambda_{t}{\rm d}x+\frac{\mu_n+\mu_p}{2}\lambda^2\int_{{\Bbb T}^3} z |{\rm div}\bar{E}^\lambda_{t}|^2{\rm d}x-\frac{\mu_n+\mu_p}{2}\int_{{\Bbb T}^3} z_t\bar{E}^\lambda \bar{E}^\lambda_{t}{\rm d}x\nonumber \\ && -\frac{\mu_n-\mu_p}{2}\int_{{\Bbb T}^3}A_t\bar{E}^\lambda_{t}{\rm d}x+\lambda^2\int_{{\Bbb T}^3}(v({\rm div}\bar{E}^\lambda+{\rm div}\varepsilon))_t\bar{E}^\lambda_{t}{\rm d}x\nonumber \\ && -\lambda^2\int_{{\Bbb T}^3} ((\bar{v}^\lambda+v){\rm div}\varepsilon)_t \bar{E}^\lambda_{t}{\rm d}x +\int_{{\Bbb T}^3} D\bar{v}^\lambda_t \bar{E}^\lambda_{t}{\rm d}x-\frac{\mu_n+\mu_p}{2}\int_{{\Bbb T}^3} {(\bar{z}^\lambda\bar{E}^\lambda)_t}\bar{E}^\lambda_{t} {\rm d}x\nonumber \\ &&-\lambda^2\int_{{\Bbb T}^3} (\bar{v}^\lambda{\rm div}\bar{E}^\lambda)_t\bar{E}^\lambda_{t} {\rm d}x +\lambda^2\frac{\mu_n-\mu_p}{2}\int_{{\Bbb T}^3} {(\bar{E}^\lambda{\rm div}\bar{E}^\lambda)_t}\bar{E}^\lambda_{t} {\rm d}x. \label{eq:eq-eest7} \end{matrix}$

估计(3.11)式右端各项. 对于线性项, 应用Cauchy-Schwarz不等式, Sobolev嵌入定理, $\bar{E}^\lambda_{t}=-\nabla\bar{\phi}^\lambda_{t}$, 线性项可以被(3.12)控制

$\begin{equation} \epsilon\parallel (\bar{E}^\lambda,\bar{E}^\lambda_{t},\bar{z}^\lambda,\bar{z}^\lambda_{t},\bar{v}^\lambda,\bar{v}^\lambda_{t}) \parallel^2 +M(\epsilon)\lambda^4 \parallel ({\rm div}\bar{E}^\lambda,{\rm div}\bar{E}^\lambda_{t}) \parallel^2+M\lambda,\label{eq:eq-eest8} \end{equation}$

对于非线性项, 应用Cauchy-Schwarz不等式和Sobolev嵌入不等式, 有

$\begin{matrix} &&-\frac{\mu_n+\mu_p}{2}\int_{{\Bbb T}^3} {(\bar{z}^\lambda\bar{E}^\lambda)_t}\bar{E}^\lambda_{t} {\rm d}x\nonumber \\ &=&-\frac{\mu_n+\mu_p}{2}\int_{{\Bbb T}^3} {(\bar{z}^\lambda_{t}\bar{E}^\lambda+\bar{z}^\lambda\bar{E}^\lambda_{t})}\bar{E}^\lambda_{t} {\rm d}x\nonumber \\ & \leq & M(\epsilon)\parallel \bar{E}^\lambda_{t} \bar{z}^\lambda \parallel^2+ M(\epsilon)\parallel \bar{E}^\lambda \bar{z}^\lambda_{t} \parallel^2+\epsilon\parallel \bar{E}^\lambda_{t} \parallel^2\nonumber \\ & \leq &M(\epsilon)\parallel \bar{z}^\lambda \parallel_{H^2}^2 \parallel \bar{E}^\lambda_{t} \parallel^2+M(\epsilon)\parallel \bar{E}^\lambda \parallel_{H^1}^2 \parallel \bar{z}^\lambda_{t} \parallel_{H^1}^2+\epsilon\parallel \bar{E}^\lambda_{t} \parallel^2\nonumber \\ &\leq&\epsilon\parallel \bar{E}^\lambda_{t} \parallel^2+M\mid\mid\mid W\mid\mid\mid^4+M\mid\mid\mid W\mid\mid\mid^2G^\lambda(t),\label{eq:eq-eest9} \end{matrix}$
$\begin{matrix} && \lambda^2\frac{\mu_n-\mu_p}{2}\int_{{\Bbb T}^3}{(\bar{E}^\lambda{\rm div}\bar{E}^\lambda)_t}\bar{E}^\lambda_{t} {\rm d}x\nonumber \\ & =&\lambda^2\frac{\mu_n-\mu_p}{2}\int_{{\Bbb T}^3}{\bar{E}^\lambda_{t}{\rm div}\bar{E}^\lambda}\bar{E}^\lambda_{t} {\rm d}x+\frac{\mu_n-\mu_p}{2}\int_{{\Bbb T}^3} \lambda^2{\bar{E}^\lambda{\rm div}\bar{E}^\lambda_{t}}\bar{E}^\lambda_{t} {\rm d}x\nonumber \\ &\leq& M(\epsilon)\lambda^4\parallel \bar{E}^\lambda_{t} {\rm div}\bar{E}^\lambda\parallel^2+M(\epsilon)\lambda^4\parallel \bar{E}^\lambda {\rm div}\bar{E}^\lambda_{t}\parallel^2+\epsilon(\frac{\mu_n-\mu_p}{2})^2\parallel \bar{E}^\lambda_{t} \parallel^2\nonumber \\ & \leq& M\lambda^4\parallel \bar{E}^\lambda\parallel_{H^2}^2 \parallel {\rm div}\bar{E}^\lambda_{t}\parallel^2+M\lambda^4\parallel \bar{E}^\lambda_{t}\parallel_{H^1}^2 \parallel {\rm div}\bar{E}^\lambda\parallel_{H^1}^2+\epsilon(\frac{\mu_n-\mu_p}{2})^2\parallel \bar{E}^\lambda_{t} \parallel^2\nonumber \\ &\leq&\epsilon\parallel \bar{E}^\lambda_{t} \parallel^2+M\mid\mid\mid W\mid\mid\mid^4,\label{eq:eq-eest10} \end{matrix}$
$\begin{matrix} && -\lambda^2 \int_{{\Bbb T}^3}({\rm div}\bar{E}^\lambda \bar{v}^\lambda)_t\bar{E}^\lambda_{t} {\rm d}x\nonumber \\ & =&-\lambda^2 \int_{{\Bbb T}^3} {\rm div}\bar{E}^\lambda \bar{v}^\lambda_{t}\bar{E}^\lambda_{t} {\rm d}x-\lambda^2 \int_{{\Bbb T}^3} {\rm div}\bar{E}^\lambda_{t} \bar{v}^\lambda\bar{E}^\lambda_{t} {\rm d}x\nonumber \\ & \leq& M(\epsilon)\lambda^4\parallel \bar{v}^\lambda_{t} {\rm div}\bar{E}^\lambda\parallel^2+M(\epsilon)\lambda^4\parallel \bar{v}^\lambda {\rm div}\bar{E}^\lambda_{t}\parallel^2+\epsilon\parallel \bar{E}^\lambda_{t} \parallel^2\nonumber \\ & \leq& M\lambda^4\parallel \bar{v}^\lambda_{t}\parallel_{H^1}^2 \parallel {\rm div}\bar{E}^\lambda\parallel_{H^1}^2+M\lambda^4\parallel \bar{v}^\lambda\parallel_{H^2}^2 \parallel {\rm div}\bar{E}^\lambda_{t}\parallel^2+\epsilon\parallel \bar{E}^\lambda_{t} \parallel^2\nonumber\\ & \leq&\epsilon\parallel \bar{E}^\lambda_{t} \parallel^2+M\mid\mid\mid W\mid\mid\mid^4.\label{eq:eq-eest11} \end{matrix}$

那么, 由不等式(3.11)-(3.15), 并限制$\lambda$足够小, 有

$\begin{matrix} && \lambda^2 \frac{\rm d}{{\rm d}t}\parallel \bar{E}^\lambda_{t} \parallel^2+k\parallel (\bar{E}^\lambda_{t},\lambda {\rm div} \bar{E}^\lambda_{t})\parallel^2\nonumber \\ & \leq & M(\epsilon)\parallel(\bar{z}^\lambda,\bar{z}^\lambda_{t},\nabla \bar{z}^\lambda_{t},\bar{E}^\lambda,{\rm div}\bar{E}^\lambda,\bar{v}^\lambda,\bar{v}^\lambda_{t})\parallel^2+M\mid\mid\mid W\mid\mid\mid^4\nonumber \\ && +M\mid\mid\mid W\mid\mid\mid^2G^\lambda(t)+M\lambda.\label{eq:eq-eest13} \end{matrix}$

对(2.13)式关于$t$求导, 两边乘以$\lambda^2{\rm div}\bar{E}^\lambda_{t}$, 关于$x$在区间${{\Bbb T}^3}$上积分, 可得

