设 $\Omega\subset{\Bbb R}^3$ 为光滑有界区域.我们主要研究如下Navier-Stokes-Fourier方程的人工可压逼近

$\begin{eqnarray}\label{1.1} \left\{\begin{array}{ll} \displaystyle \frac{\partial u}{\partial t} -\mu\Delta u +(u\cdot\nabla)u+\nabla p=0, & x\in\Omega,t>0,\\ \displaystyle \frac{\partial\theta}{\partial t} -\kappa\Delta\theta +u\cdot\nabla\theta=0, & x\in\Omega,t>0,\\ \mbox{div} u=0,\\ u=0,\theta=0,& x \in \partial\Omega,t>0,\\ u(x,0)=u_0,\theta(x,0)=\theta_0,& x \in\Omega, \end{array}\right. \end{eqnarray}$

(1.1)

其中, $u\in{\Bbb R}^3,\theta\in{\Bbb R},p\in{\Bbb R}$ 分别表示速度场, 温度场和流体的压强, $\mu$ 和 $\kappa>0$ 分别表示粘度常系数和热传导系数. Navier-Stokes-Fourier方程是热力学流体方程中最重要的方程.在过去的几十年里该模型已被大量研究, 参见文献[6-7, 9-11, 20].

人工可压逼近方法最初由Chorin^[3-4]和Temam^[14-16]提出, 主要是为了克服不可压缩约束条件 $"\mbox{div} u=0"$ 给Navier-Stokes方程数值逼近带来的困难.由于 $"\mbox{div} u=0"$ 不能确切成立, 在数值逼近的迭代过程中, 离散误差会不断积累, 随着误差积累的增多, 近似算法就会失效.为此, Chorin和Temam引进了如下扰动的可压Navier-Stokes方程

$\begin{eqnarray} \frac{\partial u^\epsilon}{\partial t}+\nabla p^\epsilon=\nu \Delta u^\epsilon-(u^\epsilon\cdot\nabla)u^\epsilon -\frac{1}{2}(\mbox{div} u^\epsilon)u^\epsilon , \end{eqnarray}$

$\begin{eqnarray} \epsilon\frac{\partial p^\epsilon}{\partial t}+\mbox{div} u^\epsilon=0. \end{eqnarray}$

在文献[16]中, Temam利用经典的Sobolev紧嵌入定理和经典分数阶导数的Lions方法^[11]克服了关于时间的紧性, 从而解决了Navier-Stokes方程在有界区域上可压逼近的收敛性的问题.

近年来利用Chorin和Temam的方法, 一些模型的人工可压逼近问题被大量研究.例如, 在文献[21-22]中, 赵才地教授和他的合作者分别研究了对流的Brinkman-Forchheimer方程和非Newtonian流体方程的可压逼近问题.我们在文献[13]中研究了Electro-hydrodynamics方程的可压逼近问题.

在本文中, 我们利用人工可压逼近的方法来研究系统(1.1)的逼近问题.考虑如下的关于参数 $\epsilon \in(0,1]$ 的扰动可压系统族

$\begin{eqnarray}\label{1.2} \left\{\begin{array}{ll} \displaystyle \frac{\partial u_\epsilon}{\partial t} -\mu\Delta u_\epsilon +(u_\epsilon\cdot\nabla)u_\epsilon+\frac{1}{2}(\mbox{div} u_\epsilon)u_\epsilon+\nabla p_\epsilon=0,\\ \displaystyle \frac{\partial\theta_\epsilon}{\partial t} -\kappa\Delta\theta_\epsilon +u_\epsilon\cdot\nabla\theta_\epsilon+\frac{1}{2}(\mbox{div} u_\epsilon)\theta_\epsilon=0,\\ \displaystyle \epsilon\frac{\partial p_\epsilon}{\partial t}+\mbox{div} u_\epsilon=0, \end{array}\right. \end{eqnarray}$

(1.2)

满足初边值条件

$\begin{eqnarray} u_\epsilon=0,\theta_\epsilon=0, x \in \partial\Omega, \end{eqnarray}$

(1.3)

$\begin{eqnarray} u_\epsilon(x,0)=u_0,\theta_\epsilon(x,0)=\theta_0,p_\epsilon(x,0)=p_0(x),x\in \Omega.\label{1.3} \end{eqnarray}$

(1.4)

当 $\epsilon=0$ 时, 易知扰动的可压Navier-Stokes-Fourier方程(1.2)-(1.4)即为不可压的Navier-Stokes-Fourier方程(1.1).因此, 我们会很自然地去考虑方程(1.2)-(1.4)的解是否存在, 当 $\epsilon\rightarrow 0^+$ 时, 此解是否收敛到不可压的Navier-Stokes-Fourier方程(1.1)的解 $(u,\theta,p)$ .在本文中, 我们将解决以上这些问题.

从物理的角度来看, 我们可以由可压流体模型推导出不可压流体模型.可压流体方程的不可压极限是流体动力学数学理论中的一个重要问题.在过去的几十年里, 很多学者已经在这一领域做出很多工作并取得很大的进展, 如: Jiang^[18-19], Lions和Masmoudi^[12], Temam^[16], Feireisl^{[1, 8]}, Chang和Kwak^[2].

本文结构安排如下:第二节, 给出了一些本文所涉及到的基础知识; 第三节, 证明了方程(1.2)-(1.4)解的存在性; 第四节, 研究当 $\epsilon\rightarrow 0^+$ 时扰动问题解的收敛性.

为了更好的描述问题, 我们先来回顾一些常用的Lebesgue空间和Sobolev空间定义及相关符号, 更多详细的介绍可参见文献[14].

设 $L^2(\Omega)$ 和 $ H_0^1(\Omega)$ 分别表示标准的Lebesgue空间和Sobolev空间.记向量值空间 $(L^2(\Omega))^3$ 和 $ (H_0^1(\Omega))^3 $ 为 $ {\Bbb L}^2(\Omega)$ 和 $ {\Bbb H}_0^1(\Omega)$ .设

$\begin{eqnarray} {\cal V}=\{\phi\in({\cal C}^\infty_0(\Omega) )^3 :\mbox{div} \phi=0\}.\end{eqnarray}$

设 ${\Bbb H},{\Bbb V}$ 分别是 ${\cal V}$ 在空间 $( L^2(\Omega))^3$ 和 $( H^1_0(\Omega))^3$ 上的闭包.为了书写的方便, 我们用 $(\cdot,\cdot)$ 和 $|\cdot|$ 分别表示空间 $L^2(\Omega),{\Bbb L}^2(\Omega)$ 和 ${\Bbb H}$ 的内积和范数, 用 $((\cdot,\cdot))$ 和 $\|\cdot\|$ 分别表示空间 $ H^1_0(\Omega),{\Bbb H}^1_0(\Omega)$ 和 ${\Bbb V}$ 的内积和范数.设 $H^{-1}(\Omega),{\Bbb H}^{-1}(\Omega)$ 和 $ {\Bbb V}'$ 分别表示 $H^1_0(\Omega),{\Bbb H}^1_0(\Omega)$ 和 ${\Bbb V}$ 的对偶空间.我们用 $\|\cdot\|_*$ 来表示 $H^{-1}(\Omega), {\Bbb H}^{-1}(\Omega)$ 和 ${\Bbb V}'$ 的范数, 且用 $\langle\cdot, \cdot\rangle$ 来表示 $H^1_0(\Omega)$ 和 $H^{-1}(\Omega)$ , ${\Bbb H}^1_0(\Omega)$ 和 ${\Bbb H}^{-1}(\Omega)$ 或 ${\Bbb V}$ 和 ${\Bbb V}'$ 之间的对偶对.

定义由 ${\Bbb V} $ 到 $ {\Bbb V}^*$ 上的"Stokes算子" ${\cal A}=-\triangle$ 为

$\begin{eqnarray} \langle {\cal A}u,v\rangle =\sum\limits_{i,j=1}^{3}\int_\Omega\partial_j u_i\partial_j v_i {\rm d}x,\forall u,v\in {\Bbb V}.\end{eqnarray}$

定义 ${\Bbb V}$ ( $H_0^1(\Omega)$ 或 ${\Bbb H}_0^1(\Omega)$ )上的三线性形式 $b(\cdot,\cdot,\cdot)$ 为

$\begin{eqnarray}b(u,v,w)=\sum\limits_{i,j=1}^{3}\int_\Omega u_i\frac{\partial v_j}{\partial x_i}w_j {\rm d}x,\forall u,v,w\in {\Bbb V} (\mbox{或} {\Bbb H}_0^1(\Omega) ),\end{eqnarray}$

且定义 $B(u,v): {\Bbb V} \times {\Bbb V} \mapsto {\Bbb V}' ({\Bbb H}_0^1(\Omega)\times {\Bbb H}_0^1(\Omega) \mapsto {\Bbb H}^{-1}(\Omega))$ , 即

$\begin{eqnarray} \langle B(u,v),w\rangle = b(u,v,w).\end{eqnarray}$

进一步, 定义三线性形式 $\hat{b}(\cdot,\cdot,\cdot)$ 为

$\begin{eqnarray}\hat{b}(u,v,w)=\frac{1}{2}[b(u,v,w)-b(u,w,v)],\forall u,v,w\in{{\Bbb V}} ,\end{eqnarray}$

且定义 $\hat{b}(u,v,w)$ 为 $ \langle \hat{B}(u,v),w\rangle .$

设 $X_0,X,X_1$ 是三个Hilbert空间, 且

$\begin{eqnarray}\label{2.5} X_0\hookrightarrow X\hookrightarrow X_1, \end{eqnarray}$

(2.1)

其中嵌入是连续的, 且

$\begin{eqnarray}\label{2.6} X_0\hookrightarrow X \mbox{是紧的}. \end{eqnarray}$

(2.2)

