一类不可压非牛顿Boussinesq方程组正则解的存在性

Existence of Regular Solution for a Class of Incompressible Non-Newton Boussinesq Equations

摘要: 在三维光滑有界区域中研究了奇异情况下的非牛顿Boussinesq方程组的周期初边值问题。应用Galerkin方法、Gronwall不等式、Aubin-Lions引理，并结合能量估计、紧性方法证明了外力项适当小的情况下，该方程组正则解的存在性。

Abstract: The periodic initial boundary value problem of non-Newtonian Boussinesq equations in singular case is studied in three-dimensional smooth bounded domain. By using Galerkin method, Gronwall inequality, Aubin-Lions lemma, energy estimation and compactness method, the existence of regu-lar solutions for the system is proved when the external force term is appropriately small.

1. 引言

Boussinesq方程组描述地球物理流，对生活中的气象预报、龙卷风等自然灾害等的研究都有积极现实意义 [1] ，该方程组在Rayleigh-Bérnard对流 [2] [3] 中扮演至关重要的角色。考虑如下一类非牛顿Boussinesq方程组的初边值问题：

$\{\begin{array}{l}{u}_t+div\left(u\otimes u\right)-divT\left(Du\right)+\nabla p=\rho e_3\theta +f,\left(x,t\right)\in \Omega \times \left(0,T\right),\\ divu=0,\left(x,t\right)\in \Omega \times \left(0,T\right),\\ {\theta }_t+u\nabla \theta -\kappa \Delta \theta =g,\left(x,t\right)\in \Omega \times \left(0,T\right),\\ {u|}_{t=0}=u_0\left(x\right),{\theta |}_{t=0}={\theta }_0\left(x\right),x\in \Omega ,\\ {u|}_{{\Gamma }_i}={u|}_{{\Gamma }_{i+3}},{\nabla u|}_{{\Gamma }_i}={\nabla u|}_{{\Gamma }_{i+3}},{p|}_{{\Gamma }_i}={p|}_{{\Gamma }_{i+3}},x\in \partial \Omega ,\\ {\theta |}_{{\Gamma }_i}={\theta |}_{{\Gamma }_{i+3}},{\nabla \theta |}_{{\Gamma }_i}={\nabla \theta |}_{{\Gamma }_{i+3}},x\in \partial \Omega .\end{array}$ (1)

其中：未知函数 $u=\left(u_1,u_2,u_3\right)$ ， $\theta =\left({\theta }_1,{\theta }_2,{\theta }_3\right)$ 以及p， $e_3$ 分别表示流体速度、温度、压力和 ${ℝ}^3$ 中的单位向量。 $\Omega ={\left(0,1\right)}^3\subset {ℝ}^3$ ， $T\left(Du\right)={\left|Du\right|}^{p-2}Du$ ， $1<p<2$ 。 $Du=\frac{1}{2}\left(\nabla u+\nabla u^{\text{T}}\right)$ 表示应力速率张量。

$f=\left(f_1,f_2,f_3\right)$ ， $g=\left(g_1,g_2,g_3\right)$ 为给定的外力项。 ${\Gamma }_i:=\partial \Omega \cap \left\{x_i=0\right\}$ 以及 ${\Gamma }_{i+3}:=\partial \Omega \cap \left\{x_i=1\right\}$ ， $i=1,2,3$ 。 $\rho$ 表示流体密度， $\kappa$ 表示热传导系数，均取常数。为讨论方便，取 $\rho =\kappa =1$ 。

在自然界和工业界中都存在着大量的非牛顿流体，如建筑界中的沥青、水泥浆、下水道中的污泥。食品工业中的奶油、蜂蜜和蛋白、大多数油类和润滑脂。高聚物熔体和溶液以及人体中的血液等。非牛顿流体的研究对这些工业的发展具有重大的现实意义。

当不考虑温度( $\theta =0$ )时，系统(1)退化为经典的不可压非牛顿流方程组，目前关于不可压非牛顿流有一系列的研究，具体参见文献 [4] - [9] 。当把温度( $\theta \ne 0$ )纳入考虑时，问题(1)成为非牛顿Boussinesq模型。由于Boussinesq方程组中温度和速度耦合项产生的困难，与不可压非牛顿流相比，关于该类方程组各类初边值解的研究还很少。文献 [10] 中对一类修正的双极粘性流体的Boussinesq近似进行了研究，并证明了当 $p>2n/\left(n+2\right)$ 时弱解的存在唯一性。文献 [11] 研究一类稳态不可压Boussinesq第一边值问题，应用迭代方法证明正则解存在唯一性。目前，关于Boussinesq方程组的研究主要集中在牛顿流模型，关于非牛顿Boussinesq方程组的研究还不多，且主要集中在弱解的存在性、唯一性等方面。本文在三维光滑有界域 $\Omega$ 中，考虑一类周期边界条件下正则解的存在性。

2. 预备知识与主要结果

本文用 $L^q\left(\Omega \right)\left(1\le q\le \infty \right)$ 表示标准的Lebesgue空间，其范数用 ${\Vert \cdot \Vert }_q$ 表示， $W^{m,q}\left(\Omega \right)$ 表示通常的Sobolev空间，其范数用 ${\Vert \cdot \Vert }_{m,q}$ 表示。设X为定义范数 ${\Vert \cdot \Vert }_X$ 的Banach空间，用 $L^p\left(\left(0,T\right);X\right)\left(1\le p\le \infty \right)$ 表示通常的Bochner空间，其中范数用 ${\Vert \cdot \Vert }_{L^p\left(0,T;X\right)}$ 表示。

引入函数空间

${\mathcal{l}}_{per}:=\left\{u\in C_{per}^{\infty }\left(\Omega \right),{\displaystyle {\int }_{\Omega }u\text{d}x=0}\right\},$

$V_{per}^q:={\mathcal{l}}_{per}$ 在 ${\Vert \cdot \Vert }_q$ -范数意义下的闭包，

$V_{per}^{m,q}:={\mathcal{l}}_{per}$ 在 ${\Vert \cdot \Vert }_{m,q}$ -范数意义下的闭包。

这里下标'per'表示该函数空间中的函数沿空间各方向都具有周期性且在所讨论区域上满足零均值条件。

下面给出问题(1)正则解的定义

定义1：假设 $f\in L^{\infty }\left(0,T;L_{per}^{\frac{4}{p}}\left(\Omega \right)\right)$ ， $g\in L^{\infty }\left(0,T;L_{per}^2\left(\Omega \right)\right)$ ， $u_0\in V_{per}^{1,2}\left(\Omega \right)$ ， ${\theta }_0\in V_{per}^{1,2}\left(\Omega \right)$ 。若 $\left(u,\theta \right)$ 满足下述条件

1) $u\in C\left(0,T;V_{per}^{1,2}\left(\Omega \right)\right)$ ， $\theta \in C\left(0,T;V_{per}^{1,2}\left(\Omega \right)\right)$ ， $u_t\in L^2\left({\Omega }_T\right)$ ， ${\theta }_t\in L^2\left({\Omega }_T\right)$ ， $u\left(0\right)=u_0$ ， $\theta \left(0\right)={\theta }_0$ ；