$\begin{matrix} &&\frac{\lambda^4}{2}\frac{\rm d}{{\rm d}t}\parallel {\rm div}\bar{E}^\lambda_{t} \parallel^2+\lambda^4\parallel \nabla{\rm div} \bar{E}^\lambda_{t}\parallel^2+\frac{\mu_n+\mu_p}{2}\lambda^2\int_{{\Bbb T}^3} z|{\rm div}\bar{E}^\lambda_{t}|^2{\rm d}x\nonumber \\ &=&\frac{\mu_n+\mu_p}{2}\int_{{\Bbb T}^3}B_t\nabla{\rm div}\bar{E}^\lambda_{t}{\rm d}x+\frac{\mu_n+\mu_p}{2}\lambda^2\int_{{\Bbb T}^3} z |{\rm div}\bar{E}^\lambda_{t}|^2{\rm d}x-\frac{\mu_n+\mu_p}{2}\int_{{\Bbb T}^3} z_t\bar{E}^\lambda\nabla{\rm div} \bar{E}^\lambda_{t}{\rm d}x\nonumber \\ &&-\frac{\mu_n-\mu_p}{2}\int_{{\Bbb T}^3}A_t\nabla{\rm div}\bar{E}^\lambda_{t}{\rm d}x+\lambda^2\int_{{\Bbb T}^3}(v({\rm div}\bar{E}^\lambda+{\rm div}\varepsilon))_t\nabla{\rm div}\bar{E}^\lambda_{t}{\rm d}x\nonumber \\ &&-\lambda^2\int_{{\Bbb T}^3} ((\bar{v}^\lambda+v){\rm div}\varepsilon)_t \nabla{\rm div}\bar{E}^\lambda_{t}{\rm d}x +\int_{{\Bbb T}^3} D\bar{v}^\lambda_t \nabla{\rm div}\bar{E}^\lambda_{t}{\rm d}x-\frac{\mu_n+\mu_p}{2}\int_{{\Bbb T}^3} {(\bar{z}^\lambda\bar{E}^\lambda)_t}\nabla{\rm div}\bar{E}^\lambda_{t} {\rm d}x\nonumber \\ &&-\lambda^2\int_{{\Bbb T}^3} (\bar{v}^\lambda{\rm div}\bar{E}^\lambda)_t\nabla{\rm div}\bar{E}^\lambda_{t} {\rm d}x +\lambda^2\frac{\mu_n-\mu_p}{2}\int_{{\Bbb T}^3} {(\bar{E}^\lambda{\rm div}\bar{E}^\lambda)_t}\nabla{\rm div}\bar{E}^\lambda_{t} {\rm d}x. \label{eq:eq-ee1} \end{matrix}$

由于方程(3.17)与方程(3.11)完全类似, 因此, 同理可得

$\begin{matrix} &&\lambda^4 \frac{\rm d}{{\rm d}t}\parallel {\rm div}\bar{E}^\lambda_{t} \parallel^2+k\lambda^2\parallel ({\rm div}\bar{E}^\lambda_{t},\lambda \nabla{\rm div} \bar{E}^\lambda_{t})\parallel^2\nonumber \\ &\leq & M(\epsilon)\parallel(\bar{z}^\lambda,\bar{z}^\lambda_{t},\nabla \bar{z}^\lambda_{t},\bar{E}^\lambda,\bar{v}^\lambda,\bar{v}^\lambda_{t})\parallel^2+M\lambda^2\parallel(\bar{E}^\lambda,\bar{E}^\lambda_{t},{\rm div}\bar{E}^\lambda)\parallel^2\nonumber \\ && +M\mid\mid\mid W\mid\mid\mid^4+M\mid\mid\mid W\mid\mid\mid^2G^\lambda(t)+M\lambda.\label{eq:eq-ee7} \end{matrix}$

对方程(2.14)关于$t$求导, 乘以$\bar{v}^\lambda_{t}$,在区间${{\Bbb T}^3}$上关于$x$积分, 分部积分,有

$\begin{matrix} &&\int_{{\Bbb T}^3}(\rho+\bar{\rho}^\lambda)\bar{v}^\lambda_{tt}\bar{v}^\lambda_t {\rm d}x+A\frac{\rm d}{{\rm d}t}\parallel\bar{\rho}^\lambda_t\parallel^2+\mu\parallel\nabla\bar{v}^\lambda_t\parallel^2+\parallel{\rm div}\bar{v}^\lambda_t\parallel^2\nonumber \\ & =&-\int_{{\Bbb T}^3}(\bar{\rho}^\lambda v_t+\bar{\rho}^\lambda v \nabla v+\rho\bar{v}^\lambda \nabla v+\rho v \nabla\bar{v}^\lambda+\rho_t\bar{v}^\lambda_t )_t\bar{v}^\lambda_t{\rm d}x\nonumber\\ && -\int_{{\Bbb T}^3}D\bar{E}^\lambda_t \bar{v}^\lambda_t {\rm d}x-\lambda^2\int_{{\Bbb T}^3}(\bar{E}^\lambda{\rm div}\varepsilon+\varepsilon{\rm div}\bar{E}^\lambda+\varepsilon{\rm div}\varepsilon)_t\bar{v}_t^\lambda {\rm d}x\nonumber \\ &&-2A\int_{{\Bbb T}^3}(\bar{\rho}^\lambda{\rm div}v+v \nabla\bar{\rho}^\lambda)_t\rho^\lambda_t{\rm d}x-2A\int_{{\Bbb T}^3}(\rho_t{\rm div}\bar{v}^\lambda+\bar{v}^\lambda \nabla\rho_t)\bar{\rho}^\lambda_t{\rm d}x\nonumber \\ && -2A\int_{{\Bbb T}^3}(\rho \bar{v}^\lambda \nabla\bar{v}^\lambda+\bar{\rho}^\lambda v \nabla\bar{v}^\lambda+\bar{\rho}^\lambda\bar{v}^\lambda\nabla\bar{v}^\lambda)_t \bar{v}_t^\lambda {\rm d}x\nonumber \\ &&-\int_{{\Bbb T}^3}((\bar{\rho}^\lambda\bar{v}^\lambda\nabla v)_t+\bar{\rho}_t^\lambda\bar{v}^\lambda_t+\lambda^2(\bar{E}^\lambda{\rm div}\bar{E}^\lambda)_t)\bar{v}^\lambda_t {\rm d}x\nonumber \\ && -2A\int_{{\Bbb T}^3}({\rm div}\bar{v}^\lambda\bar{\rho}^\lambda_t+\bar{v}^\lambda\nabla\bar{\rho}^\lambda_t)\bar{\rho}^\lambda_t{\rm d}x.\label{eq:eq-vest6} \end{matrix}$

估计方程(3.19)右端各项. 线性项可以被(3.20)式控制

$\begin{matrix} &&\epsilon\parallel(\bar{\rho}^\lambda,\bar{\rho}^\lambda_t,\nabla\bar{\rho}^\lambda,\bar{v}^\lambda,\nabla\bar{v}^\lambda,\bar{v}^\lambda_t,\nabla\bar{v}^\lambda_t,{\rm div}\bar{v}^\lambda,\bar{E}^\lambda,\bar{E}^\lambda_t)\parallel^2\nonumber \\ && +M(\epsilon)\lambda^4\parallel (\bar{E}^\lambda,\bar{E}^\lambda_t,{\rm div}\bar{E}^\lambda,{\rm div}\bar{E}^\lambda_t) \parallel^2+M\lambda.~~~~\label{eq:eq-vest7} \end{matrix}$

对于非线性项, 应用Cauchy-Schwarz不等式, Sobolev嵌入不等式, 见文献[4], 有

$\begin{matrix} &&-2A\int_{{\Bbb T}^3}(\rho_t{\rm div}\bar{v}^\lambda+\bar{v}^\lambda \nabla\rho_t)\bar{\rho}^\lambda_t{\rm d}x-2A\int_{{\Bbb T}^3}(\rho \bar{v}^\lambda \nabla\bar{v}^\lambda+\bar{\rho}^\lambda v \nabla\bar{v}^\lambda+\bar{\rho}^\lambda\bar{v}^\lambda\nabla\bar{v}^\lambda)_t \bar{v}_t^\lambda {\rm d}x\nonumber \\ &&-\int_{{\Bbb T}^3}((\bar{\rho}^\lambda\bar{v}^\lambda\nabla v)_t+\bar{\rho}_t^\lambda\bar{v}^\lambda_t+\lambda^2(\bar{E}^\lambda{\rm div}\bar{E}^\lambda)_t)\bar{v}^\lambda_t {\rm d}x-2A\int_{{\Bbb T}^3}({\rm div}\bar{v}^\lambda\bar{\rho}^\lambda_t+\bar{v}^\lambda\nabla\bar{\rho}^\lambda_t)\bar{\rho}^\lambda_t{\rm d}x\nonumber \\ &\leq& \epsilon \parallel(\bar{v}^\lambda_t,\nabla\bar{v}^\lambda_t,\bar{\rho}^\lambda_t)\parallel^2+M\parallel\bar{v}^\lambda\bar{v}^\lambda_t \parallel^2+M\parallel\bar{\rho}^\lambda\bar{v}^\lambda_t\parallel^2\nonumber \\ && +M\parallel\nabla\bar{v}^\lambda\bar{v}^\lambda_t\parallel^2+M\parallel\bar{\rho}^\lambda_t\nabla\bar{v}^\lambda\parallel^2+M\parallel\bar{v}^\lambda\bar{\rho}^\lambda_t\parallel^2+M\parallel\bar{\rho}^\lambda_t\bar{v}^\lambda_t\parallel^2\nonumber \\ && +\lambda^4M\parallel\bar{E}^\lambda_t{\rm div}\bar{E}^\lambda\parallel^2+M\lambda^4\parallel\bar{E}^\lambda{\rm div}\bar{E}^\lambda_t\parallel^2+M\parallel\bar{\rho}^\lambda_t{\rm div}\bar{v}^\lambda\parallel^2\nonumber\\ & \leq& \epsilon \parallel(\bar{v}^\lambda_t,\nabla\bar{v}^\lambda_t,\bar{\rho}^\lambda_t)\parallel^2+M\parallel\bar{v}^\lambda_t \parallel^2\parallel\bar{v}^\lambda\parallel^2_{H^2}+M\parallel\bar{v}^\lambda_t \parallel^2_{H^1}\parallel\bar{\rho}^\lambda\parallel^2_{H^1}\nonumber \\ && +M\parallel\bar{v}^\lambda_t \parallel^2_{H^1}\parallel\nabla\bar{v}^\lambda\parallel^2_{H^1}+M\parallel\bar{\rho}^\lambda_t \parallel^2_{H^1}\parallel\nabla\bar{v}^\lambda\parallel^2_{H^1}+M\parallel\bar{\rho}^\lambda_t \parallel^2\parallel\bar{v}^\lambda\parallel^2_{H^2}\nonumber \\ && +M\parallel\bar{v}^\lambda_t \parallel^2_{H^1}\parallel\bar{\rho}^\lambda_t\parallel^2_{H^1}+M\parallel\bar{\rho}^\lambda_t \parallel^2_{H^1}\parallel{\rm div}\bar{v}^\lambda\parallel^2_{H^1}\nonumber \\ && +M\lambda^4\parallel\bar{E}^\lambda_t \parallel^2_{H^1}\parallel{\rm div}\bar{E}^\lambda\parallel^2_{H^1}+M\lambda^4\parallel\bar{E}^\lambda \parallel^2_{H^2}\parallel{\rm div}\bar{E}^\lambda_t\parallel^2\nonumber\\ & \leq& \epsilon \parallel(\bar{v}^\lambda_t,\nabla\bar{v}^\lambda_t,\bar{\rho}^\lambda_t)\parallel^2+M\mid\mid\mid W\mid\mid\mid^4.\label{eq:eq-vest9} \end{matrix}$