设 $\psi(t)$ 是 ${\Bbb R}\rightarrow X_1$ 的函数, 我们用 $\hat{\psi}(\tau)$ 来表示它的Fourie变换

$\begin{eqnarray*} \hat{\psi}(\tau)=\int^{+\infty}_{-\infty}\exp(-2\pi{\rm i}t\tau)\psi(t){\rm d}t. \end{eqnarray*}$

$\psi(t)$ 关于 $t$ 的 $\gamma$ 阶导数是 $(2{\rm i}\pi\tau)^{\gamma}\hat{\psi}(\tau)$ 的Fourier逆变换, 即

$\begin{eqnarray*} \widehat{D^{\gamma}_{t}\psi(t)}=(2{\rm i}\pi\tau)^{\gamma}\hat{\psi}(\tau). \end{eqnarray*}$

对于任给的 $\gamma>0$ , 定义空间 $M^{\gamma}$ 为

$\begin{eqnarray*} M^{\gamma}=M^{\gamma}({\Bbb R};X_0,X_1)=\{\psi\in L^{2}({\Bbb R};X_0),D^{\gamma}_{t}\psi \in L^{2}({\Bbb R};X_1)\}. \end{eqnarray*}$

则空间 $M^{\gamma}$ 为Hilbert空间, 定义其范数为

$\begin{eqnarray}\|\psi\|_{M^{\gamma}}=\{\|\psi\|^2_{L^2({\Bbb R};X_0)}+\||\tau|^{\gamma}\hat{\psi}(\tau)\|^{2}_{L^2({\Bbb R};X_1)}\}^{\frac{1}{2}}.\end{eqnarray}$

对于任一有界集 $K\subset {\Bbb R}$ , $M^{\gamma}$ 的子空间 $M^{\gamma}_{K}$ 是紧包含在 $K$ 中的函数 $u\in M^{\gamma}$ 的集合, 即

$\begin{eqnarray*} M^{\gamma}_{K}=M^{\gamma}_{K}({\Bbb R};X_0,X_1)=\{\psi\in M^{\gamma},{\rm supp } \psi \subseteq K\}. \end{eqnarray*}$

我们需要如下关于时间变量 $t$ 的分数阶导数的紧性结果.

引理2.1 设 $X_0,X,X_1$ 是Hilbert空间且满足条件(2.1)和(2.2).则对于任意有界集 $K$ 和任意常数 $\gamma>0$ , 使得

$\begin{eqnarray} M^{\gamma}_{K}({\Bbb R};X_0,X_1)\hookrightarrow L^{2}({\Bbb R};X).\end{eqnarray}$

在本节中, 我们证明问题(1.2)-(1.4)弱解的存在性.

定义3.1 设 $\epsilon\in (0,1]$ 且 $u_{0}\in {\Bbb H},\theta_{0},p_0\in L^2(\Omega),$ 对于任意的 $T>0$ , 则系统(1.2)-(1.4)的弱解 $(u_\epsilon,\theta_\epsilon,p_\epsilon)$ 满足

$\begin{eqnarray} u_\epsilon\in L^{2}(0,T;{{\Bbb H}}^1_0(\Omega))\cap L^{\infty}(0,T;{\Bbb H}), \mbox{且} u'_\epsilon\in L^{\frac{4}{3}}(0,T;{{\Bbb H}}^{-1}(\Omega)),\label{3.7} \end{eqnarray}$

(3.1)

$\begin{eqnarray} \theta_\epsilon\in L^{2}(0,T;H^1_0(\Omega))\cap L^{\infty}(0,T;L^2(\Omega)), \mbox{且} \theta'_\epsilon\in L^{\frac{4}{3}}(0,T;H^{-1}(\Omega)),\label{3.8} \end{eqnarray}$

(3.2)

$\begin{eqnarray} p_{\epsilon}\in L^{2}(0,T;L^2(\Omega)),\mbox{且} p'_{\epsilon}\in L^{2}(0,T;L^2(\Omega)),\label{3.9} \end{eqnarray}$

(3.3)

且

$\begin{eqnarray} u'_\epsilon-\mu\Delta u_\epsilon +(u_\epsilon\cdot\nabla)u_\epsilon +\frac{1}{2}(\mbox{div} u_\epsilon)u_\epsilon+\nabla p_\epsilon=0 \mbox{在$ L^{\frac{4}{3}}(0,T;{\Bbb H}^{-1}(\Omega)) $中成立,}\label{3.10} \end{eqnarray}$

(3.4)

$\begin{eqnarray} \theta'_\epsilon-\kappa\Delta\theta_\epsilon +(u_\epsilon\cdot\nabla) \theta_\epsilon+\frac{1}{2}(\mbox{div} u_\epsilon)\theta_\epsilon=0 \mbox{在$ L^{\frac{4}{3}}(0,T;H^{-1}(\Omega)) $中成立,}\label{3.11} \end{eqnarray}$

(3.5)

$\begin{eqnarray} \epsilon p'_{\epsilon}+\mbox{div} u_\epsilon=0 \mbox{在$ L^{2}(0,T;L^2(\Omega))$中成立,}\label{3.12} \end{eqnarray}$

(3.6)

$\begin{eqnarray} u_\epsilon(x,0)=u_0,\theta_\epsilon(x,0)=\theta_0,p_\epsilon(x,0)=p_0, x\in \Omega.\label{3.13} \end{eqnarray}$

(3.7)

定理3.1 设 $\epsilon\in (0,1]$ , $u_{0}\in {\Bbb H},\theta_{0},p_0\in L^2(\Omega),$ 则问题(1.2)-(1.4)存在弱解 $(u_\epsilon,\theta_\epsilon,p_\epsilon)$ .

证 应用Galerkin逼近方法来证明弱解的存在性.用 $({\cal C}^\infty_0(\Omega))^3$ 中的元素 $\{\omega_i\}_{i=1}^{\infty}$ 来表示 ${\Bbb H}^1_{0}(\Omega)$ 的基底, ${\cal C}^\infty_0(\Omega)$ 中的元素 $\{\gamma_j\}_{j=1}^{\infty}$ 表示 $L^2(\Omega)$ 和 $ H^1_0(\Omega)$ 的基底.对于任一正整数 $m$ , 定义

$\begin{eqnarray}u_{\epsilon m}(t)=\sum\limits_{i=1}^m g_{im}(t)\omega_i, \theta_{\epsilon m}(t)=\sum\limits_{j=1}^m h_{jm}(t)\gamma_j, p_{\epsilon m}(t)=\sum\limits_{j=1}^m \xi_{jm}(t)\gamma_j,\end{eqnarray}$

并考虑逼近的常微分系统

$\begin{eqnarray} (u'_{\epsilon m},\omega_i)+\mu((u_{\epsilon m},\omega_i))+\hat{b}(u_{\epsilon m},u_{\epsilon m},\omega_i)+(\nabla p_{\epsilon m},\omega_i)=0,i=1,\cdots ,m,\label{3.14} \end{eqnarray}$

(3.8)

$\begin{eqnarray} (\theta'_{\epsilon m},\gamma_j)+\kappa((\theta_{\epsilon m},\gamma_j))+\hat{b}(u_{\epsilon m},\theta_{\epsilon m},\gamma_j)=0,j=1,\cdots ,m,\label{3.15} \end{eqnarray}$

(3.9)

$\begin{eqnarray} \epsilon(p'_{\epsilon m},\gamma_j) +(\mbox{div} u_{\epsilon m}(t),\gamma_j)=0,j=1,\cdots ,m,\label{3.16} \end{eqnarray}$

(3.10)

$\begin{eqnarray} u_{\epsilon m}(0)=u_{0m},\theta_{\epsilon m}(0)=\theta_{0m},p_{\epsilon m}(0)=p_{0m}, \end{eqnarray}$

(3.11)

其中, $u_{0m}$ ( $\theta_{0m}$ , $p_{0m}$ )是 $u_0$ ( $\theta_0$ , $p_0$ )到 ${\Bbb L}^2(\Omega)$ ( $L^{2}(\Omega)$ )中由 $\omega_1,\cdots ,\omega_m$ ( $\gamma_1,\cdots ,\gamma_m$ )张成的空间的正交投影.

方程(3.8)-(3.10)构成关于 $g_{1m},\cdots ,g_{mm}$ , $h_{1m},\cdots ,h_{mm}$ , $\xi_{1m},\cdots ,\xi_{mm}$ 的非线性微分系统.通过常微分方程的标准理论, 我们可以推断出常微分系统存在解.