2) ${\nabla }^{2}u\in L^2\left(0,T;L_{per}^{\frac{4}{4-p}}\left(\Omega \right)\right)$ ， ${\nabla }^2\theta \in L^2\left(0,T;L_{per}^2\left(\Omega \right)\right)$ ；

3) 对 $\forall \phi \in V_{per}^{1,p}\left(\Omega \right)$ ， $\varphi \in V_{per}^{1,2}\left(\Omega \right)$ 以及 $a.e.t\in \left(0,T\right)$ 有

${\displaystyle {\int }_{\Omega }{\partial }_t}u\phi \text{d}x+{\displaystyle {\int }_{\Omega }\left(u\cdot \nabla \right)u\phi \text{d}x}+{{\displaystyle {\int }_{\Omega }\left|Du\right|}}^{p-2}DuD\phi \text{d}x={\displaystyle {\int }_{\Omega }{e}_3}\theta \phi \text{d}x+{\displaystyle {\int }_{\Omega }f\phi \text{d}x},$

${\displaystyle {\int }_{\Omega }{\partial }_t}\theta \varphi \text{d}x+{\displaystyle {\int }_{\Omega }\left(u\cdot \nabla \right)\theta }\varphi \text{d}x+{\displaystyle {\int }_{\Omega }\nabla \theta \cdot \nabla \varphi \text{d}x}={\displaystyle {\int }_{\Omega }g\varphi \text{d}x},$

则称 $\left(u,\theta \right)$ 为问题(1)的正则解。

在后续中，为方便起见，省略下标'per'。

设 $\Lambda ,\Gamma ,M,N$ 为满足如下条件的常数，C是计算中产生的与 $p,\Omega$ 有关的常数。

$\Lambda <\min \left\{{\left(\frac{p}{10C}\right)}^{\frac{2}{3-p}},{\left(\frac{1}{2C}\right)}^2\right\},\Gamma >2C{N}^2,\left(C\Gamma +M^2\right)<\frac{p\left(p-1\right)}{10C}{\Lambda }^{p-1}$ (2)

本文的主要定理如下：

定理1：设 $5/3\le p<2$ ， $u_0,{\theta }_0,f,g$ 满足

$\begin{array}{l}f\in L^{\infty }\left(0,T;L^{\frac{4}{p}}\left(\Omega \right)\right),u_0\in V^{1,2}\left(\Omega \right),\text{ess}\underset{t\ge 0}{\sup }{\Vert f\Vert }_{L^{\frac{4}{p}}\left(\Omega \right)}\le M,{\Vert \nabla u_0\Vert }_2^2<\Lambda ,\\ g\in L^{\infty }\left(0,T;L^2\left(\Omega \right)\right),{\theta }_0\in V^{1,2}\left(\Omega \right),\text{ess}\underset{t\ge 0}{\sup }{\Vert g\Vert }_{L^2\left(\Omega \right)}\le N,{\Vert \nabla {\theta }_0\Vert }_2^2<\Gamma ,\end{array}$ (3)

则问题(1)至少存在一个定义1意义下的解 $\left(u,\theta \right)$ 。此外，对 $\forall T>0$ 解 $\left(u,\theta \right)$ 满足估计

$\underset{t\in \left[0,T\right]}{\sup }{\Vert \nabla u\left(t\right)\Vert }_2^2\le \Lambda ,\underset{t\in \left[0,T\right]}{\sup }{\Vert \nabla \theta \left(t\right)\Vert }_2^2\le \Gamma$

下面给出本文证明中用到的引理：

引理1 [5] ：对于 $u,v,w\in H_0^1\left(\Omega \right)$ ，定义 $b\left(u,v,w\right)={\displaystyle \underset{i,j=1}{\overset{3}{\sum }}{\displaystyle {\int }_{\Omega }{u}_i\left(x\right)\frac{\partial v_j}{\partial x_i}}}\left(x\right)w_j\left(x\right)\text{d}x$ ，若 $divu=0$ ，则

$b\left(u,v,w\right)=b\left(u,w,v\right),b\left(u,v,v\right)=0$

引理2：当 $0<q<1$ 且 $q^{\prime }=\frac{q}{q-1}$ 时，如果 $f\in L^q\left(\Omega \right)$ ， $0<{\displaystyle {\int }_{\Omega }{\left|g\left(x\right)\right|}^{q^{\prime }}\text{d}x}<\infty$ 有

${\displaystyle {\int }_{\Omega }\left|f\left(x\right)g\left(x\right)\right|\text{d}x}\ge {\left({\displaystyle {\int }_{\Omega }{\left|f\left(x\right)\right|}^q\text{d}x}\right)}^{\frac{1}{q}}{\left({\displaystyle {\int }_{\Omega }{\left|g\left(x\right)\right|}^{q^{\prime }}\text{d}x}\right)}^{\frac{1}{q^{\prime }}}$

3. 定理1的证明

定理1的证明分多步完成：在3.1节，我们构造问题(1)的逼近方程，并证明相应的Galerkin近似解的存在性；在3.2节，我们推导近似解的一致性先验估计；最后，在3.3节讨论近似解的收敛性，并最终证明原问题正则解的存在性结果。

3.1. 近似解的构造

由于问题(1)具有奇异性，故我们首先对问题正则化，设 $\gamma \in \left(0,1\right)$ ，记 $T_{\gamma }\left(Du\right)={\left(\gamma +{\left|Du\right|}^2\right)}^{\frac{p-2}{2}}Du$ ，考虑如下初边值问题

$\{\begin{array}{l}{u}_t+div\left(u\otimes u\right)-div{T}_{\gamma }\left(Du\right)+\nabla p=e_3\theta +f,\left(x,t\right)\in \Omega \times \left(0,T\right),\\ divu=0,\left(x,t\right)\in \Omega \times \left(0,T\right),\\ {\theta }_t+u\nabla \theta -\Delta \theta =g,\left(x,t\right)\in \Omega \times \left(0,T\right),\\ {u|}_{t=0}=u_0\left(x\right),{\theta |}_{t=0}={\theta }_0\left(x\right),x\in \Omega ,\\ {u|}_{{\Gamma }_i}={u|}_{{\Gamma }_{i+3}},{\nabla u|}_{{\Gamma }_i}={\nabla u|}_{{\Gamma }_{i+3}},{p|}_{{\Gamma }_i}={p|}_{{\Gamma }_{i+3}},x\in \partial \Omega ,\\ {\theta |}_{{\Gamma }_i}={\theta |}_{{\Gamma }_{i+3}},{\nabla \theta |}_{{\Gamma }_i}={\nabla \theta |}_{{\Gamma }_{i+3}},x\in \partial \Omega .\end{array}$ (4)

设 ${\left\{w_k\right\}}_{k\in \mathbb{N}}$ 为Stokes算子A的特征函数族， ${\lambda }_k$ 是相应的特征值且 $X_n=\left\{w_1,w_2,\cdots ,w_n\right\}$ 。设 ${\left\{v_k\right\}}_{k\in \mathbb{N}}$ 为Laplace算子 $-\Delta$ 的特征函数族， ${\mu }_k$ 是相应的特征值且 $Y_n=\left\{v_1,v_2,\cdots ,v_n\right\}$ 。定义投影算子

