﻿ 带有外力项的可压等熵Navier-Stokes方程的滞弹性逼近 Anelastic Approximation of Compressible Isentropic Navier-Stokes Equations with Exterior Force

Vol.06 No.09(2017), Article ID:23178,13 pages
10.12677/AAM.2017.69146

Anelastic Approximation of Compressible Isentropic Navier-Stokes Equations with Exterior Force

Changsheng Dou*, Li Wang, Chenxi Zhu

School of Statistics, Capital University of Economics and Business, Beijing

Received: Dec. 4th, 2017; accepted: Dec. 20th, 2017; published: Dec. 27th, 2017

ABSTRACT

In this paper, we prove the anelastic approximation limit to compressible isentropic Navier-Stokes equations with exterior force and Dirichlet boundary condition, as Mach number and Froude number go to zero. This covers the result of special force case in J. Math. Pures Appl. 88 (2007) 230-240.

Keywords:Compressible Isentropic Navier-Stokes Equations, Anelastic Approximation, Dirichlet Boundary Condition, Mach Number, Froude Number

1. 引言

$\left\{\begin{array}{l}{\rho }_{\epsilon t}+\text{div}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)=0,\text{\hspace{0.17em}}{\rho }_{\epsilon }\ge 0,\\ {\left({\rho }_{\epsilon }{u}_{\epsilon }\right)}_{t}+\text{div}\left({\rho }_{\epsilon }{u}_{\epsilon }\otimes {u}_{\epsilon }\right)-\mu \Delta {u}_{\epsilon }-\xi \nabla \text{div}{u}_{\epsilon }+\frac{\gamma {\rho }_{\epsilon }}{{\epsilon }^{2}\left(\gamma -1\right)}\nabla \left({\rho }_{\epsilon }^{\gamma -1}-{\overline{\rho }}_{\epsilon }^{\gamma -1}\right)=0.\end{array}$ (1.1)

(包含势能项 $\frac{1}{{\epsilon }^{2}}{\rho }_{\epsilon }\nabla V$ )趋于滞弹性系统的严格推导，其中 $V=gz$ 是重力势能( $z$ 为竖直分量)。文中滞弹性系

2. 主要结果

${\left({\rho }_{\epsilon },{\rho }_{\epsilon }{u}_{\epsilon }\right)\left(t,x\right)|}_{t=0}=\left({\rho }_{0\epsilon },{m}_{0\epsilon }\right)\left(x\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in \Omega ,$ (2.1)

${u}_{\epsilon }=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{on}\text{\hspace{0.17em}}\partial \Omega .$ (2.2)

$\frac{{m}_{0\epsilon }^{2}}{{\rho }_{0\epsilon }}\in {L}^{1}\left(\Omega \right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{m}_{0\epsilon }\in {L}^{2\gamma /\left(\gamma +1\right)}\left(\Omega \right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\rho }_{0\epsilon }\in {L}^{\gamma }\left(\Omega \right),$ (2.3)

${\int }_{\Omega }\frac{{\rho }_{0}^{\gamma }-\gamma {\overline{\rho }}^{\gamma -1}{\rho }_{0}+\left(\gamma -1\right){\overline{\rho }}^{\gamma }}{{\epsilon }^{2}\left(\gamma -1\right)}\text{d}x\le C$ (2.4)

$m={P}_{\overline{\rho }}m+\overline{\rho }\nabla \Psi ,$

$\text{div}{P}_{\overline{\rho }}m=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{P}_{\overline{\rho }}m\cdot n|{}_{\partial \Omega }=0.$

${\rho }_{\epsilon }\to \overline{\rho }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}{L}^{\infty }\left(0,T;{L}^{\gamma }\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{u}_{\epsilon }\to u\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}{L}^{2}\left(0,T;{H}_{0}^{1}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\rho }_{\epsilon }{u}_{\epsilon }\to \overline{\rho }u\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}{L}^{2}\left(0,T;{L}^{2}\right)$ (2.5)

${P}_{\overline{\rho }}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\to \overline{\rho }u\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}{L}^{2}\left(0,T;{L}^{2}\right)$ (2.6)

$\left\{\begin{array}{l}{\left(\overline{\rho }u\right)}_{t}+\text{div}\left(\overline{\rho }u\otimes u\right)-\mu \Delta u-\xi \nabla \text{div}u+\overline{\rho }\nabla q=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(t,x\right)\in \left(0,T\right)×\Omega ,\\ \text{div}\left(\overline{\rho }u\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(t,x\right)\in \left(0,T\right)×\Omega ,\\ u=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in \partial \Omega ,\\ u\left(0,x\right)=\frac{{P}_{\overline{\rho }}{m}_{0}}{\overline{\rho }}\left(x\right).\end{array}$ (2.7)

$\begin{array}{l}-{\int }_{0}^{T}{\int }_{\Omega }u{\partial }_{t}f\text{d}x\text{d}s-{\int }_{0}^{T}{\int }_{\Omega }\overline{\rho }u\otimes u\nabla \left(\frac{f}{\overline{\rho }}\right)\text{d}x\text{d}s\\ +{\int }_{0}^{T}{\int }_{\Omega }\nabla u:\nabla \left(\frac{f}{\overline{\rho }}\right)+\xi \text{div}u\text{div}\left(\frac{f}{\overline{\rho }}\right)\text{d}x\text{d}s={\int }_{\Omega }{P}_{\overline{\rho }}{m}_{0}\left(\frac{f}{\overline{\rho }}\right)\text{d}x.\end{array}$ (2.8)

3. 先验估计

${\int }_{\Omega }\left({\rho }_{\epsilon }\frac{{|{u}_{\epsilon }|}^{2}}{2}+\frac{{\pi }_{\epsilon }}{\gamma -\text{1}}\right)\text{d}x+{\int }_{0}^{T}{\int }_{\Omega }\mu {|\nabla {u}_{\epsilon }|}^{2}+\xi {|\text{div}{u}_{\epsilon }|}^{2}\text{d}x\text{d}s\le C,$ (3.1)