分别用 $g_{im}(t),h_{jm}(t)$ 和 $\xi_{jm}(t)$ 乘以方程(3.8), (3.9)和(3.10), 我们可以得到

$\begin{eqnarray*} &&(u'_{\epsilon m},u_{\epsilon m})+\mu\|u_{\epsilon m}\|^{2}+\hat{b}(u_{\epsilon m},u_{\epsilon m},u_{\epsilon m}) +(\nabla p_{\epsilon m},u_{\epsilon m})+(\theta'_{\epsilon m},\theta_{\epsilon m}) \\ &&+\kappa\|\theta_{\epsilon m}\|^{2}+\hat{b}(u_{\epsilon m},\theta_{\epsilon m},\theta_{\epsilon m}) +\epsilon(p'_{\epsilon m},p_{\epsilon m})+(\mbox{div} {u_{\epsilon m}},p_{\epsilon m})=0. \end{eqnarray*}$

易证 $\hat{b}(u_{\epsilon m},u_{\epsilon m},u_{\epsilon m})=0$ , $\hat{b}(u_{\epsilon m},\theta_{\epsilon m},\theta_{\epsilon m})=0$ .又由于

$\begin{eqnarray}(\nabla p_{\epsilon m},u_{\epsilon m}) + (\mbox{div} {u_{\epsilon m}},p_{\epsilon m})=0, \end{eqnarray}$

可得出

$\begin{eqnarray}\label{3.17} \frac{\rm d}{{\rm d}t}(|u_{\epsilon m}|^2+|\theta_{\epsilon m}|^2+\epsilon|p_{\epsilon m}|^2)+ 2\mu\|u_{\epsilon m}\|^2 +2\kappa\|\theta_{\epsilon m}\|^2=0. \end{eqnarray}$

(3.12)

对式上式从0到 $s$ 积分可推出

$\begin{eqnarray}\label{3.180} |u_{\epsilon m}(s)|^2+|\theta_{\epsilon m}(s)|^2+\epsilon|p_{\epsilon m}(s)|^2 &\leq&|u_{\epsilon m}(0)|^2+|\theta_{\epsilon m}(0)|^2 +\epsilon|p_{\epsilon m}(0)|^2\nonumber\\ &\leq&|u_0|^2+|\theta_0|^2+\epsilon|p_0|^2, \end{eqnarray}$

(3.13)

且

$\begin{eqnarray}\label{3.18} \sup\limits_{s\in[0,T]}\{|u_{\epsilon m}(s)|^{2}+|\theta_{\epsilon m}(s)|^{2}+\epsilon|p_{\epsilon m}(s)|^{2}\}\leq d_{1}, \quad d_{1}=|u_0|^{2}+|\theta_0|^{2}+|p_0|^{2}. \end{eqnarray}$

(3.14)

对于任一的 $T>0,$ 将(3.12)式关于 $t$ 从0到 $T$ 积分, 可得

$\begin{eqnarray} \int_0^T\|u_{\epsilon m}(t)\|^{2}{\rm d}t\leq\frac{d_1}{2\mu},\label{3.19} \end{eqnarray}$

(3.15)

$\begin{eqnarray} \int_0^T\|\theta_{\epsilon m}(t)\|^{2}{\rm d}t\leq\frac{d_1}{2\kappa}.\label{3.20} \end{eqnarray}$

(3.16)

为了在非线性项中取极限, 我们需要分别对 $u_{\epsilon m}$ 和 $\theta_{\epsilon m}$ 关于时间的分数阶导数进行估计.设

$\begin{eqnarray}\phi_{\epsilon m}(t)=\mu\triangle u_{\epsilon m}-\hat{B}(u_{\epsilon m}), \psi_{\epsilon m}(t)=\kappa\triangle \theta_{\epsilon m}-\hat{B}(u_{\epsilon m}, \theta_{\epsilon m}), \end{eqnarray}$

则(3.8)-(3.10)式可改写为

$\begin{eqnarray} (u'_{\epsilon m}(t),\omega_{k})+(\nabla p_{\epsilon m}(t),\omega_{k})=(\phi_{\epsilon m}(t),\omega_{k}),k=1,\cdots ,m, \end{eqnarray}$

$\begin{eqnarray} (\theta'_{\epsilon m}(t),\gamma_{j})=(\psi_{\epsilon m}(t),\gamma_{j}),j=1,\cdots ,m, \end{eqnarray}$

$\begin{eqnarray} \epsilon(p'_{\epsilon m}(t),\gamma_{j})+(\mbox{div} {u_{\epsilon m}(t)},\gamma_{j})=0,j=1,\cdots ,m. \end{eqnarray}$

参照文献[16] (也可参照文献[21-22])的证明思路, 我们把函数 $u_{\epsilon m}(t),\theta_{\epsilon m}(t) $ 和 $ p_{\epsilon m}(t)$ 在区间 $[0,T]$ 外全部延拓为 $0$ 且分别定义为 $\tilde{u}_{\epsilon m}(t),\tilde{\theta}_{\epsilon m}(t) $ 和 $ \tilde{p}_{\epsilon m}(t)$ .考虑微分方程的Fourier变换.

下面的关系式在 ${\Bbb R}$ 上成立.

$\begin{eqnarray} \frac{\rm d}{{\rm d}t}({\tilde{u}}_{\epsilon m}(t),\omega_{k})+(\nabla{\tilde{p}}_{\epsilon m}(t),\omega_{k})=({\tilde{\phi}}_{\epsilon m}(t),\omega_{k})+(u_{0m},\omega_{k})\delta(0)-(u_{\epsilon m}(T),\omega_{k})\delta_{(T)}, \end{eqnarray}$

$\begin{eqnarray} \frac{\rm d}{{\rm d}t}({\tilde{\theta}}_{\epsilon m}(t),\gamma_{j})=({\tilde{\psi}}_{\epsilon m}(t),\gamma_{j})+(\theta_{0m},\gamma_{j})\delta(0)-(\theta_{\epsilon m}(T),\gamma_{j})\delta(T), \end{eqnarray}$

$\begin{eqnarray} \epsilon\frac{\rm d}{{\rm d}t}({\tilde{p}}_{\epsilon m}(t),\gamma_{j})+(\mbox{div} {{\tilde{u}}_{\epsilon m}(t)},\gamma_{j})=\epsilon(p_{0m},\gamma_{j})\delta(0)-\epsilon(p_{\epsilon m}(T),\gamma_{j})\delta(T). \end{eqnarray}$

对其作Fourier变换, 得

$\begin{eqnarray} 2{\rm i}\pi\tau({\hat{u}}_{\epsilon m}(\tau),\omega_{k})+(\nabla{\hat{p}}_{\epsilon m}(\tau),\omega_{k})=\langle{\hat{\phi}}_{\epsilon m}(\tau),\omega_{k}\rangle+(u_{0m},\omega_{k})-(u_{\epsilon m}(T),\omega_{k})\exp(-2{\rm i}\pi\tau T), \end{eqnarray}$

$\begin{eqnarray} 2{\rm i}\pi\tau({\hat{\theta}}_{\epsilon m}(\tau),\gamma_{j})=\langle{\hat{\psi}}_{\epsilon m}(\tau),\gamma_{j}\rangle+(\theta_{0m},\gamma_{j})-(\theta_{\epsilon m}(T),\gamma_{j})\exp(-2{\rm i}\pi\tau T), \end{eqnarray}$

$\begin{eqnarray} 2{\rm i}\pi\tau\epsilon({\hat{p}}_{\epsilon m}(\tau),\gamma_{j})+(\mbox{div} {{\hat{u}}_{\epsilon m}(\tau)},\gamma_{j})=\epsilon(p_{0m},\gamma_{j})-\epsilon(p_{\epsilon m}(T),\gamma_{j})\exp(-2{\rm i}\pi\tau T). \end{eqnarray}$

我们对上面三个方程分别乘以 $\hat{g}_{km}(\tau),\hat{h}_{km}(\tau)$ 和 $\hat{\xi}_{lm}$ (其中 $\hat{g}_{km}(\tau),\hat{h}_{km}(\tau)$ 和 $\hat{\xi}_{lm}$ 分别是 $\tilde{g}_{km},\tilde{h}_{km}$ 和 $\tilde{\xi}_{lm}$ 的Fourier变换), 然后把所有的式子相加(其中 $k,j=1,\cdots ,m$ ), 从而有

$\begin{eqnarray}\label{3.21} & &2{\rm i}\pi \tau \{|{\hat u}_{\epsilon m}(\tau)|^2+|{\hat\theta}_{\epsilon m}(\tau)|^2+\epsilon|{\hat p}_{\epsilon m}(\tau )|^2\} + (\nabla {{\hat p}_{\epsilon m}}(\tau ),{\hat u}_{\epsilon m}(\tau ))+({\mbox{div}} {{\hat u}_{\epsilon m}}(\tau ),{{\hat p}_{\epsilon m}}(\tau )) \nonumber \\ & =&\langle {{\hat \phi}_{\epsilon m}}(\tau ),{{\hat u}_{\epsilon m}}(\tau )\rangle+ \langle {{\hat \psi }_{\epsilon m}}(\tau ),{{\hat \theta}_{\epsilon m}}(\tau )\rangle +({u_{0m}},{{\hat u}_{\epsilon m}}(\tau ))+({\theta_{0m}},{{\hat \theta}_{\epsilon m}}(\tau )) + \epsilon({p_{0m}},{{\hat p}_{\epsilon m}}(\tau ))\nonumber \\ &&-\{ {({u_{\epsilon m}}(T),{{\hat u}_{\epsilon m}}(\tau ))+({\theta_{\epsilon m}}(T),{{\hat \theta}_{\epsilon m}}(\tau ))+ \epsilon({p _{\epsilon m}}(T),{{\hat p }_{\epsilon m}}(\tau ))} \}\exp( - 2{\rm i}\pi\tau T). \end{eqnarray}$

(3.17)

由 ${{\hat u}_{m}}(\tau )|_{\partial \Omega}=0$ , 得 $(\nabla\hat{p}_{\epsilon m},\hat{u}_{\epsilon m})+({\mbox{div}} {{\hat u}_{m}},{\hat p }_{\epsilon m})=0$ .运用(3.13)和(3.14)式, 由(3.17)式可推断出

$\begin{eqnarray*} &&2\pi |\tau |\{ {|{{\hat u}_{\epsilon m}(\tau )|^2}+|{\hat \theta}_{\epsilon m}(\tau )|^2+\epsilon|{{\hat p}_{\epsilon m}(\tau )|^2}} \}\\ &\leq& |\langle{{\hat \phi }_{\epsilon m}}(\tau ),{{\hat u}_{\epsilon m}}(\tau )\rangle |+|\langle{{\hat \psi }_{\epsilon m}}(\tau ),{{\hat \theta}_{\epsilon m}}(\tau )\rangle | + |{u_{0m}}||{{\hat u}_{\epsilon m}}(\tau )| +|\theta_{0m}||{\hat \theta}_{\epsilon m}(\tau)|\\ &&+\epsilon|{{p_{0m}}} || {{{\hat p}_{\epsilon m}}(\tau )} | + |{u_{\epsilon m}}(T)||{{\hat u}_{\epsilon m}}(\tau )| +|\theta_{\epsilon m}(T)||{{\hat \theta}_{\epsilon m}}(\tau )| +\epsilon|{p _{\epsilon m}}(T)||{{\hat p}_{\epsilon m}}(\tau )| \\ &\leq &| {\langle {{\hat \phi }_{\epsilon m}}(\tau ),{{\hat u}_{\epsilon m}}(\tau )\rangle } |+2\sqrt {{d_1}} (|{{\hat u}_{\epsilon m}}(\tau )|+|{{\hat \theta}_{\epsilon m}}(\tau )| + \epsilon|{{\hat p }_{\epsilon m}}(\tau )|). \end{eqnarray*}$

