非等熵可压缩Navier-Stokes-Maxwell方程组Cauchy问题解的整体存在性

李新; 冯跃红; 王术

doi:10.11936/bjutxb2017110006

非等熵可压缩Navier-Stokes-Maxwell方程组Cauchy问题解的整体存在性

北京工业大学应用数理学院, 北京 100124

基金项目:

北京市自然科学基金资助项目 1164010

北京市自然科学基金资助项目 1132006

国家自然科学基金资助项目 11771031

国家自然科学基金资助项目 11371042

详细信息

作者简介:
李新(1981-), 女, 博士研究生, 主要从事应用偏微分方程方面的研究, E-mail:lixin91600@emails.bjut.edu.cn

中图分类号: O175.29
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出版历程
- 收稿日期: 2017-10-31
- 网络出版日期: 2022-08-03
- 发布日期: 2018-12-09
- 刊出日期: 2018-12-14

Global Existence of Solutions to Cauchy Problem for Non-isentropic Compressible Navier-Stokes-Maxwell Systems

College of Applied Sciences, Beijing University of Technology, Beijing 100124, China

摘要

摘要:
考察粘性等离子体物理中的非等熵可压缩Navier-Stokes-Maxwell方程组.借助非常数平衡解的小性以及对称子技巧，研究了三维全空间上的Cauchy问题.在初值为该平衡解的一个小摄动前提下，证明了该问题存在整体唯一光滑解.
- 非等熵可压缩Navier-Stokes-Maxwell方程组 /
- 粘性等离子体 /
- 整体光滑解 /
- 非常数平衡解
Abstract:
This paper is concerned with non-isentropic compressible Navier-Stokes-Maxwell systems arising from viscosity plasmas. By using techniques of symmetrizer and the smallness of non-constant steady-state solutions, the global existence of solutions to Cauchy problems with prepared initial data was investigated. It is shown that this problem admits globally smooth solutions near a non-constant steady state.
- non-isentropic compressible Navier-Stokes-Maxwell systems /
- viscosity plasmas /
- global smooth solution /
- non-constant steady-state solution

HTML全文

粘性等离子体可压缩Navier-Stokes-Maxwell方程组用来描述带电粒子流在电磁场中的输运现象^[1-2].它由描述粘性流体运动的非等熵可压缩Navier-Stokes(NS)方程组和描述自洽电磁场的Maxwell方程组通过电磁场中的Lorentz力耦合而成，既具有NS方程的难点，也包含电磁场方程产生的困难，研究起来极具挑战性.目前有许多关于理想(无粘)等离子体物理模型Euler-Maxwell方程组的相关研究，如数值模拟计算、模型间的小参数渐近机制、解的整体存在性以及渐近性态等方面.对于简化的一维Euler-Maxwell系统，文献[3]运用补偿列紧性的方法获得了弱解的整体存在性；关于小参数渐近机制的研究，参见文献[4]及其参考文献；对于光滑解的整体存在性及其渐近性态问题的研究，参见文献[5-9]；关于数值模拟计算等方面的研究，参见文献[10]及其参考文献；对于在非常数平衡解附近光滑解的存在性研究，参见文献[11].然而目前尚无关于粘性等离子体物理模型可压缩Navier-Stokes-Maxwell方程组在非常数平衡解附近光滑解的整体存在性的研究，而这是非常重要的，因为任何流体都是有粘性的.本文旨在回答上述问题.本文的结果对推动等离子体物理的发展提供了重要的理论依据.

1. 模型与定理

本文研究形式如下的粘性等离子体非等熵可压缩Navier-Stokes-Maxwell方程组：

$$ {\partial _t}n + \nabla \left( {nu} \right) = 0 $$

(1)

$$ {\partial _t}u + u\nabla u + \frac{{\nabla \left( {n\theta } \right)}}{n} = - \left( {E + u \times B} \right) + \frac{{\Delta u}}{n} $$

(2)

$$ {\partial _t}\theta + \frac{2}{3}\theta \nabla u + u\nabla \theta = {\theta _ * } - \theta - \frac{1}{3}{\left| u \right|^2} $$

(3)

$$ {\partial _t}E - \nabla B = nu,\nabla E = b - n $$

(4)

$$ {\partial _t}B + \nabla E = 0,\nabla B = 0 $$

(5)

式中$\left({t, x} \right) \in {\mathbb{R}^ + }{\mathbb{R}^3}$.未知变量分别为：电子流密度n>0，电子流速度$u \in {\mathbb{R}^3}$，绝对温度θ>0，电场强度$E \in {\mathbb{R}^3}$，磁场强度$B \in {\mathbb{R}^3}$常数θ_*>0表示背景温度，光滑函数b=b(x)≥const.>0表示静止的带正电离子的密度(背景密度).