$\frac{\text{d}}{\text{d}t}{\int }_{\Omega }{\rho }_{\epsilon }\frac{{|{u}_{\epsilon }|}^{2}}{2}+\frac{\gamma {\rho }_{\epsilon }{u}_{\epsilon }}{{\epsilon }^{2}\left(\gamma -1\right)}\nabla \left({\rho }_{\epsilon }^{\gamma -1}-{\overline{\rho }}_{\epsilon }^{\gamma -1}\right)\text{d}x+{\int }_{\Omega }\mu {|\nabla {u}_{\epsilon }|}^{2}+\xi {|\text{div}{u}_{\epsilon }|}^{2}\text{d}x\le 0.$

$\frac{\text{d}}{\text{d}t}{\int }_{\Omega }\left({\rho }_{\epsilon }\frac{{|{u}_{\epsilon }|}^{2}}{2}+\frac{{\rho }_{\epsilon }^{\gamma }-\gamma {\overline{\rho }}^{\gamma -1}{\rho }_{\epsilon }}{{\epsilon }^{2}\left(\gamma -1\right)}\right)\text{d}x+{\int }_{\Omega }\mu {|\nabla {u}_{\epsilon }|}^{2}+\xi {|\text{div}{u}_{\epsilon }|}^{2}\text{d}x\le 0.$

${\int }_{\Omega }\left({\rho }_{\epsilon }\frac{{|{u}_{\epsilon }|}^{2}}{2}+\frac{{\rho }_{\epsilon }^{\gamma }-\gamma {\overline{\rho }}^{\gamma -1}\rho }{{\epsilon }_{\epsilon }^{2}{\left(\gamma -1\right)}^{2}}\right)\text{d}x$ $+{\int }_{0}^{t}{\int }_{\Omega }\mu {|\nabla {u}_{\epsilon }|}^{2}+\xi {|\text{div}{u}_{\epsilon }|}^{2}\text{d}x\text{d}s\le {\int }_{\Omega }\left(\frac{{|{m}_{0\epsilon }|}^{2}}{2{\rho }_{0\epsilon }}+\frac{{\rho }_{0\epsilon }^{\gamma }-\gamma {\overline{\rho }}^{\gamma -1}}{{\epsilon }^{2}\left(\gamma -1\right)}\right)\text{d}x.$ (3.2)

$\begin{array}{l}{\int }_{\Omega }\left({\rho }_{\epsilon }\frac{{|{u}_{\epsilon }|}^{2}}{2}+\frac{{\rho }_{\epsilon }^{\gamma }-\gamma {\overline{\rho }}^{\gamma -1}{\rho }_{\epsilon }+\left(\gamma -1\right){\overline{\rho }}^{\gamma }}{{\epsilon }^{2}\left(\gamma -1\right)}\right)\text{d}x+{\int }_{0}^{t}{\int }_{\Omega }\mu {|\nabla {u}_{\epsilon }|}^{2}+\xi {|\text{div}{u}_{\epsilon }|}^{2}\text{d}x\text{d}s\\ \le {\int }_{\Omega }\left(\frac{{|{m}_{0\epsilon }|}^{2}}{2{\rho }_{0\epsilon }}+\frac{{\rho }_{0\epsilon }^{\gamma }-\gamma {\overline{\rho }}^{\gamma -1}{\rho }_{0\epsilon }+\left(\gamma -1\right){\overline{\rho }}^{\gamma }}{{\epsilon }^{2}\left(\gamma -1\right)}\right)\text{d}x.\end{array}$ (3.3)

$\begin{array}{c}{\epsilon }^{2}{\pi }_{\epsilon }={\rho }_{\epsilon }^{\gamma }-{\overline{\rho }}^{\gamma }-\gamma {\overline{\rho }}^{\gamma -1}\left({\rho }_{\epsilon }-\overline{\rho }\right)={\rho }_{\epsilon }^{\gamma }-\gamma {\overline{\rho }}^{\gamma -1}{\rho }_{\epsilon }+\left(\gamma -1\right){\overline{\rho }}^{\gamma }\\ ={\rho }_{\epsilon }^{\gamma }-{\overline{\rho }}^{\gamma }-\gamma \left({\overline{\rho }}^{\gamma -1}\epsilon {\varphi }_{\epsilon }\right).\end{array}$ (3.4)

$\begin{array}{l}{\int }_{\Omega }\left({\rho }_{\epsilon }\frac{{|{u}_{\epsilon }|}^{2}}{2}+\frac{{\pi }_{\epsilon }}{\gamma -1}\right)\text{d}x+{\int }_{0}^{t}{\int }_{\Omega }\mu {|\nabla {u}_{\epsilon }|}^{2}+\xi {|\text{div}{u}_{\epsilon }|}^{2}\text{d}x\text{d}s\\ \le {\int }_{\Omega }\left(\frac{{|{m}_{0\epsilon }|}^{2}}{2{\rho }_{0\epsilon }}+\frac{{\rho }_{0\epsilon }^{\gamma }-\gamma {\overline{\rho }}^{\gamma -1}{\rho }_{0\epsilon }+\left(\gamma -1\right){\overline{\rho }}^{\gamma }}{{\epsilon }^{2}\left(\gamma -1\right)}\right)\text{d}x.\\ \le C.\end{array}$ (3.5)

${\rho }_{\epsilon }\in {L}^{\gamma +\theta }\left(\left(0,T\right)×\Omega \right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}0<\theta \le {\theta }_{0}=\frac{2}{N}\gamma -1,$ (3.6)

$\frac{1}{\epsilon }{|\epsilon {\varphi }_{\epsilon }|}^{\gamma +\theta }\in {L}^{1}\left(\left(0,T\right)×\Omega \right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}1<\theta \le {\theta }_{0}=\frac{2}{N}\gamma -1,$ (3.7)

${\varphi }_{\epsilon }\in {L}^{1+\theta }\left(\left(0,T\right)×\Omega \right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}0<\theta \le {\theta }_{1}=1+4\left(\frac{1}{N}-\frac{1}{\gamma }\right)$ (3.8)

${\epsilon }^{\gamma -1}{|{\varphi }_{\epsilon }|}^{\gamma +\theta }\in {L}^{1}\left(\left(0,T\right)×\Omega \right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}0<\theta \le {\theta }_{1}=1+4\left(\frac{1}{N}-\frac{1}{\gamma }\right)$ (3.9)