接下来对项 $|{\langle {{\hat \phi }_{\epsilon m}}(\tau ),{{\hat u}_{\epsilon m}}(\tau )\rangle }|$ 和 $|{\langle {{\hat \psi }_{\epsilon m}}(\tau ),{{\hat \theta}_{\epsilon m}}(\tau )\rangle }|$ 分别进行估计.事实上

$\begin{eqnarray} |{\langle {{\hat \phi }_{\epsilon m}}(\tau ),{{\hat u}_{\epsilon m}}(\tau )\rangle }| &=&|\langle\mu\triangle{\hat u}_{\epsilon m}-{\hat B}({\hat u}_{\epsilon m}),{\hat u}_{\epsilon m}(\tau)\rangle|\nonumber\\ &\leq&\mu\|{{\hat u}_{\epsilon m}}(\tau )\|^2+\hat{b}({{\hat u}_{\epsilon m}}(\tau ),{{\hat u}_{\epsilon m}}(\tau ),{{\hat u}_{\epsilon m}}(\tau )) \\ &=&\mu\|{{\hat u}_{\epsilon m}}(\tau )\|^2,\label{3.22}\\ \end{eqnarray}$

(3.18)

$\begin{eqnarray} |{\langle {{\hat \psi }_{\epsilon m}}(\tau ), {{\hat \theta}_{\epsilon m}}(\tau )\rangle }| &=&|\langle\kappa\triangle{\hat \theta}_{\epsilon m}-{\hat B}({\hat u}_{\epsilon m},{\hat \theta}_{\epsilon m}),{\hat \theta}_{\epsilon m}(\tau)\rangle| \\ &\leq&\kappa\|{{\hat \theta}_{\epsilon m}}(\tau )\|^2+\hat{b}({{\hat u}_{\epsilon m}}(\tau ),{{\hat \theta}_{\epsilon m}}(\tau ),{{\hat \theta}_{\epsilon m}}(\tau ))\nonumber\\ &=&\kappa\|{{\hat \theta}_{\epsilon m}}(\tau )\|^2.\label{3.23} \end{eqnarray}$

(3.19)

结合(3.17), (3.18)和(3.19)式, 有

$\begin{eqnarray*} &&2\pi |\tau| \{ |{{\hat u}_{\epsilon m}(\tau )|^2}+|{\hat \theta}_{\epsilon m} (\tau )|^2\}\\ &\leq& \mu|{{\hat u}_{\epsilon m}}(\tau )|^{2}+\kappa|{{\hat \theta}_{\epsilon m}}(\tau )|^{2}+2\sqrt {{d_1}} (|{{\hat u}_{\epsilon m}}(\tau )|+|{{\hat \theta}_{\epsilon m}}(\tau )|+\epsilon|{{\hat p }_{\epsilon m}}(\tau )|). \end{eqnarray*}$

对于给定的 $\gamma\in(0,\frac{1}{4})$ , 我们有

$\begin{eqnarray}|\tau|^{2\gamma}\leq (2\gamma+1)\frac{1+|\tau|}{1+|\tau|^{1-2\gamma}}, \forall\tau\in{\Bbb R}. \end{eqnarray}$

因此

$\begin{eqnarray}\label{3.24} &&\int^{+\infty}_{-\infty}|\tau|^{2\gamma}\{|{{\hat u}_{\epsilon m}}(\tau )|^{2}+|{{\hat \theta}_{\epsilon m}}(\tau )|^{2}\}{\rm d}\tau\nonumber\\ &\leq&(2\gamma+1)\int^{+\infty}_{-\infty}\frac{|{{\hat u}_{\epsilon m}}(\tau )|^{2}+|{{\hat \theta}_{\epsilon m}}(\tau )|^{2}}{1+|\tau|^{1-2\gamma}}{\rm d}\tau+(2\gamma+1)\int^{+\infty}_{-\infty}\frac{|\tau|(|{{\hat u}_{\epsilon m}}(\tau )|^{2}+|{{\hat \theta}_{\epsilon m}}(\tau )|^{2})}{1+|\tau|^{1-2\gamma}}{\rm d}\tau\nonumber\\ &\leq&(2\gamma+1)\int^{+\infty}_{-\infty}\{|{{\hat u}_{\epsilon m}}(\tau )|^{2}+|{{\hat \theta}_{\epsilon m}}(\tau )|^{2}\}{\rm d}\tau+(2\gamma+1)\int^{+\infty}_{-\infty}\frac{|\tau|(|{{\hat u}_{\epsilon m}}(\tau )|^{2}+|{{\hat \theta}_{\epsilon m}}(\tau )|^{2})}{1+|\tau|^{1-2\gamma}}{\rm d}\tau. \\ \end{eqnarray}$

(3.20)

由Parseval不等式和Poincaré不等式, 有

$\begin{eqnarray}\label{3.25} &&(2\gamma+1)\int^{+\infty}_{-\infty}\{|{{\hat u}_{\epsilon m}}(\tau )|^2 +|{{\hat \theta}_{\epsilon m}}(\tau )|^2\}{\rm d}\tau\\ &=&(2\gamma+1)\int^{+\infty}_{-\infty}\{|{{\tilde u}_{\epsilon m}}(t )|^{2}+|{{\tilde \theta}_{\epsilon m}}(t )|^{2}\}{\rm d}t =(2\gamma+1)\int^{T}_{0}\{|{ u_{\epsilon m}}(t )|^{2}+|{ \theta_{\epsilon m}}(t)|^{2}\}{\rm d}t \\ &\leq& \lambda^{-2}_{1}(2\gamma+1)\int^{T}_{0}\{\|{ u_{\epsilon m}}(t )\|^{2}+\|{ \theta_{\epsilon m}}(t )\|^{2}\}{\rm d}t \leq C(\mu,\kappa,d_1,\gamma), \end{eqnarray}$

(3.21)

且

$\begin{eqnarray}\label{3.26} &&(2\gamma+1)\int^{+\infty}_{-\infty}\frac{|\tau|(|{{\hat u}_{\epsilon m}}(\tau )|^{2}+|{{\hat \theta}_{\epsilon m}}(\tau )|^{2})}{1+|\tau|^{1-2\gamma}}{\rm d}\tau\nonumber\\ &\leq&\frac{2\gamma+1}{2\pi}\int^{+\infty}_{-\infty}\frac{\mu\|{{\hat u}_{\epsilon m}}(\tau )\|^2+\kappa\|{{\hat \theta}_{\epsilon m}}(\tau )\|^2+2\sqrt{d_1}(|{{\hat u}_{\epsilon m}}(\tau )|^{2}+|{{\hat \theta}_{\epsilon m}}(\tau )|^{2} +\epsilon |{{\hat p}_{\epsilon m}}(\tau )|^{2})} {1+|\tau|^{1-2\gamma}}{\rm d}\tau\nonumber\\ &=&\frac{\mu(2\gamma+1)}{2\pi}\int^{+\infty}_{-\infty} \frac{\|{{\hat u}_{\epsilon m}}(\tau )\|^{2}}{1+|\tau|^{1-2\gamma}}{\rm d}\tau +\frac{\kappa(2\gamma+1)}{2\pi}\int^{+\infty}_{-\infty} \frac{\|{{\hat \theta}_{\epsilon m}}(\tau )\|^2} {1+|\tau|^{1-2\gamma}}{\rm d}\tau\nonumber\\ &&+\frac{(2\gamma+1)\sqrt{d_1}}{\pi}\int^{+\infty}_{-\infty}\frac{|{\hat u}_{\epsilon m}(\tau )|+|{\hat \theta}_{\epsilon m}(\tau )|+\epsilon|{{\hat p}_{\epsilon m}}(\tau )|}{1+|\tau|^{1-2\gamma}}{\rm d}\tau\nonumber\\ &\leq&\frac{\mu(2\gamma+1)}{2\pi}\int^{+\infty}_{-\infty}\|{{\hat u}_{\epsilon m}}(\tau )\|^{2}{\rm d}\tau+\frac{\kappa(2\gamma+1)}{2\pi}\int^{+\infty}_{-\infty}\|{{\hat \theta}_{\epsilon m}}(\tau )\|^{2}{\rm d}\tau\nonumber\\ &&+\frac{(2\gamma+1)\sqrt{d_1}}{\pi}\int^{+\infty}_{-\infty}\frac{ |{\hat u}_{\epsilon m}(\tau)|+|{\hat u}_{\epsilon m}(\tau )|+\epsilon| {\hat p}_{\epsilon m}(\tau )|}{1+|\tau|^{1-2\gamma}}{\rm d}\tau\\ &\leq &C\int^{+\infty}_{-\infty}(\|{{\tilde u}_{\epsilon m}}(\tau)\|^{2}+\|{{\tilde \theta}_{\epsilon m}}(\tau)\|^{2}){\rm d}\tau\nonumber\\ &&+C\bigg(\int^{+\infty}_{-\infty}\frac{{\rm d}\tau}{(1+|\tau|^{1-2\gamma})^{2}} \bigg)^{\frac{1}{2}}\bigg(\int^{+\infty}_{-\infty}|{{\tilde u}_{\epsilon m}}(t )|^{2} +|{{\tilde \theta}_{\epsilon m}}(t )|^{2}+\epsilon|{{\tilde p}_{\epsilon m}} (t)|^{2}{\rm d}t\bigg)^{\frac{1}{2}}\nonumber\\ &=&C\int^{T}_{0}(\|{u_{\epsilon m}}(t)\|^{2}+\|{\theta_{\epsilon m}}(t)\|^{2}){\rm d}t\\ &&+C\bigg(\int^{+\infty}_{-\infty}\frac{{\rm d}\tau}{(1+|\tau|^{1-2\gamma})^{2}} \bigg)^{\frac{1}{2}}\bigg(\int^{T}_{0}|{u_{\epsilon m}}(t )|^{2}+|\theta_{\epsilon m}(t)|^2+\epsilon|{p_{\epsilon m}}(t)|^{2}{\rm d}t \bigg)^{\frac{1}{2}}\nonumber\\ &\leq& C(\pi,\gamma,d_1,\mu,\kappa,T). \end{eqnarray}$