联合不等式(3.19)-(3.21), 并应用不等式(3.22),有

$\begin{equation} \int_{{\Bbb T}^3}(\rho+\bar{\rho}^\lambda)\bar{v}^\lambda_{tt}\bar{v}^\lambda_t {\rm d}x\geq \frac{\underline{\rho}}{2}\int_{{\Bbb T}^3}\bar{v}^\lambda_{tt}\bar{v}^\lambda_t {\rm d}x=\frac{\underline{\rho}}{4}\frac{\rm d}{{\rm d}t}\parallel\bar{v}^\lambda_t\parallel^2,\label{eq:rvt1} \end{equation}$

$\rho$和$\bar{\rho}^\lambda$满足不等式

$\begin{equation} \frac{\underline{\rho}}{2}\leq \rho-|\bar{\rho}^\lambda|_{L^\infty}\leq\rho+\bar{\rho}^\lambda\leq\overline{\rho}+|\bar{\rho}^\lambda|_{L^\infty}\leq2\overline{\rho},\label{eq:rvt2} \end{equation} $

其中$\underline{\rho}$和$\overline{\rho}$ 分别是下确界和上确界, 因此可得

$\begin{matrix} &&\epsilon\frac{\rm d}{{\rm d}t}\parallel(\bar{\rho}^\lambda_t,\bar{v}^\lambda_t) \parallel^2+k\parallel (\nabla\bar{v}^\lambda_t,{\rm div}\bar{v}^\lambda_t)\parallel^2 \nonumber \\ & \leq & M\parallel(\bar{v}^\lambda,\bar{v}^\lambda_t,\nabla\bar{v}^\lambda,{\rm div}\bar{v}^\lambda,\bar{\rho}^\lambda,\bar{\rho}^\lambda_t,\bar{E}^\lambda, \bar{E}^\lambda_t)\parallel^2\nonumber \\ && +M\lambda^4\parallel({\rm div}\bar{E}^\lambda,{\rm div} \bar{E}^\lambda_t)\parallel^2 +M\mid\mid\mid W\mid\mid\mid^4+M\lambda.\label{eq:eq-vest10} \end{matrix}$

联合$\delta_2$(3.10),(3.15),(3.18)和(3.24)式, 在区间$[t]$上关于$t$ 积分, 取$\delta_2$$>$$0$ 足够小, 可得不等式(3.4). 证毕

定理3.3 估计$\parallel(\nabla \bar{z}^\lambda, \bar{E}^\lambda,\lambda {\rm div}\bar{E}^\lambda, \lambda^2\nabla {\rm div}\bar{E}^\lambda,\nabla \bar{v}^\lambda,{\rm div}\bar{v}^\lambda,\bar{\rho}^\lambda)\parallel^2$,有

$\begin{matrix} &&k\parallel(\nabla \bar{z}^\lambda, \bar{E}^\lambda,\lambda {\rm div}\bar{E}^\lambda, \lambda^2\nabla {\rm div}\bar{E}^\lambda,\nabla \bar{v}^\lambda,{\rm div}\bar{v}^\lambda,\bar{\rho}^\lambda)\parallel^2\nonumber \\ & \leq& M\parallel(\bar{z}^\lambda,\bar{z}^\lambda_t,\bar{v}^\lambda,\bar{v}^\lambda_t,\bar{\rho}^\lambda,,\bar{\rho}^\lambda_t)\parallel^2+M\lambda^2\parallel(\bar{E}^\lambda,\bar{E}^\lambda_t,\lambda{\rm div}\bar{E}^\lambda,\lambda{\rm div}\bar{E}^\lambda_t)\parallel^2\nonumber \\ && +M\mid\mid\mid W\mid\mid\mid^4+M\lambda,\label{eq:lw1} \end{matrix}$

注3.2 不等式(3.25)可由不等式(3.3)和Green公式得到, 此处省略其证明.

定理3.4 在定理 2.1的假设下, 有

$\begin{matrix} &&\parallel(\nabla\bar{z}^\lambda,\lambda \bar{\rm div}{E}^\lambda,\lambda^2\nabla{\rm div}\bar{E}^\lambda,\bar\nabla{v}^\lambda,\bar\nabla{\rho}^\lambda)\parallel^2\nonumber \\ && +M \int_0^t\parallel (\triangle \bar{z}^\lambda, {\rm div}\bar{E}^\lambda,\lambda\nabla {\rm div}\bar{E}^\lambda, \lambda^2\triangle {\rm div}\bar{E}^\lambda,\triangle \bar{v}^\lambda,\nabla{\rm div}\bar{v}^\lambda,\nabla\bar{\rho}^\lambda) \parallel^2 dt\nonumber \\ & \leq&\parallel(\nabla\bar{z}^\lambda,\lambda \bar{\rm div}{E}^\lambda,\lambda^2\nabla{\rm div}\bar{E}^\lambda,\bar\nabla{v}^\lambda,\bar\nabla{\rho}^\lambda)\parallel^2(t=0)\nonumber \\ && +M\int_0^t\parallel ( \nabla\bar{z}^\lambda,\nabla\bar{v}^\lambda,\nabla\bar{\rho}^\lambda) \parallel^2{\rm d}t+M\int_0^t \mid\mid\mid W\mid\mid\mid^4(s){\rm d}s+M\lambda.\label{eq:eq-zevest3} \end{matrix}$

证 在方程(2.12)-(2.14)两边分别乘以$-\triangle\bar{z}^\lambda,-\triangle\bar{\phi}^\lambda,-\triangle\bar{v}^\lambda$, 再在方程(2.13)两边乘以$\lambda^2 \triangle{\rm div}\bar{E}^\lambda$, 应用$|\mu_n-\mu_p|$的小性, 在区间${\Bbb T}^3$上关于$x$ 积分, 分部积分, 应用Green公式, 有

$\begin{matrix} &&\frac{\rm d}{{\rm d}t}\parallel(\nabla\bar{z}^\lambda,\lambda \bar{\rm div}{E}^\lambda,\lambda^2\nabla{\rm div}\bar{E}^\lambda,\bar\nabla{v}^\lambda,\bar\nabla{\rho}^\lambda)\parallel^2\nonumber \\ && +k\parallel(\triangle \bar{z}^\lambda, {\rm div}\bar{E}^\lambda,\lambda\nabla {\rm div}\bar{E}^\lambda, \lambda^2\triangle {\rm div}\bar{E}^\lambda,\triangle \bar{v}^\lambda,\nabla{\rm div}\bar{v}^\lambda,\nabla\bar{\rho}^\lambda)\parallel^2\nonumber \\ & \leq& M\parallel(\nabla\bar{z}^\lambda,\nabla\bar{v}^\lambda,\nabla\bar{\rho}^\lambda)\parallel^2+M\mid\mid\mid W\mid\mid\mid^4+M\lambda\label{xyrv1} \end{matrix}$

在区间$[t]$上关于$t$ 积分, 可得不等式(3.26). 证毕

定理3.5 在定理 2.1的假设下, 有

$\begin{matrix} && \parallel (\nabla\bar{z}^\lambda_t,\lambda{\rm div} \bar{E}^\lambda_t,\lambda^2\nabla{\rm div} \bar{E}^\lambda_t,\nabla \bar{v}^\lambda_t,\nabla\bar{\rho}^\lambda_t)\parallel^2\nonumber \\ &&+k\int_0^t\parallel (\Delta \bar{z}^\lambda_t,{\rm div}\bar{E}^\lambda_t,\lambda \nabla{\rm div}\bar{E}^\lambda_t,\lambda^2 \Delta{\rm div}\bar{E}^\lambda_t,\Delta \bar{v}^\lambda_t,\nabla{\rm div}\bar{v}^\lambda_t) \parallel^2{\rm d}t\nonumber \\ &\leq& \parallel (\nabla\bar{z}^\lambda_t,\lambda{\rm div} \bar{E}^\lambda_t,\lambda^2\nabla{\rm div} \bar{E}^\lambda_t,\nabla \bar{v}^\lambda_t,\nabla\bar{\rho}^\lambda_t)\parallel^2(t=0)\nonumber \\ && +M\int_0^t\parallel(\bar{E}^\lambda,\bar{E}^\lambda_{t}, {\rm div}\bar{E}^\lambda,\lambda\nabla{\rm div}\bar{E}^\lambda,\bar{v}^\lambda,\nabla \bar{v}^\lambda,\bar{v}^\lambda_{t},\nabla \bar{v}^\lambda_{t},{\rm div}\bar{v}^\lambda,\nabla{\rm div}\bar{v}^\lambda)\parallel^2{\rm d}t\nonumber \\ && +M\int_{T^3}(\bar{z}^\lambda,\bar{z}^\lambda_{t},\nabla \bar{z}^\lambda,\nabla \bar{z}^\lambda_{t},\bar{\rho}^\lambda,\bar{\rho}^\lambda_t,\nabla\bar{\rho}^\lambda,\nabla\bar{\rho}^\lambda_t,\triangle\bar{\rho}^\lambda){\rm d}x\nonumber \\ && +M\int _0^t(\mid\mid\mid W\mid\mid\mid^4+\mid\mid\mid W\mid\mid\mid^2G^\lambda(s)){\rm d}s+M\lambda. \label{eq:eq-zevest4} \end{matrix}$