本文研究式(1)~(5)的Cauchy问题，初值为

$$ \left( {n,u,\theta ,E,B} \right)\left| {_{t = 0}} \right. = \left( {{n_0},{u_0},{\theta _0},{E_0},{B_0}} \right) $$

(6)

其满足相容性条件

$$ \nabla {E_0} = b - {n_0},\nabla \cdot {B_0} = 0 $$

(7)

记$\left({\tilde n\left(x \right), 0, {\theta _*}, \tilde E\left(x \right), 0} \right)$为式(1)~(5)的非常数平衡解，其中$\tilde n\left(x \right) > 0$，$\tilde E\left(x \right) \in {\mathbb{R}^3}$，其满足

$$ \frac{1}{{\tilde n}}\nabla \left( {\tilde n{\theta _ * }} \right) = - \tilde E $$

(8)

$$ \nabla \tilde E = b\left( x \right) - \tilde n $$

(9)

$$ \nabla \tilde E = 0 $$

(10)

由式(10)可知存在势函数$\tilde \phi $满足：$ - \nabla \tilde \phi = \tilde E$，于是式(8)~(10)可转化为

$$ \nabla \left( {{\theta _ * }\ln \tilde n - \tilde \phi } \right) = 0 $$

(11)

$$ - \Delta \tilde \phi = b\left( x \right) - \tilde n $$

(12)

下面考察b(x)=n_*+b(x)的情形.此处n_*为一正常数，b满足：当|x|→∞时，b(x)→0.寻找式(11)~(12)形如$\tilde n\left(x \right)$=n_*+n(x)的解.于是由式(11)可知

$$ \bar n\left( x \right) = {n_ * }\left( {\exp \left( {\frac{{\tilde \phi \left( x \right)}}{{{\theta _ * }}}} \right) - 1} \right) $$

(13)

这里假设当|x|→∞时，(n, ${\tilde \phi }$)(x)→0.进而将式(13)代入式(12)可知

$$ - \Delta \tilde \phi + \frac{{{n_ * }}}{{{\theta _ * }}}\tilde \phi = \bar b\left( x \right) - {n_ * }\left( {\exp \left( {\frac{{\tilde \phi }}{{{\theta _ * }}}} \right) - 1 - \frac{{\tilde \phi }}{{{\theta _ * }}}} \right) $$

(14)

众所周知，采用经典的Schauder不动点定理或极小化方法容易得到上述椭圆方程光滑解的存在唯一性，进而可得式(1)~(5)的非常数平衡解存在唯一性定理.

命题1 令s≥0，常数n_*, θ_*>0，b－n_*∈H^s(${\mathbb{R}^3}$).那么存在正常数δ₀使得：只要‖b－n_*‖_{H^s(${\mathbb{R}^3}$)}≤δ₀，则问题(11)~(12)存在唯一解(${\tilde n}$, ${\tilde E}$)满足

$$ \tilde n - {n_ * } \in {H^{s + 2}}\left( {{\mathbb{R}^3}} \right),\tilde E \in {H^{s + 1}}\left( {{\mathbb{R}^3}} \right) $$

及

$$ {\left\| {\tilde n - {n_ * }} \right\|_{s + 2}} + {\left\| {\tilde E} \right\|_{s + 1}} \le C{\left\| {b - {n_ * }} \right\|_s} $$

(15)

式中C为任意正常数.进而有$\left({\tilde n\left(x \right), 0, {\theta _*}, \tilde E\left(x \right), 0} \right)$为式(1)~(5)的一个非常数平衡解.

这里H^s(${\mathbb{R}^3}$)表示${\mathbb{R}^3}$上s阶常用Sobolev空间，H^s(${\mathbb{R}^3}$)=$\left\{ {f|\partial _x^\alpha f \in {L^2}\left({{\mathbb{R}^3}} \right), \left| \alpha \right| \le s} \right\}$，其范数${\left\| f \right\|_{{H^s}\left({{\mathbb{R}^3}} \right)}} = \sqrt {\sum\limits_{\left| \alpha \right| \le s} {{{\left| {{\partial ^\alpha }f} \right|}^2}} } $简记为：‖f‖_s.

接下来，期望在命题1给出的非常数平衡解$\left({\tilde n\left(x \right), 0, {\theta _*}, \tilde E\left(x \right), 0} \right)$附近建立Cauchy问题式(1)~(6)的光滑解的整体存在性.

众所周知，当n, θ>0时，方程组(1)~(5)为可对称化的双曲-抛物组.于是借助Kato^[12]以及Matsumura等^[13-14]的结论可知，只要初值光滑，Cauchy问题(1)~(6)就一定存在局部唯一光滑解.

命题2(光滑解的局部存在唯一性，参见文献[12, 15-16]) 令式(7)成立，整数s≥4. n_*, θ_*>0为任意给定常数.记$\left({\tilde n, \tilde E} \right)$为命题1给出的问题(11)~(12)的光滑解.对给定常数κ>0，初值n₀, θ₀≥2κ.则如果

$$ \left( {{n_0} - \tilde n,{u_0},{\theta _0} - {\theta _ * },{E_0} - \tilde E,{B_0}} \right) \in {H^s}\left( {{\mathbb{R}^3}} \right) $$