$P^n:\left(W^{1,2}\left(\Omega \right),{\Vert \cdot \Vert }_{1,2}\right)\to \left(X_n,{\Vert \cdot \Vert }_{1,2}\right),Q^n:\left(W^{1,2}\left(\Omega \right),{\Vert \cdot \Vert }_{1,2}\right)\to \left(Y_n,{\Vert \cdot \Vert }_{1,2}\right)$

分别如下

$P^{n}u={\displaystyle {\sum }_{k=1}^n\left(u,{\phi }_k\right)w_k},Q^n\theta ={\displaystyle {\sum }_{k=1}^n\left(\theta ,v_k\right)v_k}$

我们寻求问题(4)如下形式的解

$u^n\left(x,t\right)={\displaystyle {\sum }_{k=1}^n{c}_k^n}\left(t\right)w_k\left(x\right),{\theta }^n\left(x,t\right)={\displaystyle {\sum }_{k=1}^n{d}_k^n}\left(t\right)v_k(\; x\; )$

其中对 $k=1,2,\cdots ,n$ ，系数 $c_k^n\left(t\right),d_k^n\left(t\right)$ 满足

$\left({\partial }_t{u}^n,w_k\right)+\left(\left(u^n\cdot \nabla \right)u^n,w_k\right)+\left(T_{\gamma }\left(D{u}^n\right),D{w}_k\right)=\left(e_3{\theta }^n,w_k\right)+\left(f,w_k\right),$ (5)

$\left({\partial }_t{\theta }^n,v_k\right)+\left(\left(u^n\cdot \nabla \right){\theta }^n,v_k\right)+\left(\nabla {\theta }^n,\nabla v_k\right)=\left(g,v_k\right),$ (6)

$u^n\left(0\right)=P^n{u}_0,{\theta }^n\left(0\right)=Q^n{\theta }_0$ (7)

由Carathéodory定理可知存在 $t_n\left(0<t_n\le T\right)$ ，使得问题(5)-(7)在 $\left[0,t_n\right)$ 上存在光滑解 $u^n\left(x,t\right),{\theta }^n\left(x,t\right)$ 。接下来，我们推导 $c_k^n\left(t\right),d_k^n\left(t\right)$ 关于n的一致性先验估计，由这些估计可知 $t_n=T$ 。

3.2. 一致性先验估计

3.2.1. $\underset{t\in \left[0,T\right]}{sup}{\Vert u^n\left(t\right)\Vert }_2^2$ 与 $\underset{t\in \left[0,T\right]}{sup}{\Vert {\theta }^n\left(t\right)\Vert }_2^2$ 的估计

在式(6)两端乘以 $d_k^n\left(t\right)$ 并关于k求和，应用引理1、Hölder不等式和Young不等式，有

$\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert {\theta }^n\Vert }_2^2+{\Vert \nabla {\theta }^n\Vert }_2^2=\left(g,{\theta }^n\right)\le {\Vert g\Vert }_2{\Vert {\theta }^n\Vert }_2\le \frac{1}{2}{\Vert g\Vert }_2^2+\frac{1}{2}{\Vert {\theta }^n\Vert }_2^2,$

从而由Gronwall不等式可得

$\underset{t\in \left[0,T\right]}{\sup }{\Vert {\theta }^n\left(t\right)\Vert }_2^2+2{\Vert \nabla {\theta }^n\Vert }_{L^2\left(0,T;L^2\left(\Omega \right)\right)}^2\le C\left({\Vert {\theta }_0\Vert }_2^2+T{\Vert g\Vert }_{L^{\infty }\left(0,T;L^2\left(\Omega \right)\right)}^2\right),$ (8)

在式(5)两端乘以 $c_k^n\left(t\right)$ 并关于k求和，应用引理1，有

$\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert u^n\Vert }_2^2+{\displaystyle {\int }_{\Omega }{\left(\gamma +{\left|D{u}^n\right|}_2^2\right)}^{p-2}\text{d}x}=\left(e_3{\theta }^n,u^n\right)+\left(f,u^n\right),$

又由于

$\begin{array}{c}{\displaystyle {\int }_{\Omega }{\left|D{u}^n\right|}^p\text{d}x}={\displaystyle {\int }_{{\left|D{u}^n\right|}^2\ge \gamma }{\left|D{u}^n\right|}^p\text{d}x}+{\displaystyle {\int }_{{\left|D{u}^n\right|}^2<\gamma }{\left|D{u}^n\right|}^p\text{d}x}\\ \le 2^{\frac{2-p}{2}}{\displaystyle {\int }_{\Omega }{\left(\gamma +{\left|D{u}^n\right|}^2\right)}^{\frac{p-2}{2}}{\left|D{u}^n\right|}^2\text{d}x}+{\displaystyle {\int }_{\Omega }{\gamma }^{\frac{p}{2}}\text{d}x}\\ \le 2^{\frac{2-p}{2}}\left(T_{\gamma }\left(D{u}^n\right),D{u}^n\right)+{\gamma }^{\frac{p}{2}}\left|\Omega \right|.\end{array}$

并应用Hölder不等式、 ${\Vert u\Vert }_{\frac{4}{4-p}}\le C{\Vert Du\Vert }_p$ 和Young不等式，有

$\begin{array}{c}\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert u^n\Vert }_2^2+2^{\frac{p-2}{2}}{\displaystyle {\int }_{\Omega }{\left|D{u}^n\right|}^p\text{d}x}\le \left(e_3{\theta }^n,u^n\right)+\left(f,u^n\right)+{\gamma }^{\frac{p}{2}}\left|\Omega \right|\\ \le {\Vert {\theta }^n\Vert }_2{\Vert u^n\Vert }_2+{\Vert f\Vert }_{\frac{4}{p}}{\Vert u^n\Vert }_{\frac{4}{4-p}}+{\gamma }^{\frac{p}{2}}\left|\Omega \right|\\ \le \frac{1}{2}{\Vert {\theta }^n\Vert }_2^2+\frac{1}{2}{\Vert u^n\Vert }_2^2+C{\Vert f\Vert }_{\frac{4}{p}}{\Vert D{u}^n\Vert }_p+{\gamma }^{\frac{p}{2}}\left|\Omega \right|\\ \le \frac{1}{2}{\Vert {\theta }^n\Vert }_2^2+\frac{1}{2}{\Vert u^n\Vert }_2^2+\frac{1}{4}\cdot 2^{\frac{p-2}{2}}{\Vert D{u}^n\Vert }_p^p+C{\Vert f\Vert }_{\frac{4}{p}}^{p^{\prime }}+{\gamma }^{\frac{p}{2}}\left|\Omega \right|.\end{array}$