${\epsilon }^{1-\theta }|{\pi }_{\epsilon }-\frac{\gamma \left(\gamma -1\right)}{2}{\overline{\rho }}^{\gamma -2}{\varphi }_{\epsilon }^{2}|\in {L}^{1}\left(\left(0,T\right)×\Omega \right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}0<\theta \le {\theta }_{1}=1+4\left(\frac{1}{N}-\frac{1}{\gamma }\right)$ (3.10)

$\left\{\begin{array}{l}-\Delta u+\nabla p=g,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\int }_{\Omega }p\text{d}x=0,\\ u=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in \partial \Omega ,\\ \text{div}\left(u\right)=0.\end{array}$ (3.11)

$\begin{array}{l}{\partial }_{t}{S}_{0}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)+{S}_{0}\text{div}\left({\rho }_{\epsilon }{u}_{\epsilon }\otimes {u}_{\epsilon }\right)+{h}_{0}+\frac{1}{{\epsilon }^{2}}\left({\rho }_{\epsilon }^{\gamma }-{\overline{\rho }}^{\gamma }-\frac{1}{|\Omega |}{\int }_{\Omega }{\rho }_{\epsilon }^{\gamma }-{\overline{\rho }}^{\gamma }\text{d}x\right)\\ -\frac{\gamma }{\epsilon \left(\gamma -1\right)}{S}_{0}\left({\varphi }_{\epsilon }\nabla {\overline{\rho }}^{\gamma -1}\right)=0,\end{array}$ (3.12)

$\frac{1}{\epsilon }{\int }_{0}^{T}{\int }_{\Omega }\left({\rho }_{\epsilon }^{\gamma }-{\overline{\rho }}^{\gamma }-\frac{1}{|\Omega |}{\int }_{\Omega }{\rho }_{\epsilon }^{\gamma }-{\overline{\rho }}^{\gamma }\text{d}x\right){\rho }_{\epsilon }^{\theta }\text{d}x\text{d}t\le C.$ (3.13)

$\frac{1}{\epsilon }{\int }_{0}^{T}{\int }_{\Omega }\left({\rho }_{\epsilon }^{\gamma }-{\overline{\rho }}^{\gamma }\right)\left({\rho }_{\epsilon }^{\theta }-{\overline{\rho }}^{\theta }\right)\text{d}x\text{d}t\le C.$ (3.14)

${\partial }_{t}{\varphi }_{\epsilon }+\frac{1}{\epsilon }\text{div}\left(\overline{\rho }{u}_{\epsilon }\right)+\text{div}\left({\varphi }_{\epsilon }{u}_{\epsilon }\right)=0.$ (3.15)

${\partial }_{t}{\varphi }_{\epsilon }^{\theta }+\frac{1}{\epsilon }\theta {|ph{i}_{\epsilon }|}^{\theta -\text{1}}\text{div}\left(\overline{\rho }{u}_{\epsilon }\right)+\text{div}\left({\varphi }_{\epsilon }^{\theta }{u}_{\epsilon }\right)=\left(1-\theta \right){\varphi }_{\epsilon }^{\theta }\text{div}\left({u}_{\epsilon }\right).$ (3.16)

$\begin{array}{l}{\partial }_{t}\left[{\varphi }_{\epsilon }^{\theta }{S}_{0}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\right]+\text{div}\left[{\varphi }_{\epsilon }^{\theta }{S}_{0}\left({\rho }_{\epsilon }{u}_{\epsilon }\right){u}_{\epsilon }\right]-{\varphi }_{\epsilon }^{\theta }{u}_{\epsilon }\cdot \nabla {S}_{0}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\\ +\left[\theta {|{\varphi }_{\epsilon }|}^{\theta -1}\frac{1}{\epsilon }\text{div}\left(\overline{\rho }{u}_{\epsilon }\right)+\left(\theta -1\right){\varphi }_{\epsilon }^{\theta }\text{div}\left({u}_{\epsilon }\right)\right]{S}_{0}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\\ +\text{\hspace{0.17em}}{\varphi }_{\epsilon }^{\theta }{S}_{0}\text{div}\left({\rho }_{\epsilon }{u}_{\epsilon }\otimes {u}_{\epsilon }\right)+{\varphi }_{\epsilon }^{\theta }{h}_{0}+{\varphi }_{\epsilon }^{\theta }\frac{1}{{\epsilon }^{2}}\left({\rho }_{\epsilon }^{\gamma }-{\overline{\rho }}^{\gamma }-\frac{1}{|\Omega |}{\int }_{\Omega }{\rho }_{\epsilon }^{\gamma }-{\overline{\rho }}^{\gamma }\text{d}x\right)\\ -\text{\hspace{0.17em}}{\varphi }_{\epsilon }^{\theta }\frac{\gamma }{\epsilon \left(\gamma -1\right)}{S}_{0}\left({\varphi }_{\epsilon }\nabla {\overline{\rho }}^{\gamma -1}\right)=0.\end{array}$ (3.17)

$\begin{array}{l}\epsilon {\int }_{\Omega }\left[{\varphi }_{\epsilon }^{\theta }{S}_{0}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\right]\text{d}x-\epsilon {\int }_{\Omega }\left[{\varphi }_{\epsilon }^{\theta }{S}_{0}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\right]\left(0\right)\text{d}x-\epsilon {\int }_{0}^{T}{\int }_{\Omega }{\varphi }_{\epsilon }^{\theta }{u}_{\epsilon }\cdot \nabla {S}_{0}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\text{d}x\text{d}s\\ +{\int }_{0}^{T}{\int }_{\Omega }\left[\theta {|{\varphi }_{\epsilon }|}^{\theta -1}\text{div}\left(\overline{\rho }{u}_{\epsilon }\right)+\left(\theta -1\right){\varphi }_{\epsilon }^{\theta }\text{div}\left({u}_{\epsilon }\right)\right]{S}_{0}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\text{d}x\text{d}s\\ +{\int }_{0}^{T}{\int }_{\Omega }\epsilon {\varphi }_{\epsilon }^{\theta }\left[{S}_{0}\text{div}\left({\rho }_{\epsilon }{u}_{\epsilon }\otimes {u}_{\epsilon }\right)+{\varphi }_{\epsilon }^{\theta }{h}_{0}+\frac{\gamma }{\epsilon \left(\gamma -1\right)}{S}_{0}\left({\varphi }_{\epsilon }\nabla {\overline{\rho }}^{\gamma -1}\right)\right]\text{d}x\text{d}s\\ +\frac{1}{\epsilon }{\int }_{0}^{T}{\int }_{\Omega }{\varphi }_{\epsilon }^{\theta }\left({\rho }_{\epsilon }^{\gamma }-{\overline{\rho }}^{\gamma }-\frac{1}{|\Omega |}{\int }_{\Omega }{\rho }_{\epsilon }^{\gamma }-{\overline{\rho }}^{\gamma }\text{d}x\right)\text{d}x\text{d}s=0,\end{array}$ (3.18)