(3.22)

对于任一的 $ \gamma\in(0,\frac{1}{4})$ , 有

$\begin{eqnarray} \int^{+\infty}_{-\infty}\frac{{\rm d}\tau}{(1+|\tau|^{1-2\gamma})^{2}}<\infty. \end{eqnarray}$

我们可以推出

$\begin{eqnarray} \int^{+\infty}_{-\infty}|\tau|^{2\gamma}(|{{\hat u}_{\epsilon m}}(\tau )|^{2}+|{{\hat \theta}_{\epsilon m}}(\tau )|^{2}){\rm d}\tau\leq C, \mbox{对某个} \gamma\in(0,\frac{1}{4}). \end{eqnarray}$

当 $m\rightarrow\infty$ 时, 运用(3.13)-(3.16)式和(3.22)式, 对(3.8)-(3.10)式中关于 $m$ 取极限, 对于给定的 $\epsilon\in(0,1]$ , 我们只考虑当 $m\rightarrow\infty$ 时取极限的过程.存在着一个序列 $m'\rightarrow\infty$ , 和 $\{v_\epsilon,q_\epsilon,\varphi_\epsilon,p_\epsilon\}$ 满足

$\begin{eqnarray} u_{\epsilon m'} \mbox{在} L^{2}(0,T;{{\Bbb H}}^{1}_{0}(\Omega)) \mbox{中弱收敛于} u_\epsilon, \end{eqnarray}$

$\begin{eqnarray} u_{\epsilon m'} \mbox{在} L^{\infty}(0,T;{\Bbb L}^{2}(\Omega)) \mbox{中弱*收敛于} u_\epsilon, \end{eqnarray}$

$\begin{eqnarray} u_{\epsilon m'} \mbox{在} L^{2}(0,T;{\Bbb L}^{2}(\Omega)) \mbox{中强收敛于} u_\epsilon, \end{eqnarray}$

$\begin{eqnarray} \theta_{\epsilon m'} \mbox{在} L^{2}(0,T;H^{1}_{0}(\Omega)) \mbox{中弱收敛于} \theta_\epsilon, \end{eqnarray}$

$\begin{eqnarray} \theta_{\epsilon m'} \mbox{在} L^{\infty}(0,T;L^{2}(\Omega)) \mbox{中弱*收敛于} \theta_\epsilon, \end{eqnarray}$

$\begin{eqnarray} \theta_{\epsilon m'} \mbox{在} L^{2}(0,T;L^{2}(\Omega)) \mbox{中强收敛于} \theta_\epsilon, \end{eqnarray}$

$\begin{eqnarray} p_{\epsilon m'} \mbox{在} L^{\infty}(0,T;L^{2}(\Omega)) \mbox{中弱*收敛于} p_\epsilon. \end{eqnarray}$

取 $\psi(t)\in {\cal C}^{\infty}_{c}(0,T)$ , 我们用 $\psi(t)$ 乘以(3.8) (resp. (3.9)或(3.10))式, 然后在 $(0,T)$ 上积分, 得

$\begin{eqnarray} &&-\int^{T}_{0}(u_{\epsilon m'}(t),\omega_{k}\psi'(t)){\rm d}t +\int^{T}_{0}\hat{b}(u_{\epsilon m'}(t),u_{\epsilon m'}(t),\omega_{k}\psi(t)){\rm d}t \\ &&+\int^{T}_{0}\mu((u_{\epsilon m'}(t),\omega_{k}\psi(t))){\rm d}t +\int^{T}_{0}(\nabla p_{\epsilon m'},\omega_{k}\psi(t)){\rm d}t=0,\label{3.27}\\ \end{eqnarray}$

(3.23)

$\begin{eqnarray} &&-\int^{T}_{0}(\theta_{\epsilon m'}(t),\gamma_{j}\psi'(t)){\rm d}t +\int^{T}_{0}\hat{b}(u_{\epsilon m'}(t),\theta_{\epsilon m'}(t), \gamma_{j}\psi(t)){\rm d}t\\ &&+\int^{T}_{0}\kappa((\theta_{\epsilon m'}(t),\gamma_{j}\psi(t))){\rm d}t =0,\label{3.28} \end{eqnarray}$

(3.24)

$\begin{eqnarray} -\epsilon\int^{T}_{0}(p_{\epsilon m'}(t),\gamma_{j}\psi'(t)){\rm d}t +\int^{T}_{0}( \mbox{div} u_{\epsilon m'}(t),\gamma_{j}\psi(t)){\rm d}t=0, 1\leq k,l\leq m.\label{3.29} \end{eqnarray}$

(3.25)

接下来, 我们逐个考虑(3.23)和(3.24)式中非线性项的收敛性.首先

$\begin{eqnarray}\label{3.30} &&\bigg|\int^{T}_{0}\hat{b}(u_{\epsilon m'}(t),u_{\epsilon m'}(t),\omega_{k}\psi(t)){\rm d}t-\int^{T}_{0}\hat{b}(u_{\epsilon}(t),u_{\epsilon}(t),\omega_{k}\psi(t)){\rm d}t\bigg|\nonumber\\ &\leq&\frac{1}{2}\bigg|\int^{T}_{0}b(u_{\epsilon m'}(t),u_{\epsilon m'}(t),\omega_{k}\psi(t)){\rm d}t-\int^{T}_{0}b(u_{\epsilon}(t),u_{\epsilon}(t),\omega_{k}\psi(t)){\rm d}t\bigg|\nonumber\\ &&+\frac{1}{2}\bigg|\int^{T}_{0}b(u_{\epsilon m'}(t),\omega_{k}\psi(t),u_{\epsilon m'}(t)){\rm d}t-\int^{T}_{0}b(u_{\epsilon}(t),\omega_{k}\psi(t),u_{\epsilon}(t)){\rm d}t\bigg|\nonumber\\ &\leq& \frac{1}{2}\bigg|\int^{T}_{0}b(u_{\epsilon m'}(t)-u_{\epsilon}(t),u_{\epsilon m'}(t),\omega_{k}\psi(t)){\rm d}t|+\frac{1}{2} \bigg|\int^{T}_{0}b(u_{\epsilon}(t),u_{\epsilon m'}(t)-u_{\epsilon}(t), \omega_{k}\psi(t)){\rm d}t\bigg|\nonumber\\ &&+\frac{1}{2}\bigg|\int^{T}_{0}b(u_{\epsilon m'}(t)-u_{\epsilon}(t), \omega_{k}\psi(t),u_{\epsilon m'}(t)){\rm d}t\bigg| +\frac{1}{2}\bigg|\int^{T}_{0}b(u_{\epsilon}(t),\omega_{k}\psi(t),u_{\epsilon m'}(t)-u_{\epsilon}(t)){\rm d}t\bigg|\nonumber\\ &\triangleq &b_1+b_2+b_3+b_4, \end{eqnarray}$

(3.26)

其中, 当 $m'\rightarrow 0$ 时, 有

$\begin{eqnarray} b_1&=&\frac{1}{2}\bigg|\int^{T}_{0}b(u_{\epsilon m'}(t)-u_{\epsilon}(t),u_{\epsilon m'}(t),\omega_{k}\psi(t)){\rm d}t\bigg|\nonumber\\ &\leq&\frac{1}{2}\sup\limits_{t\in(0,T)}|\psi(t)|\sup\limits_{x\in\Omega}|\omega_k|\int^{T}_{0}|u_{\epsilon m'}(t)-u_{\epsilon}(t)||\nabla u_{\epsilon m'}(t)|{\rm d}t\nonumber\\ &\leq& C\|u_{\epsilon m'}(t)-u_{\epsilon}(t)\|_{L^{2}(0,T;{\Bbb L}^{2}(\Omega))}\|u_{\epsilon m'}(t)\|_{L^{2}(0,T;{{\Bbb H}}^{1}_0(\Omega))}\rightarrow 0,\label{3.31} \end{eqnarray}$

(3.27)

$\begin{eqnarray} b_2&=&\frac{1}{2} \bigg|\int^{T}_{0}b(u_{\epsilon}(t),u_{\epsilon m'}(t)-u_{\epsilon}(t),\omega_{k}\psi(t)){\rm d}t\bigg|\nonumber\\ &=&\frac{1}{2}\bigg|\int^{T}_{0}\mbox{div} u_\epsilon(u_{\epsilon m'}(t) -u_{\epsilon}(t))\omega_{k}\psi(t)){\rm d}t\bigg| +\frac{1}{2}\bigg|\int^{T}_{0}b(u_{\epsilon}(t),\omega_{k}\psi(t),u_{\epsilon m'}(t)-u_{\epsilon}(t)){\rm d}t \bigg|\nonumber\\ &\leq& \frac{1}{2}\sup\limits_{t\in(0,T)}|\psi(t)|\sup\limits_{x\in\Omega}|\omega_{k} |\int^{T}_{0}|u_{\epsilon m'}(t)-u_{\epsilon}(t)||\nabla u_{\epsilon}(t)|{\rm d}t\nonumber\\ &&+\frac{1}{2}\sup\limits_{t\in(0,T)}|\psi(t)|\sup\limits_{x\in\Omega}|\nabla\omega_{k}|\int^{T}_{0}|u_{\epsilon}(t)||u_{\epsilon m'}(t)-u_{\epsilon}(t)|{\rm d}t\nonumber\\ &\leq &C\|u_{\epsilon m'}(t)-u_{\epsilon}(t)\|_{L^{2}(0,T;{\Bbb L}^{2}(\Omega))}(\|u_{\epsilon}(t)\|_{L^{2}(0,T;{\Bbb H}^{1}_0(\Omega))}+\|u_{\epsilon}(t)\|_{L^{2}(0,T;{\Bbb L}^{2}(\Omega))}) \rightarrow 0,\label{3.32} \end{eqnarray}$