并对上式应用式(8)，可得

$\begin{array}{l}\frac{\text{d}}{\text{d}t}{\Vert u^n\Vert }_2^2+{\Vert D{u}^n\Vert }_p^p\le {\Vert {\theta }^n\Vert }_2^2+{\Vert u^n\Vert }_2^2+C{\Vert f\Vert }_{\frac{4}{p}}^{p^{\prime }}+\gamma \left|\Omega \right|\\ \le {\Vert u^n\Vert }_2^2+{\Vert {\theta }_0\Vert }_2^2+t{\Vert g\Vert }_{L^{\infty }\left(0,T;L^2\left(\Omega \right)\right)}^2+C{\Vert f\Vert }_{\frac{4}{p}}^{p^{\prime }}+\gamma \left|\Omega \right|.\end{array}$

故利用Gronwall不等式可得

$\begin{array}{l}\underset{t\in \left[0,T\right]}{\sup }{\Vert u^n\left(t\right)\Vert }_2^2+{\Vert D{u}^n\Vert }_{L^p\left(0,T;L^p\left(\Omega \right)\right)}^p\\ \le {\Vert u_0\Vert }_2^2+T{\Vert {\theta }_0\Vert }_2^2+\frac{1}{2}{T}^2{\Vert g\Vert }_{L^{\infty }\left(0,T;L^2\left(\Omega \right)\right)}^2+CT{\Vert f\Vert }_{L^{\infty }\left(0,T;L^{\frac{4}{p}}\left(\Omega \right)\right)}^{p^{\prime }}+{\gamma }^{\frac{p}{2}}\left|{\Omega }_T\right|.\end{array}$ (9)

3.2.2. $\underset{t\in \left[0,T\right]}{sup}{\Vert \nabla u^n\left(t\right)\Vert }_2^2$ 与 $\underset{t\in \left[0,T\right]}{sup}{\Vert \nabla {\theta }^n\left(t\right)\Vert }_2^2\le \Gamma$ 的估计

在式(5)，(6)两端分别乘以 ${\lambda }_k{c}_k^n\left(t\right),{\mu }_k{d}_k^n\left(t\right)$ 并关于k求和，得

$\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert \nabla u^n\Vert }_2^2+{\displaystyle {\int }_{\Omega }\nabla T_{\gamma }}\left(D{u}^n\right)\cdot \nabla D{u}^n\text{d}x=\left(\left(u^n\cdot \nabla \right)u^n,\Delta u^n\right)-\left(e_n{\theta }^n,\Delta u^n\right)-\left(f,\Delta u^n\right),$ (10)

$\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert \nabla {\theta }^n\Vert }_2^2+{\Vert {\nabla }^2{\theta }^n\Vert }_2^2=\left(\left(u^n\cdot \nabla \right){\theta }^n,\Delta {\theta }^n\right)-\left(g,\Delta {\theta }^n\right)$ (11)

对式(10)左端第二项进行整理可得

$\begin{array}{l}{\displaystyle {\int }_{\Omega }\nabla T_{\gamma }}\left(D{u}^n\right)\cdot \nabla D{u}^n\text{d}x\\ =\left(p-2\right){\displaystyle {\int }_{\Omega }{\left(\gamma +{\left|D{u}^n\right|}^2\right)}^{\frac{p-4}{2}}{\left|D{u}^n\right|}^2{\left|\nabla D{u}^n\right|}^2\text{d}x}+{\displaystyle {\int }_{\Omega }{\left(\gamma +{\left|D{u}^n\right|}^2\right)}^{\frac{p-2}{2}}{\left|\nabla D{u}^n\right|}^2\text{d}x}\\ \ge \left(p-1\right){\displaystyle {\int }_{\Omega }{\left(\gamma +{\left|D{u}^n\right|}^2\right)}^{\frac{p-2}{2}}{\left|\nabla D{u}^n\right|}^2\text{d}x},\end{array}$ (12)

为方便起见，记 $I=\left(p-1\right){\displaystyle {\int }_{\Omega }{\left(\gamma +{\left|D{u}^n\right|}^2\right)}^{\frac{p-2}{2}}{\left|\nabla D{u}^n\right|}^2\text{d}x}$ 。

对式(10)右端第一项进行处理，由文献 [12] 知 $\left|\left(u\cdot \nabla \right)u,\Delta u\right|\le {\Vert \nabla u\Vert }_3^3$ ，则

$\left(\left(u^n\cdot \nabla \right)u^n,\Delta u^n\right)\le {\Vert \nabla u^n\Vert }_3^3\le C{\Vert \nabla u^n\Vert }_2^{\frac{9p-12}{3p-2}}{\Vert \nabla D{u}^n\Vert }_{\frac{4}{4-p}}^{\frac{6}{3p-2}}$ (13)

事实上，当 $p\in \left[\frac{5}{3},2\right)$ 时，因为 $\frac{1}{3}=\frac{3p-4}{3p-2}\cdot \frac{1}{2}+\frac{2}{3p-2}\cdot \frac{8-3p}{12}$ ，从而由插值不等式及Sobolev嵌入定理可得

${\Vert \nabla u^n\Vert }_3\le {\Vert \nabla u^n\Vert }_2^{\frac{3p-4}{3p-2}}{\Vert \nabla u^n\Vert }_{\frac{12}{8-3p}}^{\frac{2}{3p-2}}\le C{\Vert \nabla u^n\Vert }_2^{\frac{3p-4}{3p-2}}{\Vert \nabla D{u}^n\Vert }_{\frac{4}{4-p}}^{\frac{2}{3p-2}}.$

将式(12)，(13)代入式(10)并对等式右端应用Hölder不等式，Sobolev嵌入得

$\begin{array}{c}\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert \nabla u^n\Vert }_2^2+I\le {\Vert \nabla u^n\Vert }_3^3+\left|\left(e_3{\theta }^n,\Delta u^n\right)\right|+\left|\left(f,\Delta u^n\right)\right|\\ \le C{\Vert \nabla u^n\Vert }_2^{\frac{9p-12}{3p-2}}{\Vert \nabla D{u}^n\Vert }_{\frac{4}{4-p}}^{\frac{6}{3p-2}}+{\Vert {\theta }^n\Vert }_{\frac{4}{p}}{\Vert \Delta u^n\Vert }_{\frac{4}{4-p}}+{\Vert f\Vert }_{\frac{4}{p}}{\Vert \Delta u^n\Vert }_{\frac{4}{4-p}}\\ \le C{\Vert \nabla u^n\Vert }_2{\Vert \nabla D{u}^n\Vert }_{\frac{4}{4-p}}^2+C{\Vert \nabla {\theta }^n\Vert }_2{\Vert \nabla D{u}^n\Vert }_{\frac{4}{4-p}}+{\Vert f\Vert }_{\frac{4}{p}}{\Vert \nabla D{u}^n\Vert }_{\frac{4}{4-p}},\end{array}$