$\begin{array}{l}\frac{1}{\epsilon }{\int }_{0}^{T}{\int }_{\Omega }{\varphi }_{\epsilon }^{\theta }\left({\rho }_{\epsilon }^{\gamma }-{\overline{\rho }}^{\gamma }-\frac{1}{|\Omega |}{\int }_{\Omega }{\rho }_{\epsilon }^{\gamma }-{\overline{\rho }}^{\gamma }\text{d}x\right)\text{d}x\text{d}s\\ =-\epsilon {\int }_{\Omega }\left[{\varphi }_{\epsilon }^{\theta }{S}_{0}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\right]\text{d}x+\epsilon {\int }_{\Omega }\left[{\varphi }_{\epsilon }^{\theta }{S}_{0}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\right]\left(0\right)\text{d}x+\epsilon {\int }_{0}^{T}{\int }_{\Omega }{\varphi }_{\epsilon }^{\theta }{u}_{\epsilon }\cdot \nabla {S}_{0}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\text{d}x\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{\int }_{0}^{T}{\int }_{\Omega }\left[\theta {|{\varphi }_{\epsilon }|}^{\theta -1}\text{div}\left(\overline{\rho }{u}_{\epsilon }\right)+\left(\theta -1\right){\varphi }_{\epsilon }^{\theta }\text{div}\left({u}_{\epsilon }\right)\right]{S}_{0}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\text{d}x\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{\int }_{0}^{T}{\int }_{\Omega }\epsilon {\varphi }_{\epsilon }^{\theta }\left[{S}_{0}\text{div}\left({\rho }_{\epsilon }{u}_{\epsilon }\otimes {u}_{\epsilon }\right)+{\varphi }_{\epsilon }^{\theta }{h}_{0}+\frac{\gamma }{\epsilon \left(\gamma -1\right)}{S}_{0}\left({\varphi }_{\epsilon }\nabla {\overline{\rho }}^{\gamma -1}\right)\right]\text{d}x\text{d}s\\ :=\sum _{i=1}^{5}{I}_{i}.\end{array}$ (3.19)

$\begin{array}{l}{\rho }_{\epsilon }{u}_{\epsilon }\in {L}^{2}\left(0,T;{L}^{q}\left(\Omega \right)\right),\\ \epsilon {\varphi }_{\epsilon }{u}_{\epsilon }\in {L}^{2}\left(0,T;{L}^{q}\left(\Omega \right)\right),\\ {\varphi }_{\epsilon }^{\theta -1}\in {L}^{\infty }\left(0,T;{L}^{2/\left(\theta -1\right)}\left(\Omega \right)\right),\end{array}$ (3.20)

$\frac{\theta -1}{2}+\frac{2}{q}=\frac{\theta -1}{2}+\frac{N\gamma -2\gamma +2N}{N\gamma }\le 1$ ，且有

$|{I}_{3}|=\epsilon {\int }_{0}^{T}{\int }_{\Omega }{\varphi }_{\epsilon }^{\theta }{u}_{\epsilon }\cdot \nabla {S}_{0}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\text{d}x\text{d}s\le C.$ (3.21)

${|{\pi }_{\epsilon }-\frac{\gamma \left(\gamma -1\right)}{2}{\overline{\rho }}^{\gamma -2}{\varphi }_{\epsilon }^{2}|}^{\gamma +\theta }\le \epsilon {|{\varphi }_{\epsilon }|}^{3}+{\epsilon }^{\gamma -2}{|{\varphi }_{\epsilon }|}^{\gamma }$ ，并利用(3.8)~(3.9)得，当 $0\le \beta \le \gamma -1$ 时有 ${\epsilon }^{\beta }{\varphi }_{\epsilon }^{1+\theta +\beta }$${L}^{1}\left(\left(0,T\right)×\Omega \right)$ 中有界。即可得对 $0<\theta \le {\theta }_{1}$ 时有 ${\epsilon }^{1-\theta }|{\pi }_{\epsilon }-\frac{\gamma \left(\gamma -1\right)}{2}{\overline{\rho }}^{\gamma -2}{\varphi }_{\epsilon }^{2}|$${L}^{1}\left(\left(0,T\right)×\Omega \right)$ 中有界。(3.10)得证。证毕。

4. 定理证明

${P}_{\overline{\rho }}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\to \overline{\rho }u,$ (4.1)

${L}^{2}\left(\left(0,T\right)×\Omega \right)$ 中强收敛，并且

$\overline{\rho }\nabla {\psi }_{\epsilon }\to 0,$ (4.2)

${L}^{2}\left(\left(0,T\right)×\Omega \right)$ 中弱收敛，其中 $\text{div}\left(\overline{\rho }u\right)=0$

