## 迁移率互异的可压电扩散模型的拟中性极限

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

 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

 Fund supported: The NSFC.  12102492Key Research Projects of Henan Higher Education Institutions.  22A110027Henan Postdoctoral Foundation

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

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

## 1 引言

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

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

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

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

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

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

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

## 3 能量估计

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\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}$

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

$\rho$$\bar{\rho}^\lambda满足不等式 $$\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}$$ 其中\underline{\rho}$$\overline{\rho}$ 分别是下确界和上确界, 因此可得

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\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}$

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

$\rho$$\bar{\rho}^\lambda满足下面的不等式 $$\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}$$ 其中\underline{\rho}$$\overline{\rho}$分别是下确界和上确界, 可得

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

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

## 4 证明定理

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

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

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

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

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

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

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

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Cimartti G.

Invariant regions for the Nernst-Planck equations

Ann Mat Pura Appl, 1998, 175: 93-118

Rubinstein I.

Electro-Diffusion of Ions

Li F C.

Quasi-neutral limit of the electro-diffusion model arising in electrohydrodynamics

J Diff Equations, 2009, 246: 3620-3641

Wang S, Jiang L M, Liu C D.

Quasi-neutral limit and the boundary layer problem of Planck Nernst Poisson Navier Stokes equations for electro hydrodynamics

J Diff Equations, 2009, 267: 3475-3523

Wang S, Jiang L M.

Quasi-neutral limit and the boundary layer problem of the electro diffusion model arising in electro hydrodynamics

Nonlinear Anal: RWA, 2021, 59: 103266

Yang J W, Ju Q C.

Convergence of the quantum Navier-Stokes-Poisson equations to the incompressible Euler equations for general initial data

Nonlinear Anal: RWA, 2015, 23: 148-159

Liu C, Wang Y, Yang T.

On the ill-posedness of the prandtl equations in three-dimensional space

Arch Ration Mech Anal, 2016, 220: 83-108

Brenier Y.

Convergence of the Vlasov-Poisson system to the incompressible Euler equations

Comm Part Diff Equations, 2000, 25: 737-754

Alì D, Bini D, Rionero S.

Existence and relaxation limit for smooth solution to the Euler-Poisson model for semiconductors

Siam J Math Anal, 2000, 32: 572-587

Alì G, Jüngel A.

Global smooth solutions to the multi-dimensional hydrodynamic model for two-carrier plasmas

J Diff Equations, 2003, 190: 663-685

Gasser I, Levermore C D, Markowich P, Shmeiser C.

The initial time layer problem and the quasi-neutral limit in the semiconductor drift-diffusion model

European J Appl Math, 2001, 12: 497-512

Guo Y, Strauss W.

Stability of semiconductor states with insulating and contact boundary conditions

Arch Rat Mech and Anal, 2006, 179: 1-30

Hsiao L, Li F C, Wang S.

Coupled quasi-neutral and inviscid limit of the Vlasov-Poisson-Fokker-Planck system

Commun Pure Appl Anal, 2008, 7: 579-589

Jüngel A. Peng Y J.

Ahierarchy of hydrodynamic models for plasmas: Quasi-neutral limits in the drift-diffusion equations

Asymptot Anal, 2001, 28: 49-73

Roubicek T. Nonlinear Partial Differential Equations with Applications. Basel: Birkhauser Verlag, 2005

Suzuki M.

Asymptotic stability of stationary solutions to the Euler-Poisson equations arising in plasma physics

Kinet Relat Models, 2011, 4: 569-588

Temam R. Navier-Stokes Equations Theory and Numerical Analysis. New York: North-Holland, 1977

Wang S.

Quasi-neutral limit of Euler-Poisson system with and without viscosity

Comm Part Diff Equations, 2004, 29: 419-456

Wang S.

Quasi-neutral limit of the multi-dimensional drift-diffusion-Poisson model for semiconductor with pn-junctions

Math Models Methods Appl Sci, 2006, 16: 737-757

Hsiao L, Wang S.

Quasi-neureal limit of a time dependent drift-diffusion-Poisson model for p-n junction semiconductor devices

J Diff Eqns, 2006, 225: 411-439

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