那么存在T>0使得问题(1)~(6)存在局部唯一光滑解，满足：(t, x)∈[0, T]×${\mathbb{R}^3}$，及

$$ \begin{array}{*{20}{c}} {u \in {C^1}\left( {\left[ {0,T} \right];{H^{s - 2}}\left( {{\mathbb{R}^3}} \right)} \right) \cap C\left( {\left[ {0,T} \right];{H^s}\left( {{\mathbb{R}^3}} \right)} \right)} \\ {\left( {n - \tilde n,\theta - {\theta _ * },E - \tilde E,B} \right) \in } \\ {{C^1}\left( {\left[ {0,T} \right];{H^{s - 1}}\left( {{\mathbb{R}^3}} \right)} \right) \cap C\left( {\left[ {0,T} \right];{H^s}\left( {{\mathbb{R}^3}} \right)} \right)} \end{array} $$

本文的主要结果如下：

定理1 在命题2的假设条件下，存在常数δ₀>0足够小，不依赖于任何时间t>0，使得若

$$ {\left\| {\left( {\nabla \bar n,{n_0} - \tilde n,{u_0},{\theta _0} - {\theta _ * },{E_0} - \tilde E,{B_0}} \right)} \right\|_s} \le {\delta _0} $$

则Cauchy问题(1)~(6)存在唯一整体光滑解

$$ u \in {C^1}\left( {{\mathbb{R}^ + };{H^{s - 2}}\left( {{\mathbb{R}^3}} \right)} \right) \cap C\left( {{\mathbb{R}^ + };{H^s}\left( {{\mathbb{R}^3}} \right)} \right) $$

(16)

$$ \begin{array}{*{20}{c}} {\left( {n - \tilde n,\theta - {\theta _ * },E - \tilde E,B} \right) \in } \\ {{C^1}\left( {{\mathbb{R}^ + };{H^{s - 1}}\left( {{\mathbb{R}^3}} \right)} \right) \cap C\left( {{\mathbb{R}^ + };{H^s}\left( {{\mathbb{R}^3}} \right)} \right)} \end{array} $$

(17)

并且对于任意t>0，满足

$$ \begin{array}{*{20}{c}} {\left\| {\left( {n - \tilde n,u,\theta - {\theta _ * },E - \tilde E,B} \right)\left( t \right)} \right\|_s^2 + }\\ {\int_0^t {\left\| {\left( {n - \tilde n,\nabla u,\theta - {\theta _ * }} \right)\left( \tau \right)} \right\|_s^2{\rm{d}}\tau } \le }\\ {C\left\| {\left( {{n_0} - \tilde n,{u_0},{\theta _0} - {\theta _ * },{E_0} - \tilde E,{B_0}} \right)} \right\|_s^2} \end{array} $$

(18)

注1. 方程组(1)~(5)中速度粘性项Δu/n与温度耗散项θ－θ_*在证明定理1的过程中起关键作用.

本文其余部分结构如下：第2节给出准备工作；第3节建立了粘性流体方程的耗散估计，进而获得光滑解的整体存在性；第4节陈述本文结论.

2. 准备工作

首先引入一些记号.用‖·‖和‖·‖_L^∞分别表示空间L²(${\mathbb{R}^3}$)和L^∞(${\mathbb{R}^3}$)的范数.用〈·, ·〉表示空间L²(${\mathbb{R}^3}$)上的内积.指标α=(α₁, α₂, α₃)∈${\mathbb{N}^3}$，记：${\partial ^\alpha } = \partial _{x1}^{\alpha 1}\partial _{x2}^{\alpha 2}\partial _{x3}^{\alpha 3}$, |α|=α₁+α₂+α₃.

现在设(n, u, θ, E, B)为Cauchy问题(1)~(6)的唯一局部光滑解.令

$$ n = \tilde n + {n_ * } + \rho ,\theta = {\theta _ * } + \mathit{\Theta },E = \tilde E + F $$

(19)

$$ \mathit{\boldsymbol{U}} = {\left( {\rho ,u,\mathit{\Theta }} \right)^{\rm{T}}},\mathit{\boldsymbol{W}} = {\left( {{\mathit{\boldsymbol{U}}^{\rm{T}}},F,B} \right)^{\rm{T}}} $$

(20)

这里(·)^T表示向量(·)转置.于是式(1)~(7)可改写为

$$ {\partial _t}\rho + u\nabla \rho + n\nabla u + u\nabla \bar n\left( x \right) = 0 $$

(21)

$$ \begin{array}{*{20}{c}} {{\partial _t}u + \left( {u\nabla } \right)u + \frac{\theta }{n}\nabla \rho + \nabla \mathit{\Theta } + }\\ {\frac{\mathit{\Theta }}{n}\nabla \bar n - \frac{{{\theta _ * }\rho }}{{n\tilde n}}\nabla \bar n = - \left( {F + u \times B} \right) - u} \end{array} $$

(22)

$$ {\partial _t}\mathit{\Theta } + \frac{2}{3}\theta \nabla u + u\nabla \mathit{\Theta } = - \frac{1}{3}{\left| u \right|^2} - \mathit{\Theta } $$

(23)

$$ {\partial _t}\mathit{F} - \nabla \times B = nu,\nabla F = - \rho $$

(24)

$$ {\partial _t}\mathit{B} + \nabla F = 0,\nabla B = 0 $$

(25)

其初始条件为

$$ \mathit{\boldsymbol{W}}\left| {_{t = 0}} \right. = {\mathit{\boldsymbol{W}}_0} = {\left( {{\rho _0},{u_0},{\mathit{\Theta }_0},{F_0},{B_0}} \right)^{\rm{T}}} $$

(26)

满足相容性条件

$$ \nabla {F^0} = - {\rho ^0},\nabla {B^0} = 0 $$

(27)

式中：${\rho _0} = {n_0} - \tilde n, {\mathit{\Theta }_0} = {\theta _0} - {\theta _*}, {F^0} = {E^0} - \tilde E$.