$\begin{array}{l}\frac{\gamma \rho }{{\epsilon }^{2}\left(\gamma -1\right)}\nabla \left({\rho }^{\gamma -1}-{\overline{\rho }}^{\gamma -1}\right)=\frac{1}{{\epsilon }^{2}}\nabla {\rho }^{\gamma }-\frac{\gamma }{{\epsilon }^{2}\left(\gamma -1\right)}\rho \nabla {\overline{\rho }}^{\gamma -1}\\ =\nabla {\pi }_{\epsilon }+\frac{1}{{\epsilon }^{2}}\left(\nabla {\overline{\rho }}^{\gamma }+\gamma \nabla \left({\overline{\rho }}^{\gamma -1}\left(\rho -\overline{\rho }\right)\right)\right)-\frac{\gamma }{{\epsilon }^{2}\left(\gamma -1\right)}\left(\rho -\overline{\rho }\right)\nabla {\overline{\rho }}^{\gamma -1}-\frac{1}{{\epsilon }^{2}}\nabla {\overline{\rho }}^{\gamma }\\ =\nabla {\pi }_{\epsilon }+\frac{1}{{\epsilon }^{2}}\gamma \nabla \left({\overline{\rho }}^{\gamma -1}\left(\rho -\overline{\rho }\right)\right)-\frac{\gamma }{{\epsilon }^{2}\left(\gamma -1\right)}\left(\rho -\overline{\rho }\right)\nabla {\overline{\rho }}^{\gamma -1}\\ =\nabla {\pi }_{\epsilon }+\frac{1}{{\epsilon }^{2}}\gamma \left(\gamma -1\right){\overline{\rho }}^{\gamma -2}\nabla \overline{\rho }\left(\rho -\overline{\rho }\right)+\frac{\gamma }{{\epsilon }^{2}}{\overline{\rho }}^{\gamma -1}\nabla \left(\rho -\overline{\rho }\right)-\frac{\gamma }{{\epsilon }^{2}}\left(\rho -\overline{\rho }\right){\overline{\rho }}^{\gamma -2}\nabla \overline{\rho }\end{array}$

$\begin{array}{l}=\nabla {\pi }_{\epsilon }+\frac{\gamma \left(\gamma -1\right)}{{\epsilon }^{2}\left(\gamma -2\right)}\overline{\rho }\nabla {\overline{\rho }}^{\gamma -2}\left(\rho -\overline{\rho }\right)+\frac{\gamma }{{\epsilon }^{2}}\overline{\rho }\nabla \left({\overline{\rho }}^{\gamma -2}\left(\rho -\overline{\rho }\right)\right)\\ -\frac{\gamma }{{\epsilon }^{2}}\left(\rho -\overline{\rho }\right)\nabla {\overline{\rho }}^{\gamma -2}\overline{\rho }-\frac{\gamma }{{\epsilon }^{2}\left(\gamma -2\right)}\left(\rho -\overline{\rho }\right)\overline{\rho }\nabla {\overline{\rho }}^{\gamma -2}=\nabla {\pi }_{\epsilon }+\frac{\gamma }{\epsilon }\overline{\rho }\nabla \left({\overline{\rho }}^{\gamma -2}{\varphi }_{\epsilon }\right)\end{array}$ (4.3)

(1.1)2可重写为：

${\left({\rho }_{\epsilon }{u}_{\epsilon }\right)}_{t}+\text{div}\left({\rho }_{\epsilon }{u}_{\epsilon }\otimes {u}_{\epsilon }\right)-\mu \Delta {u}_{\epsilon }-\xi \nabla \text{div}{u}_{\epsilon }+\nabla {\pi }_{\epsilon }+\frac{\gamma }{\epsilon }\overline{\rho }\nabla \left({\overline{\rho }}^{\gamma -2}{\varphi }_{\epsilon }\right)=0$ .(4.4)

$\begin{array}{l}-{\int }_{0}^{T}{\int }_{\Omega }\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\frac{{\partial }_{t}f}{\overline{\rho }}\text{d}x\text{d}s-{\int }_{0}^{T}{\int }_{\Omega }\left({\rho }_{\epsilon }{u}_{\epsilon }\otimes {u}_{\epsilon }\right):\nabla \left(\frac{f}{\overline{\rho }}\right)\text{d}x\text{d}s+{\int }_{0}^{T}{\int }_{\Omega }\mu \nabla {u}_{\epsilon }\nabla \left(\frac{f}{\overline{\rho }}\right)\\ +\xi \text{div}\left({u}_{\epsilon }\right)\text{div}\left(\frac{f}{\overline{\rho }}\right)-{\pi }_{\epsilon }\text{div}\left(\frac{f}{\overline{\rho }}\right)\text{d}x\text{d}s={\int }_{\Omega }{m}_{0\epsilon }\frac{f\left(t=0\right)}{\overline{\rho }}\text{d}x\end{array}$ (4.5)

$-{\int }_{0}^{T}{\int }_{\Omega }\left({\rho }_{\epsilon }{u}_{\epsilon }\otimes {u}_{\epsilon }\right):\nabla \left(\frac{f}{\overline{\rho }}\right)+{\pi }_{\epsilon }\text{div}\left(\frac{f}{\overline{\rho }}\right)\text{d}x\text{d}s\to -{\int }_{0}^{T}{\int }_{\Omega }\overline{\rho }u\otimes u:\nabla \left(\frac{f}{\overline{\rho }}\right)\text{d}x\text{d}s.$ (4.6)

$\text{div}\left({\rho }_{\epsilon }{u}_{\epsilon }\otimes {u}_{\epsilon }\right)+\nabla {\pi }_{\epsilon }\to \text{div}\left(\overline{\rho }u\otimes u\right)+\overline{\rho }\nabla P$ (4.7)

${\rho }_{\epsilon }{u}_{\epsilon }\otimes {\rho }_{\epsilon }{u}_{\epsilon }={P}_{\overline{\rho }}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\otimes {\rho }_{\epsilon }{u}_{\epsilon }+{Q}_{\overline{\rho }}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\otimes {P}_{\overline{\rho }}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)+{Q}_{\overline{\rho }}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\otimes {Q}_{\overline{\rho }}\left({\rho }_{\epsilon }{u}_{\epsilon }\right).$ (4.8)

${P}_{\overline{\rho }}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\otimes {\rho }_{\epsilon }{u}_{\epsilon }\to \left(\rho u\right)\otimes \left(\rho u\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{Q}_{\overline{\rho }}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\otimes {P}_{\overline{\rho }}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\to 0.$ (4.9)

$\left(I-{P}_{\overline{\rho }}\right)$ 作用(4.4)，并结合质量守恒方程可得方程组

$\left\{\begin{array}{l}{\partial }_{t}\overline{\rho }\nabla {\psi }_{\epsilon }+\frac{\gamma }{\epsilon }\overline{\rho }\nabla \left({\overline{\rho }}^{\gamma -2}{\varphi }_{\epsilon }\right)=\overline{\rho }{F}_{\epsilon },\hfill \\ \epsilon {\partial }_{t}{\varphi }_{\epsilon }+\text{div}\left(\overline{\rho }\nabla {\psi }_{\epsilon }\right)=0,\hfill \end{array}$ (4.10)