对上式左端第二项应用引理2，右端应用Young不等式得

$\begin{array}{l}\frac{1}{2}{I}_t^{\frac{2-p}{2}}\frac{\text{d}}{\text{d}t}{\Vert \nabla u^n\Vert }_2^2+\frac{p-1}{2}{\Vert \nabla D{u}^n\Vert }_{\frac{4}{4-p}}^2\\ \le C{I}_t^{\frac{2-p}{2}}{\Vert \nabla u^n\Vert }_2{\Vert \nabla D{u}^n\Vert }_{\frac{4}{4-p}}^2+\frac{C}{p-1}{I}_t^{2-p}{\Vert \nabla {\theta }^n\Vert }_2^2+\frac{1}{p-1}{I}_t{}^{2-p}{\Vert f\Vert }_{\frac{4}{p}}^2,\end{array}$ (14)

这里， $I_t={\gamma }_1+{\Vert \nabla u^n\left(t\right)\Vert }_2^2$ ， ${\gamma }_1=\gamma \left|\Omega \right|$ 。

对式(11)右端第一项应用引理1，Hölder不等式和Sobolev嵌入有

$\begin{array}{c}\left(u^n\cdot \nabla {\theta }^n,\Delta {\theta }^n\right)={\displaystyle {\int }_{\Omega }{u}^n}\cdot \nabla {\theta }^n\Delta {\theta }^n\text{d}x={\displaystyle {\int }_{\Omega }{u}_j^n}\cdot \frac{\partial {\theta }_i^n}{\partial x_j}\cdot div\left(\nabla {\theta }_i^n\right)\text{d}x\\ =-{\displaystyle {\int }_{\Omega }\frac{\partial u_j^n}{\partial x_k}}\frac{\partial {\theta }_i^n}{\partial x_j}\frac{\partial {\theta }_i^n}{\partial x_k}\text{d}x-{\displaystyle {\int }_{\Omega }{u}_j^n}\frac{{\partial }^2{\theta }_i^n}{\partial x_j\partial x_k}\frac{\partial {\theta }_i^n}{\partial x_k}\text{d}x\\ \le {\displaystyle {\int }_{\Omega }\left|\frac{\partial u_j^n}{\partial x_k}\frac{\partial {\theta }_i^n}{\partial x_j}\frac{\partial {\theta }_i^n}{\partial x_k}\right|\text{d}x}={\displaystyle {\int }_{\Omega }\left|\nabla u^n\right|}\cdot {\left|\nabla {\theta }^n\right|}^2\text{d}x\\ \le {\Vert \nabla u^n\Vert }_2{\Vert \nabla {\theta }^n\Vert }_4^2\le C{\Vert \nabla u^n\Vert }_2{\Vert {\nabla }^2{\theta }^n\Vert }_2^2.\end{array}$

将上式代入式(11)，并应用Hölder不等式，Young不等式得

$\begin{array}{l}\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert \nabla {\theta }^n\Vert }_2^2+{\Vert {\nabla }^2{\theta }^n\Vert }_2^2\le C{\Vert \nabla u^n\Vert }_2{\Vert {\nabla }^2{\theta }^n\Vert }_2^2+{\displaystyle {\int }_{\Omega }\left|g\cdot \Delta {\theta }^n\right|\text{d}x}\\ \le C{\Vert \nabla u^n\Vert }_2{\Vert {\nabla }^2{\theta }^n\Vert }_2^2+{\Vert g\Vert }_2{\Vert \Delta {\theta }^n\Vert }_2\\ \le C{\Vert \nabla u^n\Vert }_2{\Vert {\nabla }^2{\theta }^n\Vert }_2^2+{\Vert g\Vert }_2{\Vert {\nabla }^2{\theta }^n\Vert }_2\\ \le C{\Vert \nabla u^n\Vert }_2{\Vert {\nabla }^2{\theta }^n\Vert }_2^2+\frac{1}{2}{\Vert g\Vert }_2^2+\frac{1}{2}{\Vert {\nabla }^2{\theta }^n\Vert }_2^2,\end{array}$

整理后得

$\frac{\text{d}}{\text{d}t}{\Vert \nabla {\theta }^n\Vert }_2^2+{\Vert {\nabla }^2{\theta }^n\Vert }_2^2\le C{\Vert \nabla u^n\Vert }_2{\Vert {\nabla }^2{\theta }^n\Vert }_2^2+{\Vert g\Vert }_2^2$ (15)

由假设 ${\Vert \nabla u_0\Vert }_2^2<\Lambda ,{\Vert \nabla {\theta }_0\Vert }_2^2<\Gamma$ 及 ${\Vert \nabla u^n\left(t\right)\Vert }_2,{\Vert \nabla {\theta }^n\left(t\right)\Vert }_2$ 的连续性，令 $t_1,t_2$ 分别是满足 ${\Vert \nabla u^n\left(t\right)\Vert }_2^2=\Lambda$ ， ${\Vert \nabla {\theta }^n\left(t\right)\Vert }_2^2=\Gamma$ 的最小时刻，下面对 $t_1,t_2$ 分情况讨论。

1) 情形一： $t_1<t_2$ 。

由假设知 ${\Vert \nabla u^n\left(t_1\right)\Vert }_2^2=\Lambda$ ， ${\Vert \nabla {\theta }^n\left(t_1\right)\Vert }_2^2<\Gamma$ ， ${\Vert \nabla {\theta }^n\left(t_2\right)\Vert }_2^2=\Gamma$ ，由于 ${\gamma }_1$ 任意小，为方便起见，取 ${\gamma }_1=\Lambda$ 并在式(14)中取 $t=t_1$ ，应用式(3)，得

$\begin{array}{l}\frac{1}{2}{\left(2\Lambda \right)}^{\frac{2-p}{2}}\frac{\text{d}}{\text{d}t}{\Vert \nabla u^n\left(t_1\right)\Vert }_2^2+\frac{p}{5}{\Vert \nabla D{u}^n\left(t_1\right)\Vert }_{\frac{4}{4-p}}^2\\ \le C{\left(2\Lambda \right)}^{\frac{2-p}{2}}{\Lambda }^{\frac{1}{2}}{\Vert \nabla D{u}^n\left(t_1\right)\Vert }_{\frac{4}{4-p}}^2+\frac{C}{p-1}{\left(2\Lambda \right)}^{2-p}\Gamma +\frac{1}{p-1}{\left(2\Lambda \right)}^{2-p}{M}^2,\end{array}$

对上式应用式(2)，有

$\frac{1}{2}{\left(2\Lambda \right)}^{\frac{2-p}{2}}\frac{\text{d}}{\text{d}t}{\Vert \nabla u^n\left(t_1\right)\Vert }_2^2+\frac{p}{10}{\Vert \nabla D{u}^n\left(t_1\right)\Vert }_{\frac{4}{4-p}}^2-\frac{C}{p-1}{\left(2\Lambda \right)}^{2-p}\Gamma -\frac{1}{p-1}{\left(2\Lambda \right)}^{2-p}{M}^2<0,$

由Sobolev嵌入有

$\frac{1}{2}{\left(2\Lambda \right)}^{\frac{2-p}{2}}\frac{\text{d}}{\text{d}t}{\Vert \nabla u^n\left(t_1\right)\Vert }_2^2+\frac{p}{10}{C}^{-1}{\Vert \nabla u^n\left(t_1\right)\Vert }_2^2-\frac{C}{p-1}{\left(2\Lambda \right)}^{2-p}\Gamma -\frac{1}{p-1}{\left(2\Lambda \right)}^{2-p}{M}^2<0,$