证 对方程(2.12)关于$t$求导, 乘以$-\Delta \bar{z}^\lambda_{t}$, 在区间${{\Bbb T}^3}$ 上关于$x$积分, 应用Green 公式, 有

$\begin{matrix} &&\frac{1}{2}\frac{\rm d}{{\rm d}t}\parallel \nabla\bar{z}_t^\lambda \parallel^2+\frac{\mu_n+\mu_p}{2}\parallel \triangle \bar{z}_t^\lambda\parallel^2\nonumber\\ &=&-\frac{\mu_n+\mu_p}{2}\int_{{\Bbb T}^3} (A-\nabla \bar{z}^\lambda)_t\triangle \bar{z}_t^\lambda {\rm d}x-\frac{\mu_n-\mu_p}{2}\int_{{\Bbb T}^3} B_t\triangle \bar{z}^\lambda_{t}{\rm d}x\nonumber \\ &&+\int_{{\Bbb T}^3} (z\bar{v}^\lambda+\bar{z}^\lambda v)_t\triangle \bar{z}^\lambda_t{\rm d}x-\frac{\mu_n-\mu_p}{2}\int_{{\Bbb T}^3}(\bar{z}^\lambda \bar{E}^\lambda)_t\triangle \bar{z}^\lambda_t{\rm d}x\nonumber \\ &&+\lambda^2\frac{\mu_n+\mu_p}{2}\int_{{\Bbb T}^3}(\bar{E}^\lambda{\rm div}\bar{E}^\lambda)_t\triangle \bar{z}^\lambda_t{\rm d}x+\int_{{\Bbb T}^3}(\bar{z}^\lambda \bar{v}^\lambda)_t\triangle \bar{z}^\lambda_t {\rm d}x. \label{eq:eq-zest25} \end{matrix}$

估计方程(3.29)右端各项. 对于线性项, 应用Cauchy-Schwarz不等式, Sobolev嵌入定理, 见文献[4], 线性项可由(3.30)式控制

$\begin{matrix} &&\epsilon\parallel( \bar{E}^\lambda,{\rm div}\bar{E}^\lambda,\nabla{\rm div}\bar{E}^\lambda,\bar{E}^\lambda_{t},{\rm div}\bar{E}^\lambda_{t},\bar{z}^\lambda,\nabla \bar{z}^\lambda,\nabla \bar{z}^\lambda_{t},\bar{z}^\lambda_{t},\bar{v}^\lambda,\bar{v}^\lambda_{t},{\rm div}\bar{v}^\lambda,\nabla\bar{v}^\lambda_t) \parallel^2\nonumber \\ && +\epsilon\frac{\mu_n+\mu_p}{2}\parallel\Delta \bar{z}^\lambda_{t}\parallel^2+ M(\epsilon)\lambda^4\parallel (\nabla{\rm div}\bar{E}^\lambda_{t},\Delta{\rm div}\bar{E}^\lambda_{t}) \parallel^2+M\lambda.\label{eq:eq-zest26} \end{matrix}$

对于非线性项,有

$ \begin{matrix} && \int_{{\Bbb T}^3} {\rm div}{(\bar{z}^\lambda\bar{v}^\lambda)_t}\Delta \bar{z}^\lambda_{t} {\rm d}x\nonumber \\ &=&\int_{{\Bbb T}^3} \nabla \bar{z}^\lambda\bar{v}^\lambda_{t}\Delta \bar{z}^\lambda_{t} {\rm d}x+\int_{{\Bbb T}^3} \nabla \bar{z}^\lambda_{t}\bar{v}^\lambda\Delta \bar{z}^\lambda_{t} {\rm d}x+\int_{{\Bbb T}^3} \bar{z}^\lambda_{t}{\rm div}\bar{v}^\lambda\Delta \bar{z}^\lambda_{t} {\rm d}x+\int_{{\Bbb T}^3} \bar{z}^\lambda{\rm div}\bar{v}^\lambda_{t}\Delta \bar{z}^\lambda_{t} {\rm d}x\nonumber\\ &\leq &M(\epsilon)\parallel \nabla \bar{z}^\lambda\bar{v}^\lambda_{t} \parallel^2+ M(\epsilon)\parallel \nabla \bar{z}^\lambda_{t}\bar{v}^\lambda \parallel^2+ M(\epsilon)\parallel \bar{z}^\lambda_{t}{\rm div}\bar{v}^\lambda \parallel^2\nonumber \\ && + M(\epsilon)\parallel \bar{z}^\lambda{\rm div}\bar{v}^\lambda_{t} \parallel^2+\epsilon\parallel \Delta \bar{z}^\lambda_{t} \parallel^2\nonumber\\ &\leq & M(\epsilon)M_s\parallel \nabla \bar{z}^\lambda\parallel_{H^1}^2 \parallel \bar{v}^\lambda_{t} \parallel_{H^1}^2+M(\epsilon)M_s\parallel \nabla \bar{z}^\lambda_{t}\parallel^2 \parallel \bar{v}^\lambda \parallel_{H^2}^2\nonumber \\ && +M(\epsilon)M_s\parallel \bar{z}^\lambda_{t}\parallel^2_{H^2} \parallel {\rm div}\bar{v}^\lambda \parallel_{H^1}^2+M(\epsilon)M_s\parallel \bar{z}^\lambda\parallel^2_{H^2} \parallel{\rm div} \bar{v}^\lambda_t \parallel^2 +\epsilon\parallel \Delta \bar{v}^\lambda_{t} \parallel^2\nonumber \\ &\leq&\epsilon\parallel \Delta \bar{z}^\lambda_{t} \parallel^2+M\mid\mid\mid W\mid\mid\mid^4,\label{eq:eq-zest27} \end{matrix}$
$ \begin{matrix} &&-\frac{\mu_n-\mu_p}{2}\int_{{\Bbb T}^3} {\rm div}{(\bar{z}^\lambda\bar{E}^\lambda)_t}\Delta \bar{z}^\lambda_{t} {\rm d}x\nonumber\\ & =&-\frac{\mu_n-\mu_p}{2}\int_{{\Bbb T}^3} ({\nabla \bar{z}^\lambda_{t}\bar{E}^\lambda+\bar{z}^\lambda_{t}{\rm div} \bar{E}^\lambda+\nabla \bar{z}^\lambda\bar{E}^\lambda_{t}+\bar{z}^\lambda{\rm div}\bar{E}^\lambda_{t}})\Delta \bar{E}^\lambda_{t} {\rm d}x\nonumber\\ &\leq & M(\epsilon)(\parallel \nabla \bar{z}^\lambda_{t}\bar{E}^\lambda \parallel^2+\parallel \bar{z}^\lambda_{t}{\rm div} \bar{E}^\lambda \parallel^2)\nonumber \\ && +M(\epsilon)(\parallel \nabla \bar{z}^\lambda\bar{E}^\lambda_{t} \parallel^2+\parallel \bar{z}^\lambda{\rm div}\bar{E}^\lambda_{t} \parallel^2)+\epsilon\frac{\mu_n-\mu_p}{2}\parallel \Delta \bar{z}^\lambda_{t} \parallel^2\nonumber \\ & \leq& M(\epsilon)\parallel \nabla \bar{z}^\lambda_{t}\parallel^2 \parallel \bar{E}^\lambda \parallel_{H^2}^2+M(\epsilon)\parallel \bar{z}^\lambda_{t}\parallel_{H^1}^2 \parallel{\rm div} \bar{E}^\lambda \parallel_{H^1}^2\nonumber\\ && +M(\epsilon)\parallel \nabla \bar{z}^\lambda\parallel_{H^1}^2 \parallel \bar{E}^\lambda_{t} \parallel_{H^1}^2 +M(\epsilon)\parallel \bar{z}^\lambda\parallel_{H^2}^2\parallel{\rm div}\bar{E}^\lambda_{t} \parallel^2+\epsilon\frac{\mu_n-\mu_p}{2}\parallel \Delta \bar{z}^\lambda_{t} \parallel^2\nonumber \\ & \leq&\epsilon\frac{\mu_n-\mu_p}{2}\parallel \Delta \bar{z}^\lambda_{t} \parallel^2+M\mid\mid\mid W\mid\mid\mid^4+M\mid\mid\mid W\mid\mid\mid^2G^\lambda(t),\label{eq:eq-zest28} \end{matrix}$
$ \begin{matrix} && \lambda^2\frac{\mu_n+\mu_p}{2}\int_{{\Bbb T}^3}{\rm div}{(\bar{E}^\lambda{\rm div}\bar{E}^\lambda)}_t\Delta \bar{z}^\lambda_{t}{\rm d}x \nonumber \\ & =&\lambda^2\frac{\mu_n+\mu_p}{2}\int_{{\Bbb T}^3}({\rm div}{\bar{E}^\lambda_{t}{\rm div}\bar{E}^\lambda}+{\bar{E}^\lambda_{t}\nabla{\rm div}\bar{E}^\lambda}+{{\rm div}\bar{E}^\lambda{\rm div}\bar{E}^\lambda_{t}}+{\bar{E}^\lambda\nabla{\rm div}\bar{E}^\lambda_{t}}) \Delta \bar{z}^\lambda_{t} {\rm d}x \nonumber \\ &\leq& \lambda^4M(\epsilon)( \parallel {\rm div}{\bar{E}^\lambda_{t}{\rm div}\bar{E}^\lambda}\parallel^2+\parallel{\bar{E}^\lambda_{t}\nabla{\rm div}\bar{E}^\lambda}\parallel^2)\nonumber \\ &&+\lambda^4M(\epsilon)(\parallel{{\rm div}\bar{E}^\lambda{\rm div}\bar{E}^\lambda_{t}}\parallel^2+\parallel{\bar{E}^\lambda\nabla{\rm div}\bar{E}^\lambda_{t}} \parallel^2)+\epsilon\frac{\mu_n+\mu_p}{2}\parallel \Delta \bar{z}^\lambda_{t} \parallel^2\nonumber \\ &\leq& \lambda^4M(\epsilon)( \parallel {\rm div}{\bar{E}^\lambda_{t}\parallel_{H^1}^2\parallel{\rm div}\bar{E}^\lambda}\parallel_{H^1}^2+\parallel{\bar{E}^\lambda_{t}\parallel_{H^1}^2\parallel\nabla{\rm div}\bar{E}^\lambda}\parallel_{H^1}^2)\nonumber \\ && +\lambda^4M(\epsilon)(\parallel{{\rm div}\bar{E}^\lambda\parallel_{H^1}^2\parallel{\rm div}\bar{E}^\lambda_{t}}\parallel_{H^1}^2+\parallel{\bar{E}^\lambda\parallel_{H^2}^2\parallel\nabla{\rm div}\bar{E}^\lambda_{t}} \parallel^2)+\epsilon\frac{\mu_n+\mu_p}{2}\parallel \Delta \bar{z}^\lambda_{t} \parallel^2\nonumber \\ & \leq&\epsilon\frac{\mu_n+\mu_p}{2}\parallel \Delta \bar{z}^\lambda_{t} \parallel^2+M\mid\mid\mid W\mid\mid\mid^4+M\mid\mid\mid W\mid\mid\mid^2G^\lambda(t)+M\lambda^2\parallel \bar{E}^\lambda_{t}\parallel^2_{H^2}\mid\mid\mid W\mid\mid\mid^2.~~~~~~~\label{eq:eq-zest29} \end{matrix}$