(3.28)

$\begin{eqnarray} b_3&=&\frac{1}{2}\bigg|\int^{T}_{0}b(u_{\epsilon m'}(t)-u_{\epsilon}(t),\omega_{k}\psi(t),u_{\epsilon m'}(t)){\rm d}t\bigg|\nonumber\\ &\leq &\frac{1}{2}\sup\limits_{t\in(0,T)}|\psi(t)|\sup\limits_{x\in\Omega}|\nabla\omega_{k}|\int^{T}_{0}|u_{\epsilon m'}(t)-u_{\epsilon}(t)||u_{\epsilon m'}(t)|{\rm d}t\nonumber\\ &\leq &C\|u_{\epsilon m'}(t)-u_{\epsilon}(t)\|_{L^{2}(0,T;{\Bbb L}^{2}(\Omega))}\|u_{\epsilon m'}(t)\|_{L^{2}(0,T;{\Bbb L}^{2}(\Omega))}\rightarrow 0,\label{3.33} \end{eqnarray}$

(3.29)

$\begin{eqnarray} b_4&=&\frac{1}{2}\bigg|\int^{T}_{0}b(u_{\epsilon}(t),\omega_{k}\psi(t),u_{\epsilon m'}(t)-u_{\epsilon}(t)){\rm d}t\bigg|\nonumber\\ &=&\frac{1}{2}\sup\limits_{t\in(0,T)}|\psi(t)|\sup\limits_{x\in\Omega}|\nabla\omega_{k}|\int^{T}_{0}|u_{\epsilon m'}(t)-u_{\epsilon}(t)||u_{\epsilon}(t)|{\rm d}t\nonumber\\ &\leq& C\|u_{\epsilon}(t)\|_{L^{2}(0,T;{\Bbb L}^{2}(\Omega))}\|u_{\epsilon m'}(t)-u_{\epsilon}(t)\|_{L^{2}(0,T;{\Bbb L}^{2}(\Omega))}\rightarrow 0.\label{3.34} \end{eqnarray}$

(3.0)

由(3.26)-(3.30)式, 得

$\begin{eqnarray}\label{3.35} \int^{T}_{0}\hat{b}(u_{\epsilon m'}(t),u_{\epsilon m'}(t),\omega_{k}\psi(t)){\rm d}t\rightarrow \int^{T}_{0}\hat{b}(u_{\epsilon}(t),u_{\epsilon}(t),\omega_{k}\psi(t)){\rm d}t, \end{eqnarray}$

(3.31)

同理, 我们可以推出

$\begin{eqnarray}\label{3.36} \int^{T}_{0}\hat{b}(u_{\epsilon m'}(t),\theta_{\epsilon m'}(t),w_{k}\psi(t)){\rm d}t\rightarrow \int^{T}_{0}\hat{b}(u_{\epsilon}(t),\theta_{\epsilon}(t),w_{k}\psi(t)){\rm d}t, \end{eqnarray}$

(3.32)

当 $m'\rightarrow\infty$ 时, 对(3.23)-(3.25)式取极限(其中 $ 1\leq k,j\leq m$ ), 有

$\begin{eqnarray} &&-\int^{T}_{0}(u_{\epsilon}(t),\omega_{k}\psi'(t)){\rm d}t +\int^{T}_{0}\hat{b}(u_{\epsilon}(t),u_{\epsilon}(t),\omega_{k}\psi(t)){\rm d}t \end{eqnarray}$

(3.33)

$\begin{eqnarray} &&+\int^{T}_{0}\mu((u_{\epsilon}(t),\omega_{k}\psi(t))){\rm d}t +\int^{T}_{0}(\nabla p_{\epsilon},\omega_{k}\psi(t)){\rm d}t=0,\label{3.37}\\ &&-\int^{T}_{0}(\theta_{\epsilon}(t),\gamma_{j}\psi'(t)){\rm d}t +\int^{T}_{0}\hat{b}(u_{\epsilon}(t),\theta_{\epsilon}(t),\gamma_{j}\psi(t)){\rm d}t \end{eqnarray}$

(3.34)

$\begin{eqnarray} &&+ \int^{T}_{0}\kappa((\theta_{\epsilon }(t),\gamma_{j}\psi(t))){\rm d}t=0,\label{3.38}\\ &&-\epsilon\int^{T}_{0}(p_{\epsilon}(t),\gamma_{j}\psi'(t)){\rm d}t +\int^{T}_{0}( \mbox{div} u_{\epsilon }(t),\gamma_{j}\psi(t)){\rm d}t=0.\label{3.39} \end{eqnarray}$

(3.35)

注意到

$\begin{eqnarray*} &&\int^{T}_{0}\hat{b}(u_{\epsilon}(t),u_{\epsilon}(t),\omega_{k}\psi(t)){\rm d}t\\ &\leq& \frac{1}{2}\bigg|\int_0^T b(u_{\epsilon}(t),u_{\epsilon}(t), \omega_{k}\psi(t)){\rm d}t\bigg|+ \frac{1}{2} \bigg|\int_0^T b(u_{\epsilon}(t),\omega_{k}\psi(t),u_{\epsilon}(t)){\rm d}t\bigg|\\ &\leq& \frac{1}{2}\sup\limits_{t\in (0,T)}|\psi(t) \bigg\{ \int_0^T\|u_{\epsilon}(t) \|_{L^4(\Omega)}|\nabla u_{\epsilon}(t) | \|\omega_{k} \|_{L^4(\Omega)}{\rm d}t + \int_0^T\|u_{\epsilon}(t) \|^2_{L^4(\Omega)}|\nabla\omega_{k} |{\rm d}t \bigg\}\\ &\leq &C\| u_{\epsilon}\|^2_{L^2(0,T; {\Bbb H}_0^1(\Omega))}\|\omega_{k}\|. \end{eqnarray*}$

同理

$\begin{eqnarray} \int^{T}_{0}\hat{b}(u_{\epsilon}(t),\theta_{\epsilon}(t),\gamma_{j}\psi(t)){\rm d}t\leq C\| u_{\epsilon}\|_{L^2(0,T; {\Bbb H}_0^1(\Omega))}\| \theta_{\epsilon}\|_{L^2(0,T; H_0^1(\Omega))} \|\omega_{k}\|. \end{eqnarray}$

(3.36)

因此, (3.33) (resp. (3.34)或(3.35))式对于任意 $\omega_{k}$ (resp. w $_{k}$ 或 $\gamma_l$ )的有限线性组合 $\omega$ (resp. w或 $\gamma$ )成立.由连续性知, 对于任意的 $\omega\in{{\Bbb H}^{1}_{0}}(\Omega)$ , (3.33)式成立, 对于任意的 $w\in H^1_0(\Omega)$ , (3.34)式成立, 对于任意的 $\gamma\in L^{2}(\Omega)$ , (3.35)式成立.因此

$\begin{eqnarray} &&-\int^{T}_{0}(u_{\epsilon}(t),\omega\psi'(t)){\rm d}t +\int^{T}_{0}\hat{b}(u_{\epsilon}(t),u_{\epsilon}(t),\omega\psi(t)){\rm d}t\\ &&+\int^{T}_{0}\mu((u_{\epsilon}(t),\omega\psi(t))){\rm d}t +\int^{T}_{0}(\nabla p_{\epsilon},\omega\psi(t)){\rm d}t=0, \forall \omega\in{{\Bbb H}^{1}_{0}}(\Omega),\label{3.40}\\ \end{eqnarray}$

(3.37)

$\begin{eqnarray} &&-\int^{T}_{0}(\theta_{\epsilon}(t),w\psi'(t)){\rm d}t +\int^{T}_{0}\hat{b}(u_{\epsilon}(t),\theta_{\epsilon}(t),w\psi(t)){\rm d}t\\ &&+\int^{T}_{0}\kappa((\theta_{\epsilon }(t),w\psi(t))){\rm d}t=0, \forall w\in H^1_0(\Omega),\label{3.41}\\ \end{eqnarray}$

(3.38)

$\begin{eqnarray} &&-\epsilon\int^{T}_{0}(p_{\epsilon}(t),\gamma\psi'(t)){\rm d}t+\int^{T}_{0} (\mbox{div} u_{\epsilon }(t),\gamma\psi(t)){\rm d}t=0, \forall \gamma\in L^{2}(\Omega).\label{3.42} \end{eqnarray}$

(3.39)

方程(3.37)-(3.39)证明了 $\{u_{\epsilon},\theta_{\epsilon}, p_{\epsilon}\}$ 在分布意义下满足(3.4)-(3.6)式.接下来证明 $u_{\epsilon}$ , $\theta_{\epsilon}$ 和 $p_{\epsilon}$ 满足(3.7)式.为此, 我们取 $\psi(t)\in{{\cal C}^{\infty}_{c}}([0,T])$ 且 $\psi(T)=0$ , 并用 $\psi(t)$ 分别乘以方程(3.4)-(3.6), 然后在 $[0,T]$ 上积分, 对于首项利用分部积分可得