$\text{div}\left(\overline{\rho }\nabla {\psi }_{\epsilon }\otimes \nabla {\psi }_{\epsilon }\right)+\frac{\gamma }{2}\nabla \left({\overline{\rho }}^{\gamma -2}{\varphi }_{\epsilon }^{2}\right)\to \overline{\rho }\nabla P$ (4.11)

$\begin{array}{l}\text{div}\left(\overline{\rho }\nabla {\psi }_{\epsilon }\otimes \nabla {\psi }_{\epsilon }\right)=\overline{\rho }\nabla \frac{{|\nabla {\psi }_{\epsilon }|}^{2}}{2}+\text{div}\left(\overline{\rho }\nabla {\psi }_{\epsilon }\right)\nabla {\psi }_{\epsilon }\\ =\overline{\rho }\nabla \frac{{|\nabla {\psi }_{\epsilon }|}^{2}}{2}-\epsilon {\partial }_{t}{\varphi }_{\epsilon }\nabla {\psi }_{\epsilon }\\ =\overline{\rho }\nabla \frac{{|\nabla {\psi }_{\epsilon }|}^{2}}{2}-\epsilon {\partial }_{t}\left({\varphi }_{\epsilon }\nabla {\psi }_{\epsilon }\right)+\epsilon {\varphi }_{\epsilon }{\partial }_{t}\left(\nabla {\psi }_{\epsilon }\right)\\ =\overline{\rho }\nabla \frac{{|\nabla {\psi }_{\epsilon }|}^{2}}{2}-\epsilon {\partial }_{t}\left({\varphi }_{\epsilon }\nabla {\psi }_{\epsilon }\right)-\gamma \overline{\rho }\nabla \left({\overline{\rho }}^{\gamma -2}{\varphi }_{\epsilon }\right)+\epsilon {\varphi }_{\epsilon }{F}_{\epsilon }.\end{array}$ (4.12)

$\overline{\rho }\nabla \left({\overline{\rho }}^{\gamma -2}{\varphi }_{\epsilon }\right)=\left(2-\gamma \right)\overline{\rho }\nabla \left({\overline{\rho }}^{\gamma -3}\frac{{\varphi }_{\epsilon }^{2}}{2}\right)+\left(\left(\gamma -1\right){\overline{\rho }}^{\gamma -2}\frac{{\varphi }_{\epsilon }^{2}}{2}\right).$ (4.13)

$\begin{array}{l}\text{div}\left(\overline{\rho }\nabla {\psi }_{\epsilon }\otimes {\psi }_{\epsilon }\right)+\frac{\gamma \left(\gamma -1\right)}{2}\nabla \left({\overline{\rho }}^{\gamma -2}{\varphi }_{\epsilon }^{2}\right)\\ =\overline{\rho }\nabla \left(\frac{{|\nabla {\psi }_{\epsilon }|}^{2}}{2}+\frac{\gamma \left(\gamma -2\right)}{2}{\overline{\rho }}^{\gamma -3}\frac{{\varphi }_{\epsilon }^{2}}{2}\right)-\epsilon {\partial }_{t}\left({\varphi }_{\epsilon }\nabla {\psi }_{\epsilon }\right)+\epsilon {\varphi }_{\epsilon }{F}_{\epsilon }.\end{array}$ (4.14)

${‖{\varphi }_{\epsilon }\left(t,\cdot \right)\ast {\chi }_{\delta }-{\varphi }_{\epsilon }\left(t,\cdot \right)‖}_{{L}^{P}\left({L}^{2}\right)}\to 0,$ (4.15)

$\epsilon$ 是一致的。

$\left\{\begin{array}{l}-\Delta u+\overline{\rho }\nabla p=g,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{div}\left(\overline{\rho }u\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in B,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\int }_{B}p\text{d}x=0,\hfill \\ u=0,\text{on}x\in \partial B.\hfill \end{array}$ (4.16)

${\stackrel{˜}{R}}_{\delta }=I-{R}_{\delta }$ 。用 $S$ 作用(4.4)后得到的式子再用 ${\stackrel{˜}{R}}_{\delta }$ 作用得

${\partial }_{t}{\stackrel{˜}{R}}_{\delta }S\left({\rho }_{\epsilon }{u}_{\epsilon }\right)+{\stackrel{˜}{R}}_{\delta }S\text{div}\left({\rho }_{\epsilon }{u}_{\epsilon }\otimes {u}_{\epsilon }\right)+{\stackrel{˜}{R}}_{\delta }h+\frac{\gamma }{\epsilon }{\stackrel{˜}{R}}_{\delta }{f}_{\epsilon }+{\stackrel{˜}{R}}_{\delta }S\left(\nabla {\pi }_{\epsilon }\right)=0.$ (4.17)

${f}_{\epsilon }={\overline{\rho }}^{\gamma -2}{\varphi }_{\epsilon }$ ，可得

${\partial }_{t}{\stackrel{˜}{R}}_{\delta }{f}_{\epsilon }+\frac{1}{\epsilon }{\overline{\rho }}^{\gamma -2}\text{div}\left({\stackrel{˜}{R}}_{\delta }{\rho }_{\epsilon }{u}_{\epsilon }\right)+\frac{1}{\epsilon }\left[{\stackrel{˜}{R}}_{\delta },{\overline{\rho }}^{\gamma -2}\text{div}\right]\left({\rho }_{\epsilon }{u}_{\epsilon }\right)=0.$ (4.18)