联合不等式(3.29)-(3.33), 有

$\begin{matrix} &&\frac{\rm d}{{\rm d}t}\parallel\nabla \bar{z}^\lambda_{t} \parallel^2+k\parallel \Delta \bar{z}^\lambda_{t}\parallel^2\nonumber \\ &\leq& M\parallel(\bar{z}^\lambda,\bar{z}^\lambda_{t},\nabla \bar{z}^\lambda,\nabla \bar{z}^\lambda_{t},\bar{E}^\lambda,\bar{E}^\lambda_{t},{\rm div}\bar{E}^\lambda,{\rm div}\bar{E}^\lambda_{t},\nabla{\rm div}\bar{E}^\lambda,\bar{v}^\lambda,\bar{v}^\lambda_{t},{\rm div}\bar{v}^\lambda,\nabla\bar{v}^\lambda_{t})\parallel^2\nonumber \\ && +M\lambda^4\parallel(\nabla {\rm div}\bar{E}^\lambda_{t},\Delta{\rm div}\bar{E}^\lambda_{t})\parallel^2+M\mid\mid\mid W\mid\mid\mid^4+M\lambda^2\parallel E_{R,t}\parallel^2_{H^2}\mid\mid\mid W\mid\mid\mid^2\nonumber \\ &&+M\mid\mid\mid W\mid\mid\mid^2G^\lambda(t)+M\lambda.\label{eq:eq-zest32} \end{matrix}$

对方程(2.13)两端关于$t$求导, 乘以$-\Delta\bar{\phi}^\lambda_{t}={\rm div}\bar{E}^\lambda_{t}$, 在区间${{\Bbb T}^3}$ 上关于$x$ 积分, 有

$\begin{matrix} &&\frac{\lambda^2}{2}\frac{\rm d}{{\rm d}t}\parallel {\rm div}\bar{E}^\lambda_{t} \parallel^2+\lambda^2\parallel \nabla{\rm div} \bar{E}^\lambda_{t}\parallel^2+\frac{\mu_n+\mu_p}{2}\int_{{\Bbb T}^3} z|{\rm div}\bar{E}^\lambda_{t}|^2{\rm d}x\nonumber \\ &=&\frac{\mu_n+\mu_p}{2}\int_{{\Bbb T}^3}B_t\nabla{\rm div}\bar{E}^\lambda_{t}{\rm d}x+\frac{\mu_n+\mu_p}{2}\int_{{\Bbb T}^3} z |{\rm div}\bar{E}^\lambda_{t}|^2{\rm d}x-\frac{\mu_n+\mu_p}{2}\int_{{\Bbb T}^3} z_t\bar{E}^\lambda\nabla{\rm div} \bar{E}^\lambda_{t}{\rm d}x\nonumber \\ &&-\frac{\mu_n-\mu_p}{2}\int_{{\Bbb T}^3}A_t\nabla{\rm div}\bar{E}^\lambda_{t}{\rm d}x+\int_{{\Bbb T}^3}(v({\rm div}\bar{E}^\lambda+{\rm div}\varepsilon))_t\nabla{\rm div}\bar{E}^\lambda_{t}{\rm d}x\nonumber\\ &&-\int_{{\Bbb T}^3} ((\bar{v}^\lambda+v){\rm div}\varepsilon)_t \nabla{\rm div}\bar{E}^\lambda_{t}{\rm d}x +\int_{{\Bbb T}^3} D\bar{v}^\lambda_t \nabla{\rm div}\bar{E}^\lambda_{t}{\rm d}x-\frac{\mu_n+\mu_p}{2}\int_{{\Bbb T}^3} {(\bar{z}^\lambda\bar{E}^\lambda)_t}\nabla{\rm div}\bar{E}^\lambda_{t} {\rm d}x\nonumber\\ && -\int_{{\Bbb T}^3} (\bar{v}^\lambda{\rm div}\bar{E}^\lambda)_t\nabla{\rm div}\bar{E}^\lambda_{t} {\rm d}x +\lambda^2\frac{\mu_n-\mu_p}{2}\int_{{\Bbb T}^3} {(\bar{E}^\lambda{\rm div}\bar{E}^\lambda)_t}\nabla{\rm div}\bar{E}^\lambda_{t} {\rm d}x.\label{eq:eq-eest20} \end{matrix}$

估计方程(3.35)右端各项. 对线性项应用Cauchy-Schwarz不等式, Green公式以及Sobolev嵌入不等式, $\bar{E}^\lambda_t=-\nabla\bar{\phi}^\lambda_t$, 线性项可由(3.36)式控制

$\begin{matrix} &&\epsilon\parallel(\bar{z}^\lambda, \bar{z}^\lambda_t,\nabla \bar{z}^\lambda,\nabla \bar{z}^\lambda_t, \bar{E}^\lambda,{\rm div}\bar{E}^\lambda,\bar{E}^\lambda_{t},{\rm div}\bar{E}^\lambda_{t},\bar{v}^\lambda,\bar{v}^\lambda_{t},{\rm div}\bar{v}^\lambda,\nabla\bar{v}^\lambda_{t}) \parallel^2\nonumber \\ &&+M(\epsilon)\lambda^4 \parallel (\nabla{\rm div}\bar{E}^\lambda,\nabla{\rm div}\bar{E}^\lambda_{t}) \parallel^2+M\lambda.\label{eq:eq-eest21} \end{matrix}$