$\begin{eqnarray} &&-\int^{T}_{0}(u_{\epsilon}(t),\omega\psi'(t)){\rm d}t+\int^{T}_{0}\hat{b}(u_{\epsilon}(t),u_{\epsilon}(t),\omega\psi(t)){\rm d}t+\int^{T}_{0}\mu((u_{\epsilon}(t),\omega\psi(t))){\rm d}t\nonumber\\ &&+\int^{T}_{0}(\nabla p_{\epsilon},\omega\psi(t)){\rm d}t =(u_\epsilon(0),\omega)\psi(0), \forall \omega\in{{\Bbb H}^{1}_{0}}(\Omega),\label{3.43}\\ \end{eqnarray}$

(3.40)

$\begin{eqnarray} &&-\int^{T}_{0}(\theta_{\epsilon}(t),w\psi'(t)){\rm d}t+\int^{T}_{0}\hat{b}(u_{\epsilon}(t),\theta_{\epsilon}(t),w\psi(t)){\rm d}t+\int^{T}_{0}\kappa((\theta_{\epsilon }(t),w\psi(t))){\rm d}t\nonumber\\ &=&(\theta_\epsilon(0),w)\psi(0), \forall w\in H^1_0(\Omega),\label{3.44} \end{eqnarray}$

(3.41)

$\begin{eqnarray} -\epsilon\int^{T}_{0}(p_{\epsilon}(t),\gamma\psi'(t)){\rm d}t+\int^{T}_{0} ({\rm div} u_{\epsilon }(t),\gamma\psi(t)){\rm d}t =\epsilon(p_\epsilon(0),\gamma)\psi(0), \forall \gamma\in L^{2}(\Omega).\label{3.45} \end{eqnarray}$

(3.42)

通过比较(3.37)式与(3.40)式, (3.38)式与(3.41)式, (3.39)式与(3.42)式, 可得

$\begin{eqnarray*} &&(u_{\epsilon}(0)-u_0,\omega)\psi(0)=0 ,\ \forall \omega\in{{\Bbb H}^{1}_{0}}(\Omega),\\ &&(\theta_{\epsilon}(0)-\theta_0,w)\psi(0)=0,\ \forall w\in H^1_0(\Omega),\\ &&(p_{\epsilon}(0)-p_0,\gamma)\psi(0)=0,\ \forall \gamma\in L^{2}(\Omega). \end{eqnarray*}$

我们选取 $\psi(0)\neq0$ 可得

$\begin{eqnarray*} &&(u_{\epsilon}(0)-u_0,\omega)=0,\ \forall\omega\in{{\Bbb H}^{1}_{0}}(\Omega),\\ &&(\theta_{\epsilon}(0)-\theta_0,w)=0,\ \forall w\in H^1_0(\Omega),\\ &&(p_{\epsilon}(0)-p_0,\gamma)=0 ,\ \forall\gamma\in L^{2}(\Omega). \end{eqnarray*}$

因此, (3.7)式成立.

在本节中, 我们证明扰动的可压Navier-Stokes-Fourier方程的解收敛到不可压方程的解.

定义4.1 设 $u_{0}\in {\Bbb H},\theta_{0}\in L^2(\Omega).$ 对于任意的 $T>0$ , 系统(1.1)的弱解 $(u,\theta)$ 满足

$\begin{eqnarray} u\in L^{2}(0,T;{\Bbb V})\cap L^{\infty}(0,T;{\Bbb H}), \mbox{且} u'\in L^{\frac{4}{3}}(0,T;{\Bbb V}'), \end{eqnarray}$

$\begin{eqnarray} \theta\in L^{2}(0,T;H^1_0)\cap L^{\infty}(0,T;L^2), \mbox{且} \theta'\in L^{\frac{4}{3}}(0,T;H^{-1}(\Omega)), \end{eqnarray}$

$\begin{eqnarray} \frac{\partial u}{\partial t} +\mu{\cal A} u +B(u,u)=0 \mbox{在 $L^{\frac{4}{3}}(0,T;{\Bbb V}')$中成立}, \end{eqnarray}$

$\begin{eqnarray} \frac{\partial\theta}{\partial t} -\kappa\Delta\theta +(u\cdot\nabla)\theta=0 \mbox{在$ L^{\frac{4}{3}}(0,T; H^{-1}(\Omega)) $中成立}, \end{eqnarray}$

$\begin{eqnarray} u(x,0)=u_{0},\theta(x,0)=\theta_{0}. \end{eqnarray}$

由参考文献[5]知, 对于给定的 $\mu,\kappa>0$ 且任一 $T>0$ , 问题(1.1)至少存在一个弱解.

定理4.1 对于任意的 $\epsilon\in(0,1]$ , $u_0\in {\Bbb H},\theta_0,p_0\in L^{2}(\Omega)$ , 当 $\epsilon\rightarrow 0^{+}$ 时, 问题(1.2)-(1.4)的解 $\{u_{\epsilon},\theta_{\epsilon},p_{\epsilon}\}$ 按以下意义收敛到问题(1.1)的解 $\{u,\theta,p\}$ .

$\begin{eqnarray} u_{\epsilon}\rightarrow u, \mbox{在$ L^{\infty}(0,T;{\Bbb L}^{2}(\Omega))$中弱*收敛},\label{4.1} \end{eqnarray}$

(4.1)

$\begin{eqnarray} \theta_{\epsilon}\rightarrow \theta, \mbox{在$ L^{\infty}(0,T;L^{2}(\Omega))$中弱收敛},\label{4.2} \end{eqnarray}$

(4.2)

$\begin{eqnarray} u_{\epsilon}\rightarrow u, \mbox{在$ L^{2}(0,T;{{\Bbb H}^{1}_{0}}(\Omega)) $中弱收敛},\label{4.3} \end{eqnarray}$

(4.3)

$\begin{eqnarray} \theta_{\epsilon}\rightarrow \theta , \mbox{在$ L^{2}(0,T;{H^{1}_{0}}(\Omega)) $中弱收敛},\label{4.4} \end{eqnarray}$

(4.4)

$\begin{eqnarray} u_{\epsilon}\rightarrow u , \mbox{在$ L^{2}(0,T;{\Bbb L}^{2}(\Omega)) $中强收敛},\label{4.5} \end{eqnarray}$

(4.5)

$\begin{eqnarray} \theta_{\epsilon}\rightarrow \theta , \mbox{在$ L^{2}(0,T;L^2(\Omega)) $中强收敛},\label{4.6} \end{eqnarray}$

(4.6)

$\begin{eqnarray} \nabla p_\epsilon\rightarrow \nabla p ,\mbox{在$ L^{\frac{4}{3}}(0,T;{\Bbb H}^{-1})$中弱收敛}.\label{4.7} \end{eqnarray}$

(4.7)

证 由(3.14)-(3.16)式, (3.20)-(3.22)式, 对于任一的$\epsilon\in(0, 1]$, 我们有

$\begin{eqnarray} \|u_{\epsilon}\|_{L^{\infty}(0,T;{\Bbb L}^{2}(\Omega))}\leq\liminf\limits_{m\rightarrow\infty}\|u_{\epsilon m}\|_{L^{\infty}(0,T;{\Bbb L}^{2}(\Omega))}\leq\sqrt{d_1},\label{4.8} \end{eqnarray}$

(4.8)

$\begin{eqnarray} \|\theta_{\epsilon}\|_{L^{\infty}(0,T;L^{2}(\Omega))}\leq\liminf\limits_{m\rightarrow\infty}\|\theta_{\epsilon m}\|_{L^{\infty}(0,T;L^{2}(\Omega))}\leq\sqrt{d_1},\label{4.9} \end{eqnarray}$

(4.9)

$\begin{eqnarray} \|u_{\epsilon}\|_{L^{2}(0,T;{{\Bbb H}^{1}_{0}}(\Omega))}\leq\liminf\limits_{m\rightarrow\infty}\|u_{\epsilon m}\|_{L^{2}(0,T;{{\Bbb H}^{1}_{0}}(\Omega))}\leq\sqrt{\frac{d_1}{2\mu}},\label{4.10} \end{eqnarray}$

(4.10)

$\begin{eqnarray} \|\theta_{\epsilon}\|_{L^{2}(0,T;{H^{1}_{0}}(\Omega))}\leq\liminf\limits_{m\rightarrow\infty}\|\theta_{\epsilon m}\|_{L^{2}(0,T;{H^{1}_{0}}(\Omega))}\leq\sqrt{\frac{d_1}{2\kappa}},\label{4.11} \end{eqnarray}$

(4.11)

$\begin{eqnarray} \sqrt{\epsilon}\|p_\epsilon\|_{L^{\infty}(0,T;L^{2}(\Omega))}\leq\liminf\limits_{m\rightarrow\infty}\sqrt{\epsilon}\|p_\epsilon m\|_{L^{\infty}(0,T;L^{2}(\Omega))}\leq\sqrt{d_1},\label{4.12} \end{eqnarray}$

(4.12)

$\begin{eqnarray} \int^{+\infty}_{-\infty}|\tau|^{2\gamma}(|{\hat{u}_{\epsilon}}(\tau)|^{2}+|{\hat{\theta}_{\epsilon}}(\tau)|^{2}){\rm d}\tau\leq\liminf\limits_{m\rightarrow\infty}\int^{+\infty}_{-\infty}|\tau|^{2\gamma} (|{\hat{u}_{\epsilon m}}(\tau)|^{2}+|{\hat{\theta}_{\epsilon m}}(\tau)|^{2}){\rm d}\tau\leq C.\label{4.13} \end{eqnarray}$

(4.13)