此外，NS方程(21)~(23)可写为矩阵形式

$$ {\partial _t}\mathit{\boldsymbol{U}} + \sum\limits_{j = 1}^3 {{\mathit{\boldsymbol{A}}_j}{\partial _j}\mathit{\boldsymbol{U}} + \mathit{\boldsymbol{MU}}} = {\mathit{\boldsymbol{K}}_1} + {\mathit{\boldsymbol{K}}_2} $$

(28)

式中

$$ {\mathit{\boldsymbol{A}}_j} = \left( {\begin{array}{*{20}{c}} {{u_j}}&{n\mathit{\boldsymbol{e}}_j^{\rm{T}}}&0\\ {\frac{\theta }{n}{\mathit{\boldsymbol{e}}_j}}&{{u_j}{\mathit{\boldsymbol{I}}_3}}&{{\mathit{\boldsymbol{e}}_j}}\\ 0&{\frac{2}{3}\theta \mathit{\boldsymbol{e}}_j^{\rm{T}}}&{{u_j}} \end{array}} \right),j = 1,2,3 $$

(29)

$$ \mathit{\boldsymbol{M}} = \left( {\begin{array}{*{20}{c}} 0&{{{\left( {\nabla \bar n} \right)}^{\rm{T}}}}&0\\ { - \frac{{{\theta _ * }}}{{n\tilde n}}}&0&{\frac{{\nabla \bar n}}{n}}\\ 0&0&0 \end{array}} \right) $$

$$ {\mathit{\boldsymbol{K}}_1} = - {\left( {0,F + u \times B,\frac{1}{3}{{\left| u \right|}^2}} \right)^{\rm{T}}} $$

$$ {\mathit{\boldsymbol{K}}_2} = {\left( {0,\frac{{\Delta u}}{n}, - \mathit{\Theta }} \right)^{\rm{T}}} $$

式中：(e₁, e₂, e₃)为${\mathbb{R}^3}$的标准正交基，I₃为3×3单位矩阵.

易知，当n_*+n+ρ, θ_*+Θ>0时，式(28)关于U是可对称化双曲-抛物组.事实上，由常数n_*, θ_*≥const.>0及n, ρ, Θ非常接近零，可得n_*+n+ρ, θ_*+Θ≥const.>0.

令T>0，W是式(21)~(25)定义在[0, T]上的光滑解，初值为W₀，解的存在性由命题2给出.定义

$$ {\omega _T} = \mathop {\sup }\limits_{0 \le t \le T} {\left\| \mathit{\boldsymbol{W}} \right\|_s} $$

(30)

于是，由连续嵌入H^s(${\mathbb{R}^3}$)$ \subset $L^∞(${\mathbb{R}^3}$), s≥2，可得：存在常数C_*>0使得

$$ {\left\| f \right\|_{{L^\infty }}} \leqslant {C_ * }{\left\| f \right\|_s},\forall f \in {H^s}\left( {{\mathbb{R}^3}} \right) $$

3. 主要结果的证明

本节将在非常数平衡解$\left({\tilde n\left(x \right), 0, {\theta _*}, \tilde E\left(x \right), 0} \right)$附近建立Cauchy问题(1)~(7)的小摄动光滑解的整体存在唯一性.

3.1 一致先验估计

注意到，在光滑解的意义下，问题(1)~(7)等价于问题(21)~(27).于是，基于命题2已经给出了光滑解的局部存在性这一事实，证明定理1的关键在于建立一致先验估计^[17-22].为此，需要建立以下2个引理.首先，引入能量泛函E_s(t)与耗散泛函H_s(t)如：

$$ {E_s}\left( t \right) = \left\| {\mathit{\boldsymbol{W}}\left( t \right)} \right\|_s^2,{H_s}\left( t \right) = \left\| {\left( {\rho ,\nabla u,\mathit{\Theta }} \right)\left( t \right)} \right\|_s^2 $$

(31)

记常数C>0不依赖任意时间t>0和T，在不同的地方值可以不同.