对非线性项, 应用Cauchy-Schwarz不等式和Green公式, 有

$\begin{matrix} &&-\frac{\mu_n+\mu_p}{2}\int_{{\Bbb T}^3} {\rm div}{(\bar{z}^\lambda\bar{E}^\lambda)_t}{\rm div}\bar{E}^\lambda_{t} {\rm d}x\nonumber \\ & =&-\frac{\mu_n+\mu_p}{2}\int_{{\Bbb T}^3} {(\bar{z}^\lambda_{t}{\rm div}\bar{E}^\lambda+\nabla \bar{z}^\lambda\bar{E}^\lambda_{t}+\nabla \bar{z}^\lambda_{t}\bar{E}^\lambda+\bar{z}^\lambda{\rm div}\bar{E}^\lambda_{t})}{\rm div}\bar{E}^\lambda_{t} {\rm d}x\nonumber\\ & \leq& \epsilon\parallel {\rm div}\bar{E}^\lambda_{t} \parallel^2+M(\epsilon)(\|\bar{z}^\lambda_{t}{\rm div}\bar{E}^\lambda\|^2+\|\nabla \bar{z}^\lambda\bar{E}^\lambda_{t}\|^2+\|\nabla \bar{z}^\lambda_{t}\bar{E}^\lambda\|^2+\|\bar{z}^\lambda{\rm div}\bar{E}^\lambda_{t}\|^2)\nonumber \\ & \leq& \epsilon\parallel {\rm div}\bar{E}^\lambda_{t} \parallel^2+M(\epsilon)M_s\|\bar{z}^\lambda_{t}\|_{H^2}^2\|{\rm div}\bar{E}^\lambda\|^2+M(\epsilon)M_s\|\nabla \bar{z}^\lambda\|_{H^1}^2\|\bar{E}^\lambda_{t}\|_{H^1}^2\nonumber \\ &&+M(\epsilon)M_s\|\nabla \bar{z}^\lambda_{t}\|_{H^1}^2\|\bar{E}^\lambda\|_{H^1}^2+M(\epsilon)M_s\|\bar{z}^\lambda\|_{H^2}^2\|{\rm div}\bar{E}^\lambda_{t}\|^2\nonumber \\ &\leq &\epsilon\parallel {\rm div}\bar{E}^\lambda_{t} \parallel^2+M\mid\mid\mid W\mid\mid\mid^2G^\lambda(t)+M \mid\mid\mid W\mid\mid\mid^4,\label{eq:eq-eest22} \end{matrix}$
$\begin{matrix} && \lambda^2\frac{\mu_n-\mu_p}{2}\int_{{\Bbb T}^3}{{\rm div}(\bar{E}^\lambda{\rm div}\bar{E}^\lambda)_t}{\rm div}\bar{E}^\lambda_{t} {\rm d}x\nonumber \\ & =&\lambda^2\frac{\mu_n-\mu_p}{2}\int_{{\Bbb T}^3} (\nabla{\rm div}\bar{E}^\lambda \bar{E}^\lambda_{t}+\frac 32{\rm div}\bar{E}^\lambda{\rm div}\bar{E}^\lambda_{t}){\rm div}\bar{E}^\lambda_{t}{\rm d}x\nonumber \\ & \le&\epsilon \|{\rm div}\bar{E}^\lambda_{t}\|^2+M(\epsilon)\lambda^4(\|\nabla{\rm div}\bar{E}^\lambda \bar{E}^\lambda_{t}\|^2+\|{\rm div}\bar{E}^\lambda{\rm div}\bar{E}^\lambda_{t}\|^2)\nonumber \\ & \le&\epsilon \|{\rm div}\bar{E}^\lambda_{t}\|^2+M(\epsilon)M_s\lambda^4(\|\nabla{\rm div}\bar{E}^\lambda\|^2\| \bar{E}^\lambda_{t}\|_{H^2}^2+\|{\rm div}\bar{E}^\lambda\|_{H^1}\|{\rm div}\bar{E}^\lambda_{t}\|_{H^1}^2)\nonumber\\ & \leq &\epsilon\parallel {\rm div}\bar{E}^\lambda_{t} \parallel^2+M\lambda^2\parallel \bar{E}^\lambda_{t}\parallel^2_{H^2}\mid\mid\mid W\mid\mid\mid^2+M\mid\mid\mid W\mid\mid\mid^2G^\lambda(t),\label{eq:eq-eest23} \end{matrix}$
$\begin{matrix} && -\lambda^2 \int_{{\Bbb T}^3} {\rm div}(\bar{v}^\lambda{\rm div}\bar{E}^\lambda )_t{\rm div}\bar{E}^\lambda_{t} {\rm d}x\nonumber \\ & =&-\lambda^2 \int_{{\Bbb T}^3}( \bar{v}^\lambda_{t}\nabla{\rm div}\bar{E}^\lambda+\bar{v}^\lambda\nabla{\rm div}\bar{E}^\lambda_{t}+{\rm div}\bar{v}^\lambda_{t}{\rm div}\bar{E}^\lambda+{\rm div}\bar{v}^\lambda{\rm div}\bar{E}^\lambda_t){\rm div}\bar{E}^\lambda_{t} {\rm d}x\nonumber \\ &\leq& M(\epsilon)\lambda^4\parallel \bar{v}^\lambda_{t} \nabla {\rm div}\bar{E}^\lambda\parallel^2+M(\epsilon)\lambda^4\parallel \bar{v}^\lambda \nabla {\rm div}\bar{E}^\lambda_t\parallel^2\nonumber \\ && +M(\epsilon)\lambda^4\parallel {\rm div}\bar{v}^\lambda_{t} {\rm div}\bar{E}^\lambda\parallel^2+M(\epsilon)\lambda^4\parallel {\rm div}\bar{v}^\lambda {\rm div}\bar{E}^\lambda_t\parallel^2+\epsilon\parallel{\rm div}E_{R,t}\parallel^2\nonumber \\ &\leq & M(\epsilon)\lambda^4\parallel \bar{v}^\lambda_{t}\parallel_{H^2}^2 \parallel \nabla{\rm div}\bar{E}^\lambda\parallel^2+M(\epsilon)\lambda^4\parallel \bar{v}^\lambda\parallel_{H^2}^2 \parallel \nabla{\rm div}\bar{E}^\lambda+t\parallel^2\nonumber \\ &&M(\epsilon)\lambda^4\parallel {\rm div}\bar{v}^\lambda\parallel_{H^1}^2 \parallel {\rm div}\bar{E}^\lambda\parallel^2_{H^1}+M(\epsilon)\lambda^4\parallel {\rm div}\bar{v}^\lambda\parallel_{H^1}^2 \parallel {\rm div}\bar{E}^\lambda\parallel^2_{H^1}+\epsilon\parallel {\rm div}\bar{E}^\lambda_{t} \parallel^2\nonumber \\ & \leq& \epsilon\parallel {\rm div}\bar{E}^\lambda_{t} \parallel^2+M\mid\mid\mid W\mid\mid\mid^4+M\mid\mid\mid W\mid\mid\mid^2G^\lambda(t).\label{eq:eq-eest24} \end{matrix}$

联合不等式(3.35)-(3.39), 并假设$\lambda$足够小, 可得

$\begin{matrix} &&\lambda^2\frac{\rm d}{{\rm d}t}\parallel {\rm div}\bar{E}^\lambda_{t} \parallel^2+k\parallel ({\rm div}\bar{E}^\lambda_{t},\lambda \nabla{\rm div} \bar{E}^\lambda_{t})\parallel^2\nonumber \\ & \leq & M(\epsilon)\parallel(\bar{z}^\lambda,\bar{z}^\lambda_{t},\nabla\bar{z}^\lambda,\nabla \bar{z}^\lambda_{t},\bar{E}^\lambda,\bar{v}^\lambda,\bar{v}^\lambda_{t},{\rm div}\bar{v}^\lambda,{\rm div}\bar{v}^\lambda_{t})\parallel^2+M\lambda^2\parallel(\bar{E}^\lambda,\bar{E}^\lambda_{t},{\rm div}\bar{E}^\lambda)\parallel^2\nonumber \\ &&+M\mid\mid\mid W\mid\mid\mid^4+M\mid\mid\mid W\mid\mid\mid^2G^\lambda(t)+M\lambda.\label{eq:eq-eest26} \end{matrix}$

对方程(2.13)关于$t$求导, 乘以$\lambda^2\Delta{\rm div}\bar{E}^\lambda_{t}$, 在区间${{\Bbb T}^3}$上关于$x$积分, 有

$\begin{matrix} &&\frac{\lambda^4}{2}\frac{\rm d}{{\rm d}t}\parallel\nabla {\rm div}\bar{E}^\lambda_{t} \parallel^2+\lambda^4\parallel \triangle{\rm div} \bar{E}^\lambda_{t}\parallel^2+\frac{\mu_n+\mu_p}{2}\lambda^2\int_{{\Bbb T}^3} z|\nabla{\rm div}\bar{E}^\lambda_{t}|^2{\rm d}x\nonumber \\ & =&\frac{\mu_n+\mu_p}{2}\int_{{\Bbb T}^3}{\rm div}B_t\triangle{\rm div}\bar{E}^\lambda_{t}{\rm d}x+\frac{\mu_n+\mu_p}{2}\lambda^2\int_{{\Bbb T}^3} z |\nabla{\rm div}\bar{E}^\lambda_{t}|^2{\rm d}x\nonumber\\ && -\frac{\mu_n+\mu_p}{2}\int_{{\Bbb T}^3} z_t\bar{E}^\lambda\triangle{\rm div} \bar{E}^\lambda_{t}{\rm d}x-\frac{\mu_n-\mu_p}{2}\int_{{\Bbb T}^3}{\rm div}A_t\triangle{\rm div}\bar{E}^\lambda_{t}{\rm d}x\nonumber\\ && +\lambda^2\int_{{\Bbb T}^3}(v({\rm div}\bar{E}^\lambda+{\rm div}\varepsilon))_t\triangle{\rm div}\bar{E}^\lambda_{t}{\rm d}x-\lambda^2\int_{{\Bbb T}^3} ((\bar{v}^\lambda+v){\rm div}\varepsilon)_t \triangle{\rm div}\bar{E}^\lambda_{t}{\rm d}x\nonumber \\ &&+\int_{{\Bbb T}^3} D\bar{v}^\lambda_t \triangle{\rm div}\bar{E}^\lambda_{t}{\rm d}x-\frac{\mu_n+\mu_p}{2}\int_{T^3} {(\bar{z}^\lambda\bar{E}^\lambda)_t}\triangle{\rm div}\bar{E}^\lambda_{t} {\rm d}x\nonumber\\ &&-\lambda^2\int_{{\Bbb T}^3} (\bar{v}^\lambda{\rm div}\bar{E}^\lambda)_t\triangle{\rm div}\bar{E}^\lambda_{t} {\rm d}x +\lambda^2\frac{\mu_n-\mu_p}{2}\int_{{\Bbb T}^3} {(\bar{E}^\lambda{\rm div}\bar{E}^\lambda)_t}\triangle{\rm div}\bar{E}^\lambda_{t} {\rm d}x.\label{eq:eq-ee15} \end{matrix}$

由于方程(3.41)与方程(3.35)完全类似, 同理可得

$\begin{matrix} &&\lambda^4 \frac{\rm d}{{\rm d}t}\parallel\nabla{\rm div} \bar{E}^\lambda_{t} \parallel^2+k\lambda^2\parallel( \nabla{\rm div} \bar{E}^\lambda_{t},\lambda\Delta {\rm div} \bar{E}^\lambda_{t}) \parallel^2\nonumber \\ & \leq & M\parallel(\bar{z}^\lambda,\bar{z}^\lambda_{t},\nabla \bar{z}^\lambda,\nabla \bar{z}^\lambda_{t}, \bar{v}^\lambda,\bar{v}^\lambda_{t})\parallel^2+ M\lambda^4\parallel(\nabla{\rm div}\bar{E}^\lambda,{\rm div}\bar{E}^\lambda_{t})\parallel^2\nonumber \\ && +M\lambda^2\parallel(\bar{E}^\lambda,\bar{E}^\lambda_{t},{\rm div}\bar{E}^\lambda)\parallel^2+\epsilon\frac{\mu_n+\mu_p}{2}\parallel\Delta{\rm div}\bar{E}^\lambda_{t}\parallel^2\nonumber \\ && +M\lambda^2\parallel \bar{E}^\lambda_{t}\parallel^2_{H^2}\mid\mid\mid W\mid\mid\mid^2+ M\mid\mid\mid W\mid\mid\mid^4+M\mid\mid\mid \mid W\mid\mid\mid^2G^\lambda(t)+M\lambda.\label{eq:eq-ee217} \end{matrix}$

对方程(2.14)关于$t$求导, 乘以$-\Delta \bar{v}^\lambda_{t}$, 在区间${{\Bbb T}^3}$上关于$x$积分, 应用分部积分, 有