由(4.8)-(4.13)式, 存在序列 $\{\epsilon_m\}\subset (0,1]$ ( $\epsilon_m\rightarrow 0^{+}$ 当 $m\rightarrow\infty$ 时) $u_{*}\in L^{\infty}(0,T;{\Bbb L}^{2}(\Omega)) \cap L^{2}(0,T;{{\Bbb H}^{1}_{0}}(\Omega))$ , $\theta_*\in L^{\infty}(0,T;L^{2}(\Omega)) \cap L^{2}(0,T;H^{1}_{0}(\Omega))$ 且 $p_*\in L^{\infty}(0,T;L^{2}(\Omega))$ , 当 $\epsilon_m\rightarrow 0^{+}$ 时满足

$\begin{eqnarray} u_{\epsilon_m}\rightarrow u_{*}, \mbox{在$ L^{\infty}(0,T;{\Bbb L}^{2}(\Omega)) $中弱*收敛},\label{4.14} \end{eqnarray}$

(4.14)

$\begin{eqnarray} \theta_{\epsilon_m}\rightarrow \theta_{*} , \mbox{在$ L^{\infty}(0,T;L^{2}(\Omega)) $中弱*收敛},\label{4.15} \end{eqnarray}$

(4.15)

$\begin{eqnarray} u_{\epsilon_m}\rightarrow u_{*} , \mbox{在$ L^{2}(0,T;{{\Bbb H}^{1}_{0}}(\Omega)) $中弱收敛},\label{4.16} \end{eqnarray}$

(4.16)

$\begin{eqnarray} \theta_{\epsilon_m}\rightarrow \theta_{*} , \mbox{在$ L^{2}(0,T;{H^{1}_{0}}(\Omega)) $中弱收敛},\label{4.17} \end{eqnarray}$

(4.17)

$\begin{eqnarray} u_{\epsilon_m}\rightarrow u_{*}, \mbox{在$ L^{2}(0,T;{\Bbb L}^{2}(\Omega)) $中强收敛},\label{4.18} \end{eqnarray}$

(4.18)

$\begin{eqnarray} \theta_{\epsilon_m}\rightarrow \theta_{*} , \mbox{在$ L^{2}(0,T;L^{2}(\Omega)) $中强收敛},\label{4.19} \end{eqnarray}$

(4.19)

$\begin{eqnarray} \sqrt{\epsilon_m}p_{\epsilon_m}\rightarrow\chi , \mbox{在$ L^{\infty}(0,T;L^{2}(\Omega)) $中弱*收敛}.\label{4.20} \end{eqnarray}$

(4.20)

当 $\epsilon_m\rightarrow 0^{+}$ 时, 对(3.6)式在分布意义下取极限, 得

$\begin{eqnarray}\label{4.21} \epsilon_m\frac{\rm d}{{\rm d}t}(p_{\epsilon_m},q)=\sqrt{\epsilon_m}\sqrt{\epsilon_m}\frac{\rm d}{{\rm d}t}(p_{\epsilon_m},q)\rightarrow 0,\forall q\in L^{2}(\Omega). \end{eqnarray}$

(4.21)

结合(4.21)和(3.6)式有

$\begin{eqnarray*} (\mbox{div} u_{*},q)=0,\forall q\in L^{2}(\Omega). \end{eqnarray*}$

从上式可以推断出 $\mbox{div} u_{*}=0$ , 从而有

$\begin{eqnarray} u_{*}\in L^{\infty}(0,T;{\Bbb L}^{2}(\Omega)) \cap L^{2}(0,T;{{\Bbb H}^{1}_{0}}(\Omega)), \end{eqnarray}$

$\begin{eqnarray} \theta_{*}\in L^{\infty}(0,T;L^2(\Omega)) \cap L^{2}(0,T;H^{1}_{0}(\Omega)). \end{eqnarray}$

现设 $\omega\in{\cal V}$ , $w\in{\cal W}$ .考虑到

$\begin{eqnarray} (\nabla p_{\epsilon},\omega)=(p_\epsilon,\mbox{div} \omega)=0. \end{eqnarray}$

方程(3.4)和(3.5)可写为

$\begin{eqnarray} \frac{\rm d}{{\rm d}t}\langle u_\epsilon,\omega\rangle-\langle \mu\triangle u_{\epsilon}(t),\omega\rangle+\hat{b}(u_{\epsilon},u_{\epsilon},\omega)=0,\label{4.22} \end{eqnarray}$

(4.22)

$\begin{eqnarray} \frac{\rm d}{{\rm d}t}\langle \theta_\epsilon,w\rangle-\langle\kappa\triangle \theta_{\epsilon}(t),w\rangle+\hat{b}(u_{\epsilon},\theta_{\epsilon},w)=0.\label{4.23} \end{eqnarray}$

(4.23)

若 $\psi$ 是 $[0,T]$ 上的一个连续可微的标量函数, 且 $\psi(T)=0$ , 我们用 $\psi(t)$ 分别乘以方程(4.22), (4.23), 再关于 $t$ 进行积分.则由分部积分可得

$\begin{eqnarray} &&-\int^{T}_{0}(u_{\epsilon}(t),\omega\psi'(t)){\rm d}t +\int^{T}_{0}\hat{b}(u_{\epsilon}(t),u_{\epsilon}(t),\omega\psi(t)){\rm d}t \\ &&+\int^{T}_{0}\mu((u_{\epsilon}(t),\omega\psi(t))){\rm d}t =(u_0,\omega)\psi(0),\label{4.24}\\ \end{eqnarray}$

(4.24)

$\begin{eqnarray} &&-\int^{T}_{0}(u_{\epsilon}(t),\omega\psi'(t)){\rm d}t +\int^{T}_{0}\hat{b}(u_{\epsilon}(t),u_{\epsilon}(t),\omega\psi(t)){\rm d}t \\ &&-\int^{T}_{0}(\theta_{\epsilon}(t),w\psi'(t)){\rm d}t+\int^{T}_{0}\hat{b}(u_{\epsilon}(t),\theta_{\epsilon}(t), w\psi(t)){\rm d}t\\ &&+\int^{T}_{0}\kappa((\theta_{\epsilon }(t),w\psi(t))){\rm d}t =(\theta_0,w)\psi(0).\label{4.25} \end{eqnarray}$

(4.25)

类似于(3.31)-(3.32)式的推导, 利用(4.14)-(4.20)式的收敛性可得

$\begin{eqnarray} \int^{T}_{0}\hat{b}(u_{\epsilon m}(t),u_{\epsilon m}(t),\omega_{k}\psi(t)){\rm d}t\rightarrow \int^{T}_{0}\hat{b}(u_*(t),u_*(t),\omega_{k}\psi(t)){\rm d}t, \end{eqnarray}$

$\begin{eqnarray} \int^{T}_{0}\hat{b}(u_{\epsilon m}(t),\theta_{\epsilon m}(t),w_{k}\psi(t)){\rm d}t\rightarrow \int^{T}_{0}\hat{b}(u_*(t),\theta_*(t),w_{k}\psi(t)){\rm d}t. \end{eqnarray}$

注意到 $\mbox{div} u_{*}=0$ 且

$\begin{eqnarray}\hat{b}(u_{\epsilon}(t),\theta_{\epsilon}(t),\omega\psi(t))=b(u_{\epsilon}(t),\theta_{\epsilon}(t),\omega\psi(t)),\end{eqnarray}$

$\begin{eqnarray}\hat{b}(u_{\epsilon}(t),u_{\epsilon}(t),\omega\psi(t))=b(u_{\epsilon}(t),u_{\epsilon}(t),\omega\psi(t)).\end{eqnarray}$

在(4.24)和(4.25)式中取极限, 可得

$\begin{eqnarray} &&-\int^{T}_{0}(u_*(t),\omega\psi'(t)){\rm d}t+\int^{T}_{0}\hat{b}(u_*,u_*(t),\omega\psi(t)){\rm d}t\\ &&+\int^{T}_{0}\mu((u_*(t),\omega\psi(t))){\rm d}t =(u_0,\omega)\psi(0),\label{4.26}\\ \end{eqnarray}$

(4.26)

$\begin{eqnarray} &&-\int^{T}_{0}(\theta_*(t),w\psi'(t)){\rm d}t+\int^{T}_{0}\hat{b}(u_*(t),\theta_*(t),w\psi(t)){\rm d}t\\ &&+\int^{T}_{0}\kappa((\theta_*(t),w\psi(t))){\rm d}t =(\theta_0,w)\psi(0).\label{4.27} \end{eqnarray}$

(4.27)

易证明 $\{u_{*},\theta_{*}\}$ 是问题(1.1)的解.接下来证明(4.1)-(4.7)式的收敛性.注意到(4.1)-(4.4)式分别可由(4.8)-(4.11)式推导得到.更进一步, (4.5)式可以由(4.13), (4.14), (4.16)式和紧嵌入定理(参照文献[17])得出, 类似地, (4.6)式可由(4.13), (4.15), (4.17)式和紧嵌入定理得出.我们只需证明(4.7)式的收敛性.为此将(3.4)式写作

$\begin{eqnarray}\label{4.28} \nabla p_{\epsilon_m}=-u'_{\epsilon_m}-\hat{B}(u_{\epsilon_m})+\mu\Delta u_{\epsilon_m}. \end{eqnarray}$

(4.28)

由 $u_{\epsilon_m}$ 的收敛性知, 当 $\epsilon_m\rightarrow 0^+$ 时, (4.28)式的右端在空间 $L^{\frac{4}{3}}(0,T;{\Bbb H}^{-1})$ 中弱收敛到

$\begin{eqnarray}\label{4.29} -u'_{*}-\hat{B}(u_{*})+\mu\Delta u_{*}. \end{eqnarray}$

(4.29)

注意到(4.29)式即为 $\nabla p$ .因此

$\begin{eqnarray} \nabla p_\epsilon\rightarrow \nabla p ~ \mbox{在 $L^{\frac{4}{3}}(0,T;{\Bbb H}^{-1}) $中弱收敛}, \end{eqnarray}$

从而(4.7)式得证.