引理1 在定理1的条件下，存在δ₀>0充分小，若

$$ {\omega _T} + {\left\| {\nabla \bar n} \right\|_s} \le \delta \le {\delta _0} $$

(32)

则对任意t∈[0, T]成立

$$ \begin{array}{*{20}{c}} {\frac{{\rm{d}}}{{{\rm{d}}t}}{E_s}\left( t \right) + \left\| {\left( {\nabla u,\mathit{\Theta }} \right)} \right\|_s^2 \le }\\ {C{{\left\| {\nabla \bar n} \right\|}_s}\left\| \rho \right\|_s^2 + CE_s^{\frac{1}{2}}\left( t \right){H_s}\left( t \right)} \end{array} $$

(33)

证明：对于α∈${\mathbb{N}^3}$，|α|≤s，对式(28)求${\partial ^\alpha }$，然后左乘上对称子A₀后有

$$ \begin{array}{*{20}{c}} {{\mathit{\boldsymbol{A}}_0}{\partial _t}{\partial ^\alpha }\mathit{\boldsymbol{U}} + \sum\limits_{j = 1}^3 {{{\mathit{\boldsymbol{\tilde A}}}_j}{\partial _j}{\partial ^\alpha }\mathit{\boldsymbol{U}}} + {\mathit{\boldsymbol{A}}_0}{\partial ^\alpha }\left( {\mathit{\boldsymbol{MU}}} \right) = }\\ {{\mathit{\boldsymbol{A}}_0}{\partial ^\alpha }\left( {{\mathit{\boldsymbol{K}}_1} + {\mathit{\boldsymbol{K}}_2}} \right) + {h_\alpha }} \end{array} $$

(34)

其中

$$ {\mathit{\boldsymbol{A}}_0} = \left( {\begin{array}{*{20}{c}} {\frac{\theta }{n}{u_j}}&0&0\\ 0&{n{\mathit{\boldsymbol{I}}_3}}&0\\ 0&0&{\frac{3}{2}\frac{n}{\theta }} \end{array}} \right) $$

$$ {{\mathit{\boldsymbol{\tilde A}}}_j} = {\mathit{\boldsymbol{A}}_0}{\mathit{\boldsymbol{A}}_j} = \left( {\begin{array}{*{20}{c}} {\frac{\theta }{n}{u_j}}&{\theta e_j^{\rm{T}}}&0\\ {\theta {e_j}}&{n{u_j}{\mathit{\boldsymbol{I}}_3}}&{n{e_j}}\\ 0&{ne_j^{\rm{T}}}&{\frac{3}{2}\frac{n}{\theta }{u_j}} \end{array}} \right) $$

$$ {h_\alpha } = - \sum\limits_{j = 1}^3 {{\mathit{\boldsymbol{A}}_0}\left( {{\partial ^\alpha }\left( {{\mathit{\boldsymbol{A}}_j}{\partial _j}\mathit{\boldsymbol{U}}} \right) - {\mathit{\boldsymbol{A}}_j}{\partial ^\alpha }{\partial _j}\mathit{\boldsymbol{U}}} \right)} $$

显然，当|α|=0时，h_α=0.

令式(34)两端分别与$2{\partial ^\alpha }\mathit{\boldsymbol{U}}$在空间L²(${\mathbb{R}^3}$)上做内积可得

$$ \begin{array}{*{20}{c}} {\frac{{\rm{d}}}{{{\rm{d}}t}}\left\langle {{\mathit{\boldsymbol{A}}_0}{\partial ^\alpha }\mathit{\boldsymbol{U}},{\partial ^\alpha }\mathit{\boldsymbol{U}}} \right\rangle = }\\ {2\left\langle {{h_\alpha },{\partial ^\alpha }\mathit{\boldsymbol{U}}} \right\rangle + \left\langle {{\rm{div}}\mathit{\boldsymbol{A}}\;{\partial ^\alpha }\mathit{\boldsymbol{U}},{\partial ^\alpha }\mathit{\boldsymbol{U}}} \right\rangle - }\\ {2\left\langle {{\mathit{\boldsymbol{A}}_0}\;{\partial ^\alpha }\left( {\mathit{\boldsymbol{MU}}} \right),{\partial ^\alpha }\mathit{\boldsymbol{U}}} \right\rangle + }\\ {2\left\langle {{\mathit{\boldsymbol{A}}_0}\;{\partial ^\alpha }\left( {{\mathit{\boldsymbol{K}}_1} + {\mathit{\boldsymbol{K}}_2}} \right),{\partial ^\alpha }\mathit{\boldsymbol{U}}} \right\rangle } \end{array} $$

(35)

此处

$$ {\rm{div}}\;\mathit{\boldsymbol{A}} = {\partial _t}{\mathit{\boldsymbol{A}}_0} + \sum\limits_{j = 1}^3 {{\partial _j}{{\mathit{\boldsymbol{\tilde A}}}_j}} $$

(36)

下面估计式(35)右端各项.关于第1项，由h_α的定义，Moser型不等式以及Sobolev连续嵌入定理可得

$$ 2\left\langle {{h_\alpha },{\partial ^\alpha }\mathit{\boldsymbol{U}}} \right\rangle \le C{\left\| {\left( {\rho ,u,\mathit{\Theta }} \right)} \right\|_s}{H_s}\left( t \right) $$

(37)

关于第2项，由小性条件(32)以及Moser型不等式知

$$ \begin{array}{*{20}{c}} {\left\langle {{\rm{div}}\;\mathit{\boldsymbol{A}}\;{\partial ^\alpha }\mathit{\boldsymbol{U}},{\partial ^\alpha }\mathit{\boldsymbol{U}}} \right\rangle \le }\\ {C{{\left\| {\left( {\rho ,u,\mathit{\Theta }} \right)} \right\|}_s}\left\| {\nabla \left( {\rho ,u,\mathit{\Theta }} \right)} \right\|_{s - 1}^2} \end{array} $$