即

$\frac{1}{2}{\left(2\Lambda \right)}^{\frac{2-p}{2}}\frac{\text{d}}{\text{d}t}{\Vert \nabla u^n\left(t_1\right)\Vert }_2^2+\frac{p}{10}{C}^{-1}\Lambda -\frac{C}{p-1}{\left(2\Lambda \right)}^{2-p}\Gamma -\frac{1}{p-1}{\left(2\Lambda \right)}^{2-p}{M}^2<0,$

再次应用式(2)，得

$\frac{1}{2}{\left(2\Lambda \right)}^{\frac{2-p}{2}}\frac{\text{d}}{\text{d}t}{\Vert \nabla u^n\left(t_1\right)\Vert }_2^2<0$ ，即 ${\Vert \nabla u^n\left(t\right)\Vert }_2^2<\Lambda$ 。

在式(15)中令 $t=t_2$ ，应用式(3)和上式，得

$\frac{\text{d}}{\text{d}t}{\Vert \nabla {\theta }^n\left(t_2\right)\Vert }_2^2+{\Vert {\nabla }^2{\theta }^n\left(t_2\right)\Vert }_2^2\le C{\Lambda }^{\frac{1}{2}}{\Vert {\nabla }^2{\theta }^n\left(t_2\right)\Vert }_2^2+N^2,$

对上式应用式(2)，得

$\frac{\text{d}}{\text{d}t}{\Vert \nabla {\theta }^n\left(t_2\right)\Vert }_2^2+\frac{1}{2}{\Vert {\nabla }^2{\theta }^n\left(t_2\right)\Vert }_2^2-N^2<0,$

由Sobolev嵌入进而有

$\frac{\text{d}}{\text{d}t}{\Vert \nabla {\theta }^n\left(t_2\right)\Vert }_2^2+\frac{1}{2}{C}^{-1}{\Vert \nabla {\theta }^n\left(t_2\right)\Vert }_2^2-N^2<0,$

即

$\frac{\text{d}}{\text{d}t}{\Vert \nabla {\theta }^n\left(t_2\right)\Vert }_2^2+\frac{1}{2}{C}^{-1}\Gamma -N^2<0,$

再次应用式(2)，得

$\frac{\text{d}}{\text{d}t}{\Vert \nabla {\theta }^n\left(t_2\right)\Vert }_2^2<0$ ，即 ${\Vert \nabla {\theta }^n\left(t\right)\Vert }_2^2<\Gamma$ 。

2) 情形二： $t_1>t_2$ 。

与 $t_1<t_2$ 的证明类似的步骤，可以证得 ${\Vert \nabla u^n\left(t\right)\Vert }_2^2<\Lambda ,{\Vert \nabla {\theta }^n\left(t\right)\Vert }_2^2<\Gamma$ 。

3) 情形三： $t_1=t_2$ 。

与 $t_1<t_2$ 的证明类似的步骤，证得 ${\Vert \nabla u^n\left(t\right)\Vert }_2^2<\Lambda ,{\Vert \nabla {\theta }^n\left(t\right)\Vert }_2^2<\Gamma$ 。

综合上述，对任意的 $T>0$ ，都有

$\underset{t\in \left[0,T\right]}{\sup }{\Vert \nabla u\left(t\right)\Vert }_2^2\le \Lambda ,\underset{t\in \left[0,T\right]}{\sup }{\Vert \nabla \theta \left(t\right)\Vert }_2^2\le \Gamma$ (16)

3.2.3. ${\Vert u^n\Vert }_{L^2\left(0,T;W^{2,\frac{4}{4-p}}\left(\Omega \right)\right)}$ 与 ${\Vert {\theta }^n\Vert }_{L^2\left(0,T;W^{2,2}\left(\Omega \right)\right)}$ 的估计

由式(14)和式(15)知

$\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert \nabla u^n\left(t\right)\Vert }_2^2+\frac{p}{10}{\Vert \nabla D{u}^n\left(t\right)\Vert }_{\frac{4}{4-p}}^2\le \frac{C}{p-1}{\left({\gamma }_1+\Lambda \right)}^{\frac{2-p}{2}}\Gamma +\frac{1}{p-1}{\left({\gamma }_1+\Lambda \right)}^{\frac{2-p}{2}}{\Vert f\Vert }_{\frac{4}{p}}^2,$ (17)

$\frac{\text{d}}{\text{d}t}{\Vert \nabla {\theta }^n\left(t\right)\Vert }_2^2+\frac{1}{2}{\Vert {\nabla }^2{\theta }^n\left(t\right)\Vert }_2^2\le {\Vert g\Vert }_2^2$ (18)

对式(17)，(18)在 $\left(0,T\right)$ 上积分得

${\Vert \nabla u^n\left(T\right)\Vert }_2^2+{\displaystyle {\int }_0^T{\Vert \nabla D{u}^n\left(t\right)\Vert }_{\frac{4}{4-p}}^2\text{d}t}<C$ (19)

${\Vert \nabla {\theta }^n\left(T\right)\Vert }_2^2+{\displaystyle {\int }_0^T{\Vert {\nabla }^2{\theta }^n\left(t\right)\Vert }_2^2\text{d}t}<C$ (20)

由式(19)，(20)得

${\Vert u^n\Vert }_{L^2\left(0,T;W^{2,\frac{4}{4-p}}\left(\Omega \right)\right)}\le C$ (21)

${\Vert {\theta }^n\Vert }_{L^2\left(0,T;W^{2,2}\left(\Omega \right)\right)}\le C$ (22)

3.2.4. ${\Vert u_t^n\Vert }_{L^2\left(0,T;L^2\left(\Omega \right)\right)}$ 与 ${\Vert {\theta }_t^n\Vert }_{L^2\left(0,T;L^2\left(\Omega \right)\right)}$ 的估计

在式(5)，(6)两端分别乘以 $\frac{\text{d}}{\text{d}t}{c}_k^n\left(t\right),\frac{\text{d}}{\text{d}t}{d}_k^n\left(t\right)$ 并关于k求和，得

${\Vert u_t^n\Vert }_2^2+\left(T_{\gamma }\left(D{u}^n\right),D{u}_t^n\right)=-\left(u^n\cdot \nabla u^n,u_t^n\right)+\left(e_3{\theta }^n,u_t^n\right)+\left(f,u_t^n\right)$ (23)

对式(23)，(24)右端应用Hölder不等式、Sobolev嵌入、Young不等式，和式(16)得

${\Vert {\theta }_t^n\Vert }_2^2+\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert \nabla {\theta }^n\Vert }_2^2=-\left(u^n\cdot \nabla {\theta }^n,{\theta }_t^n\right)+\left(g,{\theta }_t^n\right)$ (24)