$\begin{matrix} && \int_{{\Bbb T}^3}(\rho+\bar{\rho}^\lambda)\bar{v}^\lambda_{tt}\triangle\bar{v}^\lambda_t {\rm d}x+A\frac{\rm d}{{\rm d}t}\parallel\nabla\bar{\rho}^\lambda_t\parallel^2+\mu\parallel\triangle\bar{v}^\lambda_t\parallel^2+\parallel\nabla{\rm div}\bar{v}^\lambda_t\parallel^2\nonumber \\ & =&-\int_{{\Bbb T}^3}(\bar{\rho}^\lambda v_t+\bar{\rho}^\lambda v \nabla v+\rho\bar{v}^\lambda \nabla v+\rho v \nabla\bar{v}^\lambda+\rho_t\bar{v}^\lambda_t )_t\triangle\bar{v}^\lambda_t{\rm d}x-\int_{{\Bbb T}^3}D\bar{E}^\lambda_t\triangle \bar{v}^\lambda_t {\rm d}x\nonumber \\ && -\lambda^2\int_{{\Bbb T}^3}(\bar{E}^\lambda{\rm div}\varepsilon+\varepsilon{\rm div}\bar{E}^\lambda+\varepsilon{\rm div}\varepsilon)_t\triangle\bar{v}_t^\lambda {\rm d}x-2A\int_{{\Bbb T}^3}(\bar{\rho}^\lambda{\rm div}v+v \nabla\bar{\rho}^\lambda)_t\nabla\rho^\lambda_t{\rm d}x\nonumber \\ && -2A\int_{{\Bbb T}^3}(\rho_t{\rm div}\bar{v}^\lambda+\bar{v}^\lambda \nabla\rho_t)\nabla\bar{\rho}^\lambda_t{\rm d}x-2A\int_{{\Bbb T}^3}(\rho \bar{v}^\lambda \nabla\bar{v}^\lambda+\bar{\rho}^\lambda v \nabla\bar{v}^\lambda+\bar{\rho}^\lambda\bar{v}^\lambda\nabla\bar{v}^\lambda)_t \triangle\bar{v}_t^\lambda {\rm d}x\nonumber \\ &&-\int_{{\Bbb T}^3}((\bar{\rho}^\lambda\bar{v}^\lambda\nabla v)_t+\bar{\rho}_t^\lambda\bar{v}^\lambda_t+\lambda^2(\bar{E}^\lambda{\rm div}\bar{E}^\lambda)_t)\triangle\bar{v}^\lambda_t {\rm d}x-2A\int_{{\Bbb T}^3}({\rm div}\bar{v}^\lambda\bar{\rho}^\lambda_t+\bar{v}^\lambda\nabla\bar{\rho}^\lambda_t)\nabla\bar{\rho}^\lambda_t{\rm d}x.\label{eq:eq-vest16} \end{matrix}$

估计方程(3.43)右端各项. 对线性项应用Cauchy-Schwarz不等式以及Sobolev嵌入定理, 线性项可由(3.44)式控制

$\begin{matrix} M \parallel &&(\bar{v}^\lambda,\nabla \bar{v}^\lambda,\bar{v}^\lambda_{t},\nabla \bar{v}^\lambda_{t},{\rm div}\bar{v}^\lambda,\nabla{\rm div}\bar{v}^\lambda, \Delta \bar{v}^\lambda_{t},\nabla{\rm div}\bar{v}^\lambda_t,\bar{\rho}^\lambda,\bar{\rho}^\lambda_t,\nabla\bar{\rho}^\lambda,\nabla\bar{\rho}^\lambda_t,\triangle\bar{\rho}^\lambda)\parallel^2\nonumber \\ && +M\lambda^2\parallel( \bar{E}^\lambda,{\rm div}\bar{E}^\lambda)\parallel^2+M(\epsilon)\lambda^4\parallel(\bar{E}^\lambda_{t},{\rm div}\bar{E}^\lambda_{t}) \parallel^2+M\lambda.\label{eq:eq-vest17} \end{matrix}$

对非线性项, 应用方程(2.15)的导数, Cauchy-Schwarz不等式Sobolev嵌入定理, 有

$\begin{matrix} &&-2A\int_{{\Bbb T}^3}(\bar{\rho}^\lambda{\rm div}v+v \nabla\bar{\rho}^\lambda)_t\nabla\rho^\lambda_t{\rm d}x-2A\int_{{\Bbb T}^3}(\rho_t{\rm div}\bar{v}^\lambda+\bar{v}^\lambda \nabla\rho_t)\nabla\bar{\rho}^\lambda_t{\rm d}x\nonumber \\ && -2A\int_{{\Bbb T}^3}(\rho \bar{v}^\lambda \nabla\bar{v}^\lambda+\bar{\rho}^\lambda v \nabla\bar{v}^\lambda +\bar{\rho}^\lambda\bar{v}^\lambda\nabla\bar{v}^\lambda)_t\triangle \bar{v}_t^\lambda {\rm d}x -\int_{{\Bbb T}^3}((\bar{\rho}^\lambda\bar{v}^\lambda\nabla v)_t+\bar{\rho}_t^\lambda\bar{v}^\lambda_t)\triangle\bar{v}^\lambda_t {\rm d}x\nonumber \\ && -\lambda^2\int_{{\Bbb T}^3}(\bar{E}^\lambda{\rm div}\bar{E}^\lambda)_t)\triangle\bar{v}^\lambda_t {\rm d}x -2A\int_{{\Bbb T}^3}({\rm div}\bar{v}^\lambda\bar{\rho}^\lambda_t +\bar{v}^\lambda\nabla\bar{\rho}^\lambda_t)\nabla\bar{\rho}^\lambda_t{\rm d}x\nonumber \\ &\leq& \epsilon \parallel(\triangle\bar{v}^\lambda_t,\nabla{\rm div}\bar{v}^\lambda,\nabla{\rm div}\bar{v}^\lambda_t,\nabla\bar{\rho}^\lambda,\nabla\bar{\rho}^\lambda_t)\parallel^2+M\parallel\bar{v}^\lambda\nabla\bar{v}^\lambda_t \parallel^2+M\parallel\bar{\rho}^\lambda\bar{v}^\lambda_t\parallel^2\nonumber \\ && +M\parallel\nabla\bar{v}^\lambda\bar{v}^\lambda_t\parallel^2+M\parallel\bar{\rho}^\lambda_t\nabla\bar{v}^\lambda\parallel^2+M\parallel\bar{v}^\lambda\bar{\rho}^\lambda_t\parallel^2\nonumber \\ && +M\parallel\nabla\bar{\rho}^\lambda\nabla\bar{v}^\lambda_t\parallel^2+M\parallel\bar{\rho}^\lambda_t{\rm div}\bar{v}^\lambda\parallel^2+M\parallel\bar{\rho}^\lambda_t\nabla\bar{\rho}^\lambda\parallel^2+M\parallel\bar{\rho}^\lambda_t\bar{v}^\lambda_t\parallel^2\nonumber \\ && +M\parallel\nabla\bar{\rho}^\lambda_t\nabla\bar{v}^\lambda\parallel^2+M\parallel\nabla\bar{\rho}^\lambda{\rm div}\bar{v}^\lambda_t\parallel^2+M\parallel\nabla\bar{\rho}^\lambda{\rm div}\bar{v}^\lambda\parallel^2+M\parallel\nabla\bar{\rho}^\lambda\bar{v}^\lambda_t\parallel^2\nonumber \\ && +M\parallel\bar{\rho}^\lambda\nabla\bar{v}^\lambda_t\parallel^2+\lambda^4M\parallel\bar{E}^\lambda_t{\rm div}\bar{E}^\lambda\parallel^2+M\lambda^4\parallel\bar{E}^\lambda{\rm div}\bar{E}^\lambda_t\parallel^2\nonumber \\ & \leq &\epsilon \parallel(\triangle\bar{v}^\lambda_t,\nabla{\rm div}\bar{v}^\lambda,\nabla{\rm div}\bar{v}^\lambda_t,\nabla\bar{\rho}^\lambda,\nabla\bar{\rho}^\lambda_t)\parallel^2+M\parallel\nabla\bar{v}^\lambda_t \parallel^2\parallel\bar{v}^\lambda\parallel^2_{H^2}\nonumber \\ && +M\parallel\bar{v}^\lambda_t \parallel^2_{H^1}\parallel\bar{\rho}^\lambda\parallel^2_{H^1}+M\parallel\bar{v}^\lambda_t \parallel^2_{H^1}\parallel\nabla\bar{v}^\lambda\parallel^2_{H^1}+M\parallel\bar{\rho}^\lambda_t \parallel^2_{H^1}\parallel\nabla\bar{v}^\lambda\parallel^2_{H^1}\nonumber \\ && +M\parallel\bar{\rho}^\lambda_t \parallel^2\parallel\bar{v}^\lambda\parallel^2_{H^2}+M\parallel\bar{v}^\lambda_t \parallel^2_{H^1}\parallel\bar{\rho}^\lambda_t\parallel^2_{H^1}+M\parallel\bar{\rho}^\lambda_t \parallel^2_{H^1}\parallel{\rm div}\bar{v}^\lambda\parallel^2_{H^1}\nonumber \\ && +M\parallel\bar{\rho}^\lambda_t \parallel^2_{H^1}\parallel\nabla\bar{\rho}^\lambda\parallel^2_{H^1}+M\parallel\nabla\bar{\rho}^\lambda_t \parallel^2_{H^1}\parallel{\rm div}\bar{v}^\lambda\parallel^2_{H^1}+M\parallel\nabla\bar{\rho}^\lambda_t \parallel^2_{H^1}\parallel\nabla\bar{v}^\lambda\parallel^2_{H^1}\nonumber \\ && +M\parallel\nabla\bar{\rho}^\lambda_t \parallel^2_{H^1}\parallel{\rm div}\bar{v}^\lambda_t\parallel^2_{H^1}+M\parallel\nabla\bar{\rho}^\lambda \parallel^2_{H^1}\parallel{\rm div}\bar{v}^\lambda\parallel^2_{H^1}+M\parallel\triangle\bar{\rho}^\lambda\parallel^2\parallel\bar{v}^\lambda_t\parallel^2_{H^2}\nonumber \\ && +M\lambda^4\parallel\bar{E}^\lambda_t \parallel^2_{H^1}\parallel{\rm div}\bar{E}^\lambda\parallel^2_{H^1}+M\lambda^4\parallel\bar{E}^\lambda \parallel^2_{H^2}\parallel{\rm div}\bar{E}^\lambda_t\parallel^2\nonumber \\ & \leq& \epsilon \parallel(\triangle\bar{v}^\lambda_t,\nabla{\rm div}\bar{v}^\lambda,\nabla{\rm div}\bar{v}^\lambda_t,\nabla\bar{\rho}^\lambda,\nabla\bar{\rho}^\lambda_t)\parallel^2+M\mid\mid\mid W\mid\mid\mid^4.\label{eq:eq-vest19} \end{matrix}$