(38)

关于第3项，由M的定义、小性条件(32)、Moser型不等式、Cauchy-Schwarz不等式可知

$$ \begin{array}{*{20}{c}} {\left| {\left\langle {{\mathit{\boldsymbol{A}}_0}{\partial ^\alpha }\left( {\mathit{\boldsymbol{MU}}} \right),{\partial ^\alpha }\mathit{\boldsymbol{U}}} \right\rangle } \right| \le }\\ {\varepsilon \left\| {\nabla u} \right\|_s^2 + C{{\left\| {\nabla \bar n} \right\|}_s}\left\| \rho \right\|_s^2} \end{array} $$

(39)

关于式(35)的最后一项，借助Leibniz公式可得

$$ \begin{array}{*{20}{c}} {2\left\langle {{\mathit{\boldsymbol{A}}_0}{\partial ^\alpha }\left( {{\mathit{\boldsymbol{K}}_1} + {\mathit{\boldsymbol{K}}_2}} \right),{\partial ^\alpha }\mathit{\boldsymbol{U}}} \right\rangle \le }\\ { - 2\left\langle {{\partial ^\alpha }\left( {nu} \right),{\partial ^\alpha }\mathit{\boldsymbol{F}}} \right\rangle - 2{{\left\| {{\partial ^\alpha }\nabla u} \right\|}^2} - }\\ {3\left\langle {\frac{n}{\theta },{{\left| {{\partial ^\alpha }\mathit{\Theta }} \right|}^2}} \right\rangle + CE_s^{\frac{1}{2}}\left( t \right)\left\| {\left( {\rho ,\nabla u} \right)} \right\|_s^2} \end{array} $$

(40)

另外，Maxwell方程组(24)~(25)做能量估计可得

$$ \frac{{\rm{d}}}{{{\rm{d}}t}}\left( {{{\left\| {{\partial ^\alpha }\mathit{\boldsymbol{F}}} \right\|}^2} + {{\left\| {{\partial ^\alpha }\mathit{\boldsymbol{B}}} \right\|}^2}} \right) = 2\left\langle {{\partial ^\alpha }\left( {nu} \right),{\partial ^\alpha }\mathit{\boldsymbol{F}}} \right\rangle $$

(41)

于是，合并式(35)~(41)可得

$$ \begin{array}{*{20}{c}} {\frac{{\rm{d}}}{{{\rm{d}}t}}\left( {\left\langle {{\mathit{\boldsymbol{A}}_0}{\partial ^\alpha }\mathit{\boldsymbol{U}},{\partial ^\alpha }\mathit{\boldsymbol{U}}} \right\rangle + {{\left\| {{\partial ^\alpha }\left( {\mathit{\boldsymbol{F}},\mathit{\boldsymbol{B}}} \right)} \right\|}^2}} \right) + }\\ {2{{\left\| {{\partial ^\alpha }\nabla u} \right\|}^2} + 3\left\langle {\frac{n}{\theta },{{\left| {{\partial ^\alpha }\mathit{\Theta }} \right|}^2}} \right\rangle \le }\\ {\varepsilon \left\| {\nabla u} \right\|_s^2 + C{{\left\| {\nabla \bar n} \right\|}_s}\left\| \rho \right\|_s^2 + CE_s^{\frac{1}{2}}\left( t \right){H_s}\left( t \right)} \end{array} $$

(42)

借助$\left\langle {{\mathit{\boldsymbol{A}}_0}\; \; {\partial ^\alpha }\mathit{\boldsymbol{U}}, {\partial ^\alpha }\mathit{\boldsymbol{U}}} \right\rangle + {\left\| {{\partial ^\alpha }\left({\mathit{\boldsymbol{F}}, \mathit{\boldsymbol{B}}} \right)} \right\|^2}$与${\left\| {{\partial ^\alpha }\mathit{\boldsymbol{W}}} \right\|^2}$的等价性，式(42)关于指标α求和，|α|≤s并取ε>0充分小，可知式(33)成立.证毕.

引理2 在引理1的条件下，对任意t∈[0, T]，有

$$ \frac{{\rm{d}}}{{{\rm{d}}t}}{E_s}\left( t \right) + {H_s}\left( t \right) \le C\left( {{{\left\| {\nabla \bar n} \right\|}_s} + E_s^{\frac{1}{2}}\left( t \right)} \right){H_s}\left( t \right) $$

(43)