$\begin{array}{l}{\partial }_{t}\left({\stackrel{˜}{R}}_{\delta }{f}_{\epsilon }{\stackrel{˜}{R}}_{\delta }S\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\right)+\frac{1}{\epsilon }{\overline{\rho }}^{\gamma -2}\text{div}\left({\stackrel{˜}{R}}_{\delta }\left({\rho }_{\epsilon }{u}_{\epsilon }\right){\stackrel{˜}{R}}_{\delta }S\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\right)\\ -\frac{1}{\epsilon }{\overline{\rho }}^{\gamma -2}{\stackrel{˜}{R}}_{\delta }\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\cdot \nabla {\stackrel{˜}{R}}_{\delta }S\left({\rho }_{\epsilon }{u}_{\epsilon }\right)+\frac{1}{\epsilon }\left[{\stackrel{˜}{R}}_{\delta },{\overline{\rho }}^{\gamma -2}\text{div}\right]\left({\rho }_{\epsilon }{u}_{\epsilon }\right){\stackrel{˜}{R}}_{\delta }S\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\\ +{\stackrel{˜}{R}}_{\delta }{f}_{\epsilon }\left({\stackrel{˜}{R}}_{\delta }S\text{div}\left({\rho }_{\epsilon }{u}_{\epsilon }\otimes {u}_{\epsilon }\right)+{\stackrel{˜}{R}}_{\delta }h\right)+\frac{\gamma }{\epsilon }{|{\stackrel{˜}{R}}_{\delta }{f}_{\epsilon }|}^{2}+{\stackrel{˜}{R}}_{\delta }{f}_{\epsilon }{\stackrel{˜}{R}}_{\delta }S\left(\nabla {\pi }_{\epsilon }\right)=0.\end{array}$ (4.19)

${\int }_{\text{0}}^{T}{\int }_{{B}_{\frac{r}{2}}}\gamma {|{\stackrel{˜}{R}}_{\delta }{f}_{\epsilon }|}^{2}\text{d}x\text{d}s\to 0,$ (4.20)

$\epsilon$ 是一致的。

$\begin{array}{l}{{\int }_{\text{0}}^{T}{\int }_{{B}_{\frac{r}{2}}}\gamma \Phi |{\stackrel{˜}{R}}_{\delta }{f}_{\epsilon }|}^{2}\text{d}x\text{d}s\\ =\epsilon {\int }_{{B}_{\frac{r}{2}}}\left({\stackrel{˜}{R}}_{\delta }{f}_{0\epsilon }{\stackrel{˜}{R}}_{\delta }S\left({m}_{0\epsilon }\right)\right)\text{d}x+{\int }_{\text{0}}^{T}{\int }_{{B}_{\frac{r}{2}}}\left({\stackrel{˜}{R}}_{\delta }\left({\rho }_{\epsilon }{u}_{\epsilon }\right){\stackrel{˜}{R}}_{\delta }S\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\right):\nabla \left({\overline{\rho }}^{\gamma -2}\Phi \right)\text{d}x\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{0}^{T}{\int }_{{B}_{\frac{r}{2}}}\left(\Phi {\overline{\rho }}^{\gamma -2}{\stackrel{˜}{R}}_{\delta }\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\cdot \nabla {\stackrel{˜}{R}}_{\delta }S\left({\rho }_{\epsilon }{u}_{\epsilon }\right)-\left[{\stackrel{˜}{R}}_{\delta },{\overline{\rho }}^{\gamma -2}\text{div}\right]\left({\rho }_{\epsilon }{u}_{\epsilon }\right){\stackrel{˜}{R}}_{\delta }S\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\right)\text{d}x\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\epsilon {\int }_{0}^{T}{\int }_{{B}_{\frac{r}{2}}}\Phi \left({\stackrel{˜}{R}}_{\delta }{f}_{\epsilon }\left({\stackrel{˜}{R}}_{\delta }S\text{div}\left({\rho }_{\epsilon }{u}_{\epsilon }\otimes {u}_{\epsilon }\right)+{\stackrel{˜}{R}}_{\delta }h\right)+{\stackrel{˜}{R}}_{\delta }{f}_{\epsilon }{\stackrel{˜}{R}}_{\delta }S\left(\nabla {\pi }_{\epsilon }\right)\right)\text{d}x\text{d}s.\end{array}$ (4.21)

${‖{\varphi }_{\epsilon }\left(t,\cdot \right)\ast {\chi }_{\delta }-{\varphi }_{\epsilon }\left(t,\cdot \right)‖}_{{L}^{2}\left({L}^{2}\right)}\to 0,$ (4.22)

${‖{u}_{\epsilon }\left(t,\cdot \right)\ast {\chi }_{\delta }-{u}_{\epsilon }\left(t,\cdot \right)‖}_{{L}^{2}\left({L}^{2}\right)}\to 0.$ (4.23)

${‖{\rho }_{\epsilon }{u}_{\epsilon }\left(t,\cdot \right)\ast {\chi }_{\delta }-{\rho }_{\epsilon }{u}_{\epsilon }\left(t,\cdot \right)‖}_{{L}^{2}\left({L}^{2}\right)}\to 0.$ (4.24)

$f$${C}_{\text{0}}^{\infty }\left(\Omega \right)$ 中的试验函数，并且满足 $\text{div}\left(f\right)=0$ ，i.e. ${P}_{\overline{\rho }}f=f$ 。由动量方程(4.4)可知：对 $0 ，有

$\begin{array}{l}{\int }_{\Omega }\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\left(\frac{f}{\overline{\rho }}\right)\left(t\right)\text{d}x-{\int }_{\Omega }\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\left(\frac{f}{\overline{\rho }}\right)\left(s\right)\text{d}x\\ ={\int }_{s}^{t}{\int }_{\Omega }\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\frac{{\partial }_{t}f}{\overline{\rho }}\text{d}x\text{d}s+{\int }_{s}^{t}{\int }_{\Omega }\left({\rho }_{\epsilon }{u}_{\epsilon }\otimes {u}_{\epsilon }\right):\nabla \left(\frac{f}{\overline{\rho }}\right)\text{d}x\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }-{\int }_{s}^{t}{\int }_{\Omega }\mu \nabla {u}_{\epsilon }\nabla \left(\frac{f}{\overline{\rho }}\right)+\xi \text{div}\left({u}_{\epsilon }\right)\text{div}\left(\frac{f}{\overline{\rho }}\right)-{\pi }_{\epsilon }\text{div}\left(\frac{f}{\overline{\rho }}\right)\text{d}x\text{d}s\\ ={\int }_{s}^{t}{\int }_{\Omega }\left({\rho }_{\epsilon }{u}_{\epsilon }\otimes {u}_{\epsilon }\right):\nabla \left(\frac{f}{\overline{\rho }}\right)\text{d}x\text{d}s\\ -{\int }_{s}^{t}{\int }_{\Omega }\mu \nabla {u}_{\epsilon }\nabla \left(\frac{f}{\overline{\rho }}\right)+\xi \text{div}\left({u}_{\epsilon }\right)\text{div}\left(\frac{f}{\overline{\rho }}\right)-{\pi }_{\epsilon }\text{div}\left(\frac{f}{\overline{\rho }}\right)\text{d}x\text{d}s.\end{array}$ (4.25)