$\begin{array}{c}{\Vert u_t^n\Vert }_2^2+\left(T_{\gamma }\left(D{u}^n\right),D{u}_t^n\right)\le {\Vert u^n\Vert }_{\infty }{\Vert \nabla u^n\Vert }_2{\Vert u_t^n\Vert }_2+{\Vert {\theta }^n\Vert }_2{\Vert u_t^n\Vert }_2+{\Vert f\Vert }_{\frac{4}{p}}{\Vert u_t^n\Vert }_{\frac{4}{4-p}}\\ \le {\Vert u^n\Vert }_{\infty }^2{\Vert \nabla u^n\Vert }_2^2+\frac{1}{2}{\Vert u_t^n\Vert }_2^2+{\Vert {\theta }^n\Vert }_2^2+{\Vert f\Vert }_{\frac{4}{p}}{\Vert u_t^n\Vert }_2\\ \le {\Vert u^n\Vert }_{\infty }^2{\Vert \nabla u^n\Vert }_2^2+\frac{1}{2}{\Vert u_t^n\Vert }_2^2+C{\Vert \nabla {\theta }^n\Vert }_2^2+{\Vert f\Vert }_{\frac{4}{p}}^2+\frac{1}{4}{\Vert u_t^n\Vert }_2^2\\ \le C\Lambda {\Vert \nabla D{u}^n\Vert }_{\frac{4}{4-p}}^2+\frac{3}{4}{\Vert u_t^n\Vert }_2^2+C\Gamma +{\Vert f\Vert }_{\frac{4}{p}}^2\end{array}$ (25)

$\begin{array}{c}{\Vert {\theta }_t^n\Vert }_2^2+\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert \nabla {\theta }^n\Vert }_2^2\le {\Vert u^n\Vert }_{\infty }{\Vert \nabla {\theta }^n\Vert }_2{\Vert {\theta }_t^n\Vert }_2+{\Vert g\Vert }_2{\Vert {\theta }_t^n\Vert }_2\\ \le {\Vert u^n\Vert }_{\infty }^2{\Vert \nabla {\theta }^n\Vert }_2^2+\frac{1}{2}{\Vert {\theta }_t^n\Vert }_2^2+{\Vert g\Vert }_2^2\\ \le C\Gamma {\Vert \nabla D{u}^n\Vert }_{\frac{4}{4-p}}^2+\frac{1}{2}{\Vert {\theta }_t^n\Vert }_2^2+{\Vert g\Vert }_2^2\end{array}$ (26)

对式(25)，(26)两端在 $\left(0,T\right)$ 上积分，并应用式(3)，式(19)得

$\begin{array}{l}\frac{1}{4}{\displaystyle {\int }_0^T{\Vert u_t^n\left(t\right)\Vert }_2^2\text{d}t}+\frac{1}{p}{\displaystyle {\int }_0^T\frac{\text{d}}{\text{d}t}}{\Vert {\left(\gamma +{\left|D{u}^n\left(t\right)\right|}^2\right)}^{\frac{1}{2}}\Vert }_p^p\text{d}t\\ \le C\Lambda {\displaystyle {\int }_0^T{\Vert \nabla D{u}^n\left(t\right)\Vert }_{\frac{4}{4-p}}^2\text{d}t}+C\Gamma T+{\displaystyle {\int }_0^T{\Vert f\left(t\right)\Vert }_{\frac{4}{p}}^2\text{d}t},\end{array}$

$\frac{1}{4}{\displaystyle {\int }_0^T{\Vert u_t^n\left(t\right)\Vert }_2^2\text{d}t}+{\Vert {\left(\gamma +{\left|D{u}^n\left(T\right)\right|}^2\right)}^{\frac{1}{2}}\Vert }_p^p\le C$ (27)

$\frac{1}{2}{\displaystyle {\int }_0^T{\Vert {\theta }_t^n\left(t\right)\Vert }_2^2\text{d}t}+\frac{1}{2}\frac{\text{d}}{\text{d}t}{\displaystyle {\int }_0^T{\Vert \nabla {\theta }^n\Vert }_2^2\text{d}t}\le C\Gamma {\displaystyle {\int }_0^T{\Vert \nabla D{u}^n\Vert }_{\frac{4}{4-p}}^2\text{d}t}+{\displaystyle {\int }_0^T{\Vert g\Vert }_2^2\text{d}t},$

$\frac{1}{2}{\displaystyle {\int }_0^T{\Vert {\theta }_t^n\left(t\right)\Vert }_2^2\text{d}t}+\frac{1}{2}{\Vert \nabla {\theta }^n\left(T\right)\Vert }_2^2\le C$ (28)

由式(27)，(28)可得

${\Vert u_t^n\Vert }_{L^2\left(0,T;L^2\left(\Omega \right)\right)}\le C,$ (29)

${\Vert {\theta }_t^n\Vert }_{L^2\left(0,T;L^2\left(\Omega \right)\right)}\le C$ (30)

对所有的 $T>0$ 和 $E\subseteq \Omega$ ，应用Hölder不等式有

${\displaystyle {\int }_0^T{\displaystyle {\int }_E{\left|T_{\gamma }\left(D{u}^n\right)\right|}^{p^{\prime }}\text{d}x\text{d}t}}\le {\displaystyle {\int }_0^T{\displaystyle {\int }_E{\left|D{u}^n\right|}^p\text{d}x\text{d}t}}\le T{\left|E\right|}^{\frac{2-p}{2}}{\Lambda }^{\frac{p}{2}}$ (31)

知 $T_{\gamma }\left(D{u}^n\right)\in L^{p^{\prime }}\left(\Omega \right)$ 。

3.3. 近似解的收敛性和存在性证明

由式(21)，(22)，(29)，(30)可得下列收敛性

$u^n\rightharpoonup u^{\gamma }$ 弱收敛于 $L^2\left(0,T;W^{2,\frac{4}{4-p}}\left(\Omega \right)\right),$ (32)

${\theta }^n\rightharpoonup {\theta }^{\gamma }$ 弱收敛于 $L^2\left(0,T;W^{2,2}\left(\Omega \right)\right),$ (33)

$u_t^n\rightharpoonup u_t^{\gamma }$ 弱收敛于 $L^2\left(0,T;L^2\left(\Omega \right)\right),$ (34)

${\theta }_t^n\rightharpoonup {\theta }_t^{\gamma }$ 弱收敛于 $L^{\text{2}}\left(0,T;L^2\left(\Omega \right)\right)$ (35)

由于 $W^{2,\frac{4}{4-p}}\left(\Omega \right)$ ↪↪ $W^{1,2}\left(\Omega \right)$ ， $W^{2,2}\left(\Omega \right)$ ↪↪ $W^{1,2}\left(\Omega \right)$ ，利用Aubin-Lions引理可得

$u^n\to u^{\gamma }$ 强收敛于 $L^2\left(0,T;W^{1,2}\left(\Omega \right)\right),$ (36)

${\theta }^n\to {\theta }^{\gamma }$ 强收敛于 $L^2\left(0,T;W^{1,2}\left(\Omega \right)\right)$ (37)