证明：对于α∈${\mathbb{N}^3}$，|α|≤s－1，对式(22)求${\partial ^\alpha }$，然后在空间L²(${\mathbb{R}^3}$)上与${\partial ^\alpha }\nabla \rho $内积可得

$$ \begin{array}{*{20}{c}} {\left\langle {\frac{n}{\theta },{{\left| {\nabla {\partial ^\alpha }\rho } \right|}^2}} \right\rangle + \left| {\nabla {\partial ^\alpha }\rho ,{\partial ^\alpha }\mathit{\boldsymbol{F}}} \right| = }\\ { - \frac{{\rm{d}}}{{{\rm{d}}t}}\left\langle {\nabla {\partial ^\alpha }\rho ,{\partial ^\alpha }u} \right\rangle + \left\langle {\nabla {\partial ^\alpha }{\partial _t}\rho ,{\partial ^\alpha }u} \right\rangle - }\\ {{I_1}\left( t \right) - \sum\limits_{\beta < \alpha } {C_\alpha ^\beta {I_{2\beta }}\left( t \right)} } \end{array} $$

(44)

其中

$$ \begin{array}{*{20}{c}} {{I_1}\left( t \right) = \left\langle {u\nabla {\partial ^\alpha }u,\nabla {\partial ^\alpha }\rho } \right\rangle + \left\langle {\nabla {\partial ^\alpha }\mathit{\Theta },\nabla {\partial ^\alpha }\rho } \right\rangle - }\\ {\left\langle {\frac{{{\partial ^\alpha }\Delta u}}{n},\nabla {\partial ^\alpha }\rho } \right\rangle + \left\langle {{\partial ^\alpha }u\mathit{\boldsymbol{B}} + \nabla {\partial ^\alpha }\rho } \right\rangle - }\\ {\left\langle {\frac{\mathit{\Theta }}{n}{\partial ^\alpha }\left( {\nabla \bar n} \right),\nabla {\partial ^\alpha }\rho } \right\rangle + \left\langle {\frac{{{\theta _ * }\rho }}{{n\tilde n}}{\partial ^\alpha }\left( {\nabla \bar n} \right),\nabla {\partial ^\alpha }\rho } \right\rangle } \end{array} $$

$$ \begin{array}{*{20}{c}} {{I_{2\beta }}\left( t \right) = }\\ {\left\langle {{\partial ^{\alpha - \beta }}\left( {\frac{\theta }{n}} \right)\nabla {\partial ^\beta }\rho ,\nabla {\partial ^\alpha }\rho } \right\rangle + \left\langle {{\partial ^{\alpha - \beta }}u\nabla {\partial ^\beta }u,\nabla {\partial ^\alpha }\rho } \right\rangle - }\\ {\left\langle {{\partial ^{\alpha - \beta }}\left( {\frac{1}{n}} \right){\partial ^\beta }\Delta u,\nabla {\partial ^\alpha }\rho } \right\rangle + \left\langle {{\partial ^\beta }u \times {\partial ^{\alpha - \beta }}\mathit{\boldsymbol{B}},\nabla {\partial ^\alpha }\rho } \right\rangle + }\\ {\left\langle {\left( {{\partial ^{\alpha - \beta }}\left( {\frac{{{\theta _ * }\rho }}{{n\tilde n}}} \right) - {\partial ^{\alpha - \beta }}\left( {\frac{\mathit{\Theta }}{n}} \right)} \right){\partial ^\beta }\left( {\nabla \bar n} \right),\nabla {\partial ^\alpha }\rho } \right\rangle } \end{array} $$

首先，由n, θ≥const.>0可知$\frac{\theta }{n} \ge \frac{1}{C}$.进而，由式(24)的第2部分以及分部积分公式可得

$$ \left\langle {\frac{\theta }{n},{{\left| {\nabla {\partial ^\alpha }\rho } \right|}^2}} \right\rangle + \left\langle {\nabla {\partial ^\alpha }\rho ,{\partial ^\alpha }F} \right\rangle \ge \frac{1}{C}\left\| {{\partial ^\alpha }\rho } \right\|_1^2 $$

(45)

其次，由分部积分公式和Leibniz公式可知

$$ \begin{array}{*{20}{c}} {\left\langle {\nabla {\partial ^\alpha }{\partial _t}\rho ,{\partial ^\alpha }u} \right\rangle \le }\\ {C\left\| {\nabla u} \right\|_{s - 1}^2 + {{\left\| {\left( {\rho ,u} \right)} \right\|}_s}\left\| {\left( {\rho ,\nabla u} \right)} \right\|_s^2} \end{array} $$

(46)

然后，由小性条件(32)、Cauchy-Schwarz不等式及分部积分公式可知

$$ \begin{array}{*{20}{c}} {\left| {{I_1}\left( t \right) + {I_2}\left( t \right)} \right| \le }\\ {\varepsilon \left\| \rho \right\|_s^2 + C\left\| {\left( {\nabla u,\mathit{\Theta }} \right)} \right\|_s^2 + C{{\left\| {\nabla \bar n} \right\|}_s}\left\| \rho \right\|_s^2 + }\\ {C{{\left\| {\left( {\rho ,u,\mathit{\Theta },\mathit{\boldsymbol{B}}} \right)} \right\|}_s}\left\| {\left( {\rho ,\nabla u} \right)} \right\|_s^2} \end{array} $$

(47)