联合不等式(3.43)-(3.45), 并应用不等式(3.46)

$\begin{equation} \int_{{\Bbb T}^3}(\rho+\bar{\rho}^\lambda)\bar{v}^\lambda_{tt}\triangle\bar{v}^\lambda_t {\rm d}x\geq \frac{\underline{\rho}}{2}\int_{{\Bbb T}^3}\bar{v}^\lambda_{tt}\triangle\bar{v}^\lambda_t {\rm d}x=\frac{\underline{\rho}}{4}\frac{\rm d}{{\rm d}t}\parallel\nabla\bar{v}^\lambda_t\parallel^2,\label{eq:rvt3} \end{equation}$

$\rho$和$\bar{\rho}^\lambda$满足下面的不等式

$\begin{equation} \frac{\underline{\rho}}{2}\leq \rho-|\bar{\rho}^\lambda|_{L^\infty}\leq\rho+\bar{\rho}^\lambda\leq\overline{\rho}+|\bar{\rho}^\lambda|_{L^\infty}\leq2\overline{\rho},\label{eq:rvt4} \end{equation} $

其中$\underline{\rho}$和$\overline{\rho}$分别是下确界和上确界, 可得

$\begin{matrix} && \epsilon\frac{\rm d}{{\rm d}t}\parallel(\nabla\bar{\rho}^\lambda_t,\nabla\bar{v}^\lambda_t) \parallel^2+k\parallel (\triangle\bar{v}^\lambda_t,\nabla{\rm div}\bar{v}^\lambda_t)\parallel^2 \nonumber \\ & \leq & M\parallel(\bar{v}^\lambda,\bar{v}^\lambda_t,\nabla\bar{v}^\lambda,{\rm div}\bar{v}^\lambda,\nabla\bar{v}^\lambda_t,\nabla{\rm div}\bar{v}^\lambda,\bar{\rho}^\lambda,\bar{\rho}^\lambda_t,\nabla\bar{\rho}^\lambda,\nabla\bar{\rho}^\lambda_t,\triangle\bar{\rho}^\lambda,\bar{E}^\lambda, \bar{E}^\lambda_t)\parallel^2\nonumber \\ && +M\lambda^4\parallel({\rm div}\bar{E}^\lambda,{\rm div} \bar{E}^\lambda_t)\parallel^2 +M\mid\mid\mid W\mid\mid\mid^4+M\lambda.\label{eq:eq-vest20} \end{matrix}$

联合$\delta_4$(3.32),(3.39),(3.42)以及(3.48)式, 在区间$[t]$上关于$t$ 积分,取$\delta_4$$>$$0$足够小, 可得不等式(3.28). 证毕

定理3.6 估计$\parallel(\triangle \bar{z}^\lambda, {\rm div}\bar{E}^\lambda,\lambda\nabla {\rm div}\bar{E}^\lambda, \lambda^2\triangle {\rm div}\bar{E}^\lambda,\triangle \bar{v}^\lambda,\nabla{\rm div}\bar{v}^\lambda,\nabla\bar{\rho}^\lambda)\parallel^2$, 有

$\begin{matrix} &&\parallel(\triangle \bar{z}^\lambda, {\rm div}\bar{E}^\lambda,\lambda\nabla {\rm div}\bar{E}^\lambda, \lambda^2\triangle {\rm div}\bar{E}^\lambda,\triangle \bar{v}^\lambda,\nabla{\rm div}\bar{v}^\lambda,\nabla\bar{\rho}^\lambda)\parallel^2\nonumber \\& \leq& M\parallel(\nabla\bar{z}^\lambda,\nabla\bar{v}^\lambda,\nabla\bar{\rho}^\lambda,\nabla\bar{z}^\lambda_t,\nabla\bar{v}^\lambda_t,\nabla\bar{\rho}^\lambda_t)\parallel^2\nonumber \\ && \lambda\parallel({\rm div}\bar{E}^\lambda,\lambda\nabla{\rm div}\bar{E}^\lambda,{\rm div}\bar{E}^\lambda_t,\lambda\nabla{\rm div}\bar{E}^\lambda_t)\parallel^2+M\mid\mid\mid W\mid\mid\mid^4+M\lambda. \label{eq:lw7} \end{matrix}$

注3.3 不等式(3.49)可由不等式(3.26)和 Green公式而得.

4 证明定理

在定理的假设下, 由标准的椭圆正则性理论, 有

$\begin{equation} \parallel \partial_t^i\bar{z}^\lambda \parallel_{H^2}^2\leq M(\parallel \partial_t^i\bar{z}^\lambda \parallel^2+\parallel \Delta \partial_t^i\bar{z}^\lambda \parallel^2),i=0,1,\label{eq:eq-pf1} \end{equation} $
$\begin{equation} \parallel \partial_t^i\bar{v}^\lambda \parallel_{H^2}^2\leq M(\parallel \partial_t^i\bar{v}^\lambda \parallel^2+\parallel \Delta \partial_t^i\bar{v}^\lambda \parallel^2),i=0,1,\label{eq:eq-pf3} \end{equation}$
$\begin{equation} \parallel \partial_t^i\bar{\rho}^\lambda \parallel_{H^2}^2\leq M(\parallel \partial_t^i\bar{\rho}^\lambda \parallel^2+\parallel \Delta \partial_t^i\bar{\rho}^\lambda \parallel^2),i=0,1,\label{eq:eq-pf4} \end{equation} $
$\begin{equation} \parallel \partial_t^i\bar{E}^\lambda \parallel_{H^2}^2\leq M(\parallel \partial_t^i\bar{E}^\lambda \parallel^2+\parallel \Delta \partial_t^i{\rm div}\bar{T}^\lambda \parallel^2), i=0,1~~s=1,2.\label{eq:eq-pf6} \end{equation} $

由$\Gamma^\lambda(t)$和$\mid\mid\mid W\mid\mid\mid$的表达式, 并应用不等式(4.1)-(4.4), 知存在独立于$\lambda$的两个正常数$C_1$和$C_2$, 使得

$\begin{equation} C_1\mid\mid\mid W\mid\mid\mid^2\leq\Gamma^\lambda(t)\leq C_2\mid\mid\mid W\mid\mid\mid^2.\label{eq:equiv-1} \end{equation} $

应用不等式(4.5), 并计算(3.2)+$\delta$(3.4)+(3.25)+$\delta_1$(3.26)+$\delta_2$(3.28)+(3.49), 取$\delta,\delta_1,\delta_2,\lambda$足够小, 可得

$\begin{matrix} \Gamma^\lambda(t)+\int_0^tG^\lambda(s){\rm d}s &\leq & M\tilde{\Gamma}^\lambda(t=0)+M\int_0^t(\Gamma^\lambda(s)+(\Gamma^\lambda(s))^2){\rm d}s\nonumber\\ && +M(\Gamma^\lambda(t))^2 +M\int_0^t\Gamma^\lambda(s)G^\lambda(s){\rm d}s+M\lambda,\label{eq:txg3-1} \end{matrix}$

其中

$\begin{matrix} \tilde{\Gamma^\lambda}(0)&=&\parallel(\bar{z}^\lambda,\bar{v}^\lambda,\bar{\rho}^\lambda,\bar{z}_t^\lambda,\bar{v}_t^\lambda,\bar{\rho}_t^\lambda,\nabla\bar{z}^\lambda,\nabla\bar{v}^\lambda,\nabla\bar{\rho}^\lambda,\nabla\bar{z}_t^\lambda,\nabla\bar{v}_t^\lambda,\nabla\bar{\rho}_t^\lambda)\parallel^2(t=0)\nonumber \\ &&+\lambda^2\parallel (\bar{E}^\lambda,\lambda{\rm div}\bar{E}^\lambda,\bar{E}_t^\lambda,\lambda{\rm div}\bar{E}_t^\lambda,\lambda\nabla{\rm div}\bar{E}^\lambda,\lambda\nabla{\rm div}\bar{E}_t^\lambda) \parallel^2(t=0).\label{eq:txg4-1} \end{matrix}$

不等式(4.6)是一个$\lambda$ -加权的Growall型熵积分不等式, 因此有下面的结论.

定理4.1 假设

$\begin{equation} \tilde{\Gamma^\lambda}(t=0)\leq M \lambda, \label{eq:init-s} \end{equation}$

其中$M$是独立于$\lambda$的正常数. 那么对于任意的$T\in (0,T_{\max})$, $T_{\max}\leq\infty$, 存在正常数$\lambda_0\ll1$, 使得对于任意的$\lambda\leq \lambda_0,\delta\in (0,1), 0\leq t\leq T$不等式

$\begin{equation} \Gamma^\lambda(t)\leq M \lambda^{1-\delta}\label{eq:con-res} \end{equation}$

成立.

注4.1 此定理的证明类似于文献[引理10], 此处省略其证明.

按照初值的假设, 易知(4.8)式成立, 应用定理3.5, 知不等式(4.9)成立. 由(4.9)式, 可得不等式(2.24). 定理得证.

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