利用 $L^2\left(\Omega \right)$ 范数下半连续性及式(16)有

${\Vert \nabla u^{\gamma }\left(t\right)\Vert }_2^2\le \Lambda ,{\Vert \nabla {\theta }^{\gamma }\left(t\right)\Vert }_2^2\le \Gamma$ (38)

由式(32)，(34)知 $u^{\gamma }\in L^2\left(0,T;W^{2,\frac{4}{4-p}}\left(\Omega \right)\right)$ ， $u_t^{\gamma }\in L^2\left(0,T;L^2\left(\Omega \right)\right)$ ，进而可得

$u^{\gamma }\in C\left(0,T;V^{1,2}\left(\Omega \right)\right)$ (39)

由式(33)，(35)知 ${\theta }^{\gamma }\in L^2\left(0,T;W^{2,2}\left(\Omega \right)\right)$ ， ${\theta }_t^{\gamma }\in L^2\left(0,T;L^2\left(\Omega \right)\right)$ ，有

${\theta }^{\gamma }\in C\left(0,T;V^{1,2}\left(\Omega \right)\right)$ (40)

由式(36)，(37)知，对 $\gamma >0$ 有

$\nabla u^n\to \nabla u^{\gamma }$ $a.e.$ 强收敛于 ${\Omega }_T$ $\nabla {\theta }^n\to \nabla {\theta }^{\gamma }$ $a.e.$ 强收敛于 ${\Omega }_T$ (41)

由式(41)以及 $T_{\gamma }\left(D{u}^n\right)$ 的连续性知，对所有 $\gamma >0$

$T_{\gamma }\left(D{u}^n\right)\to T_{\gamma }\left(D{u}^{\gamma }\right)$ $a.e.$ 强收敛于 ${\Omega }_T,$ (42)

因此，根据式(31)，应用Vitali’s收敛定理得

$\underset{n\to \infty }{\lim }{\displaystyle {\int }_{{\Omega }_T}{T}_{\gamma }}\left(D{u}^n\right):D\Phi ={\displaystyle {\int }_{{\Omega }_T}{T}_{\gamma }}\left(D{u}^{\gamma }\right):D\Phi ,\Phi \in L^p\left(0,T;W^{1,p}\left(\Omega \right)\right)$ (43)

由式(5)，(6)知，对于固定的 $n\ge m$ 和 $w\in X_m,v\in Y_m$ ，有

$\begin{array}{l}\left(u_t^n,w\right)+\left(\left(u^n\cdot \nabla \right)u^n,w\right)+\left(T_{\gamma }\left(D{u}^n\right),Dw\right)=\left(e_3{\theta }^n,w\right)+\left(f,w\right),\\ \left({\theta }_t^n,v\right)+\left(\left(u^n\cdot \nabla \right){\theta }^n,v\right)+\left(\nabla {\theta }^n,\nabla v\right)=\left(g,v\right).\end{array}$ (44)

在式(41)两端乘以 $\eta \in C^1\left(0,T;ℝ\right)$ ，关于t在 $\left(0,T\right)$ 上积分有

$\begin{array}{l}{\displaystyle {\int }_0^T\left(u_t^n,\eta w\right)+\left(\left(u^n\cdot \nabla \right)u^n,\eta w\right)+\left(T_{\gamma }\left(D{u}^n\right),\eta Dw\right)\text{d}t}={\displaystyle {\int }_0^T\left(e_3{\theta }^n,\eta w\right)\text{d}t}+{\displaystyle {\int }_0^T\left(f,\eta w\right)\text{d}t},\\ {\displaystyle {\int }_0^T\left({\theta }_t^n,\eta v\right)+\left(\left(u^n\cdot \nabla \right){\theta }^n,\eta v\right)+\left(\nabla {\theta }^n,\eta \nabla v\right)\text{d}t}={\displaystyle {\int }_0^T\left(g,\eta v\right)\text{d}t}.\end{array}$ (45)

在式(45)中令 $n\to \infty$ 取极限，对任意的 $\eta \in C^1\left(0,T;ℝ\right)$ ，由式(32)-(35)和式(43)有

$\begin{array}{l}{\displaystyle {\int }_0^T\left(u_t^{\gamma },\eta w\right)+\left(\left(u^{\gamma }\cdot \nabla \right)u^{\gamma },\eta w\right)+\left(T_{\gamma }\left(D{u}^{\gamma }\right),\eta Dw\right)\text{d}t}={\displaystyle {\int }_0^T\left(e_3{\theta }^{\gamma },\eta w\right)\text{d}t}+{\displaystyle {\int }_0^T\left(f,\eta w\right)\text{d}t},\\ {\displaystyle {\int }_0^T\left({\theta }_t^{\gamma },\eta v\right)+\left(\left(u^{\gamma }\cdot \nabla \right){\theta }^{\gamma },\eta v\right)+\left(\nabla {\theta }^{\gamma },\eta \nabla v\right)\text{d}t}={\displaystyle {\int }_0^T\left(g,\eta v\right)\text{d}t}.\end{array}$ (46)

由式(39)的正则性知，对 $u^{\gamma }\in C\left(\left[0,T\right);L^3\left(\Omega \right)\right)$ 有

$\begin{array}{l}\left(u_t^{\gamma },\phi \right)+\left(\left(u^{\gamma }\cdot \nabla \right)u^{\gamma },\phi \right)+\left(T_{\gamma }\left(D{u}^{\gamma }\right),D\phi \right)=\left(e_3{\theta }^{\gamma },\phi \right)+\left(f,\phi \right),\phi \in V^{1,p}\left(\Omega \right),\\ \left({\theta }_t^{\gamma },\psi \right)+\left(\left(u^{\gamma }\cdot \nabla \right){\theta }^{\gamma },\psi \right)+\left(\nabla {\theta }^{\gamma },\nabla \psi \right)=\left(g,\psi \right),\psi \in V^{1,2}\left(\Omega \right).\end{array}$ (47)

令 $\gamma \to 0$ 取极限，采用与上述 $n\to \infty$ 同样的方法，我们得到

$\begin{array}{l}\left(u_t,\phi \right)+\left(\left(u\cdot \nabla \right)u,\phi \right)+\left(T_{\gamma }\left(Du\right),D\phi \right)=\left(e_3\theta ,\phi \right)+\left(f,\phi \right),\phi \in V^{1,p}\left(\Omega \right),\\ \left({\theta }_t,\psi \right)+\left(\left(u\cdot \nabla \right)\theta ,\psi \right)+\left(\nabla \theta ,\nabla \psi \right)=\left(g,\psi \right),\psi \in V^{1,2}\left(\Omega \right).\end{array}$

定理1得证。

4. 总结

本文在三维空间中考虑一类不可压非牛顿Boussinesq方程组的周期初边值问题，给出了证明正则解存在性的详细步骤，采用Galerkin方法构造近似解序列 $\left\{u^n,{\theta }^n\right\}$ ，通过对近似解序列 $\left\{u^n,{\theta }^n\right\}$ 进行一致估计以及收敛性的证明并利用紧性方法证明其解的存在性。

基金项目

吉林省教育厅科学研究项目(项目号：JJKH20230790KJ)。

NOTES

^*通讯作者。