合并式(44)~(47)可得

$$ \begin{array}{*{20}{c}} {\frac{1}{C}\left\| {{\partial ^\alpha }\rho } \right\|_1^2 + \frac{{\rm{d}}}{{{\rm{d}}t}}\left\langle {\nabla {\partial ^\alpha }\rho ,{\partial ^\alpha }u} \right\rangle \le }\\ {\varepsilon \left\| \rho \right\|_s^2 + C\left\| {\left( {\nabla u,\mathit{\Theta }} \right)} \right\|_s^2 + C{{\left\| {\nabla \bar n} \right\|}_s}\left\| \rho \right\|_s^2 + }\\ {C{{\left\| {\left( {\rho ,u,\mathit{\Theta },\mathit{\boldsymbol{B}}} \right)} \right\|}_s}\left\| {\left( {\rho ,\nabla u} \right)} \right\|_s^2} \end{array} $$

(48)

式(48)关于|α|≤s－1求和并取ε>0充分小可得

$$ \begin{array}{*{20}{c}} {\left\| \rho \right\|_s^2 + c\frac{{\rm{d}}}{{{\rm{d}}t}}\sum\limits_{\left| \alpha \right| \le s - 1} {\left\langle {\nabla {\partial ^\alpha }\rho ,{\partial ^\alpha }u} \right\rangle } \le }\\ {C\left\| {\left( {\nabla u,\mathit{\Theta }} \right)} \right\|_s^2 + C{{\left\| {\nabla \bar n} \right\|}_s}\left\| \rho \right\|_s^2 + }\\ {C{{\left\| {\left( {\rho ,u,\mathit{\Theta },\mathit{\boldsymbol{B}}} \right)} \right\|}_s}\left\| {\left( {\rho ,\nabla u} \right)} \right\|_s^2} \end{array} $$

其中c>0为一小常数，满足

$$ c\sum\limits_{\left| \alpha \right| \le s - 1} {\left\langle {\nabla {\partial ^\alpha }\rho ,{\partial ^\alpha }u} \right\rangle } \le \frac{1}{2}{E_s}\left( t \right) $$

故而联合引理1的式(33)可得式(43)成立.证毕.

3.2 整体存在性的证明

由引理2可知，若C₂(‖$\nabla \bar n$‖+ω_T) < 1，可知式(43)右端项可以被左端项控制，于是有

$$ {\left\| {\mathit{\boldsymbol{W}}\left( t \right)} \right\|_s} \le C_1^{\frac{1}{2}}{\left\| {{\mathit{\boldsymbol{W}}_0}} \right\|_s},\forall t \in \left[ {0,T} \right] $$

因此，只要‖$\nabla \bar n$‖_s+‖W₀‖_s≤$\frac{1}{2}{\delta _0}$，这里δ₀满足

$$ C_1^{\frac{1}{2}}{\delta _0} < \min \left\{ {\frac{{\min \left\{ {{n_ * } + {{\left\| {\bar n} \right\|}_{{L^\infty }}},{\theta _ * }} \right\}}}{{2{C_ * }}},\frac{1}{{{C_2}}}} \right\} $$

就可以保证

$$ \begin{array}{*{20}{c}} {{C_2}\left( {{{\left\| {\nabla \bar n} \right\|}_s} + {\omega _T}} \right) < 1}\\ {{{\left\| {\nabla \bar n} \right\|}_s} + {\omega _T} \le \frac{{\min \left\{ {{n_ * } + {{\left\| {\bar n} \right\|}_{{L^\infty }}},{\theta _ * }} \right\}}}{{2{C_ * }}}} \end{array} $$

于是，解的整体存在性可由命题2给出的解的局部存在性结论结合标准的连续性方法获得.证毕.

4. 结论

1) 借助非常数平衡解的小性以及对称子技巧，在初值为一个非常数平衡解的小摄动前提下，建立了Cauchy问题光滑解的一致先验估计.

2) 利用对称双曲-抛物组光滑解的局部解存在性理论，并结合标准的连续性讨论方法，证明了该问题在一个非常数平衡解存在唯一渐近稳定的整体光滑解.

3) 将非常数平衡解附近光滑解的整体存在性理论推广至粘性真实流体，并对粘性等离子体物理的发展提供必要理论依据.

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1. 模型与定理
2. 准备工作
3. 主要结果的证明
3.1 一致先验估计
3.2 整体存在性的证明
4. 结论

非等熵可压缩Navier-Stokes-Maxwell方程组Cauchy问题解的整体存在性

作者简介: 李新(1981-), 女, 博士研究生, 主要从事应用偏微分方程方面的研究, E-mail:lixin91600@emails.bjut.edu.cn

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出版历程

Global Existence of Solutions to Cauchy Problem for Non-isentropic Compressible Navier-Stokes-Maxwell Systems

1. 模型与定理

2. 准备工作

3. 主要结果的证明

3.1 一致先验估计

3.2 整体存在性的证明

4. 结论

期刊类型引用(2)

其他类型引用(0)

计量

出版历程

目录

1. 模型与定理

2. 准备工作

3. 主要结果的证明

3.1 一致先验估计

3.2 整体存在性的证明

4. 结论

作者简介:
李新(1981-), 女, 博士研究生, 主要从事应用偏微分方程方面的研究, E-mail:lixin91600@emails.bjut.edu.cn