${Q}_{\overline{\rho }}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\otimes {Q}_{\overline{\rho }}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)-{Q}_{\overline{\rho }}{R}_{\delta }\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\otimes {Q}_{\overline{\rho }}{R}_{\delta }\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\to 0,$ (4.26)

$\epsilon$ 是一致的。因此，我们要证明 ${Q}_{\overline{\rho }}{R}_{\delta }\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\otimes {Q}_{\overline{\rho }}{R}_{\delta }\left({\rho }_{\epsilon }{u}_{\epsilon }\right)$ 的极限，并刻画

$\text{div}\left(\frac{{Q}_{\overline{\rho }}{R}_{\delta }\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\otimes {Q}_{\overline{\rho }}{R}_{\delta }\left({\rho }_{\epsilon }{u}_{\epsilon }\right)}{\overline{\rho }}\right)$

$\frac{\gamma }{\text{2}}\nabla \left({\overline{\rho }}^{\gamma -\text{2}}{\left({\varphi }_{\epsilon }\right)}^{\text{2}}\right)-\frac{\gamma }{\text{2}}\nabla \left({\overline{\rho }}^{\gamma -\text{2}}{\left({R}_{\delta }{\varphi }_{\epsilon }\right)}^{\text{2}}\right)\to {r}_{\delta }\to 0.$ (4.27)

$\text{div}\left(\frac{{Q}_{\overline{\rho }}{R}_{\delta }\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\otimes {Q}_{\overline{\rho }}{R}_{\delta }\left({\rho }_{\epsilon }{u}_{\epsilon }\right)}{\overline{\rho }}\right)+\frac{\gamma }{2}\nabla \left({\overline{\rho }}^{\gamma -2}{\left({R}_{\delta }{\varphi }_{\epsilon }\right)}^{2}\right)\to \overline{\rho }\nabla P+{r}_{\delta },$ (4.28)

${R}_{\delta }$ 作用(4.10)得

$\left\{\begin{array}{l}{\partial }_{t}{R}_{\delta }{Q}_{\overline{\rho }}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)+\frac{\gamma }{\epsilon }\overline{\rho }\nabla \left({\overline{\rho }}^{\gamma -2}{R}_{\delta }{\varphi }_{\epsilon }\right)=\overline{\rho }{F}_{\epsilon ,\delta }-\frac{\gamma }{\epsilon }\left[{R}_{\delta },\overline{\rho }\nabla {\overline{\rho }}^{\gamma -2}\right]{\varphi }_{\epsilon },\hfill \\ \epsilon {\partial }_{t}{R}_{\delta }{\varphi }_{\epsilon }+\text{div}\left({R}_{\delta }{Q}_{\overline{\rho }}\left({\rho }_{\epsilon }{u}_{\epsilon }\right)\right)=0,\hfill \end{array}$ (4.29)

$\left[{R}_{\delta },\overline{\rho }\nabla {\overline{\rho }}^{\gamma -2}\right]{\varphi }_{\epsilon }=\left[{R}_{\delta },\overline{\rho }\right]\nabla \left({\overline{\rho }}^{\gamma -2}{\varphi }_{\epsilon }\right)+\overline{\rho }\nabla \left[{R}_{\delta },{\overline{\rho }}^{\gamma -2}\right]{\varphi }_{\epsilon },$ (4.30)

${‖{R}_{\delta }\left(fg\right)-g{R}_{\delta }\left(f\right)‖}_{{L}^{2}}\le C{‖f‖}_{{L}^{2\delta }},{‖{R}_{\delta }\left(fg\right)-g{R}_{\delta }\left(f\right)‖}_{{H}^{1}}\le C{‖f‖}_{{L}^{2}},$ (4.31)

$\frac{{R}_{\delta }\left(fg\right)-g{R}_{\delta }\left(f\right)}{\delta }\to 0,$ (4.32)

${R}_{\delta }\left(fg\right)-g{R}_{\delta }\left(f\right)\to 0.$ (4.33)

(2) 对 $f\in {H}^{-1}\left({R}^{N}\right)$$g\in {L}^{2}\left({R}^{N}\right)$ ，有

${‖{R}_{\delta }\left(fg\right)-g{R}_{\delta }\left(f\right)‖}_{{L}^{2}}\le C{‖f‖}_{{H}^{-1}},$ (4.34)

$\frac{{R}_{\delta }\left(fg\right)-g{R}_{\delta }\left(f\right)}{\delta }\to 0.$ (4.35)

$\begin{array}{l}\text{div}\left(\overline{\rho }\nabla {\psi }_{\epsilon ,\delta }\otimes \nabla {\psi }_{\epsilon ,\delta }\right)+\frac{\gamma \left(\gamma -1\right)}{2}\nabla \left({\overline{\rho }}^{\gamma -2}{\varphi }_{\epsilon ,\delta }^{2}\right)\\ =\overline{\rho }\nabla \left(\frac{{|\nabla {\psi }_{\epsilon ,\delta }|}^{2}}{2}+\frac{\gamma \left(\gamma -2\right)}{2}{\overline{\rho }}^{\gamma -3}{\varphi }_{\epsilon ,\delta }^{2}\right)-\epsilon {\partial }_{t}\left({\varphi }_{\epsilon ,\delta }\nabla {\psi }_{\epsilon ,\delta }\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+\epsilon {\varphi }_{\epsilon }{F}_{\epsilon ,\delta }+\frac{\gamma }{\overline{\rho }}\left[{R}_{\delta },\overline{\rho }\nabla {\overline{\rho }}^{\gamma -2}\right]{\varphi }_{\epsilon }{\varphi }_{\epsilon ,\delta }.\end{array}$ (4.36)

Anelastic Approximation of Compressible Isentropic Navier-Stokes Equations with Exterior Force[J]. 应用数学进展, 2017, 06(09): 1207-1219. http://dx.doi.org/10.12677/AAM.2017.69146

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