一类三维趋化–斯托克斯方程组正则化问题解的先验估计

A Prior Estimate of Solutions to Regularization Problems of a Class of Three-Dimensional Chemotaxis-Stokes Equations

摘要: 本文研究了一类带流体耦合的三维趋化–斯托克斯方程，该模型刻画了流体环境中的生物趋化现象。大量的生物实验观测表明，在趋化流体模型中，重力对细胞运动的影响和趋化本身对流体的交互作用都应该放在方程中同时考虑。本文将在 的条件下利用权函数的方法建立正则化问题解的先验估计，为进一步研究解的定性理论做好准备。

Abstract: In this paper, we study a class of three-dimensional chemotaxis-Stokes equations with fluid cou-pling, which describes the phenomenon of chemotaxis in the fluid environment. Numerous bio-logical experimental observations have shown that both the effect of gravity on bacterial motility and the interaction of chemotaxis itself on the fluid should be placed in the equations for simulta-neous consideration in the chemotaxis fluid model. In this paper, we will establish a prior estimate of the solution of the regularization problem under the condition of by using the method of weight function, in order to prepare for the further study of the qualitative theory of the solution.

1. 引言

趋化方程是一类刻画细胞自我组织和趋化运动规律的数学模型 [1] [2] ，该模型已经有大量的理论研究成果(参考文献 [3] )。实际生物实验观测发现 [1] ，细胞所处的流体环境对于细胞趋化运动行为模式有着不可忽视的影响。为此，文献 [4] 提出了如下经典的趋化–流体(chemotaxis-fluid)耦合模型：

$\{\begin{array}{l}{n}_t+u\cdot \nabla n=\Delta n-\nabla \cdot \left(n\nabla c\right),\\ c_t+u\cdot \nabla c=\Delta c-nc,\\ u_t+\kappa \left(u\cdot \nabla \right)u=\Delta u+\nabla P+n\nabla \phi ,\\ \nabla \cdot u=0,\end{array}$

其中，函数 $n=n\left(x,t\right)$ 表示细胞(一类大肠杆菌)密度、 $c=c\left(x,t\right)$ 代表化学物质(氧气)浓度， $u=u\left(x,t\right)$ 和P分别表示流体(水)速度场和相应的压力；参数 $\kappa \in R$ 与非线性流体对流项的强度有关； $\phi$ 表示足够光滑的重力势函数。

与不带流体的趋化系统类似，该类数学模型始终是通过由化学物质扩散和消耗、趋化性和粘性流体动力学组成的数学结构来解释定量关系，利用基本物理定律建立方程，而数学工作的展开主要是围绕经典解或弱解的适定性等问题进行 [5] 。基于更进一步考虑重力对细胞运动的影响和趋化本身对流体的交互作用，参考文献 [1] ，本文研究了如下趋化–斯托克斯方程组：

$\{\begin{array}{ll}{n}_t+u\cdot \nabla n=\Delta n-\nabla \cdot \left(nS\left(x,n,c\right)\cdot \nabla c\right)+\nabla \cdot \left(n\nabla \varphi \right),\hfill & x\in \Omega ,t>0,\hfill \\ c_t+u\cdot \nabla c=\Delta c-nc,\hfill & x\in \Omega ,t>0,\hfill \\ u_t+\nabla P=\Delta u-n\nabla \varphi +nS\left(x,n,c\right)\cdot \nabla c,\hfill & x\in \Omega ,t>0,\hfill \\ \nabla \cdot u=0,\hfill & x\in \Omega ,t>0,\hfill \\ \left(\nabla n-nS\left(x,n,c\right)\cdot \nabla c\right)\cdot v=\nabla c\cdot v=0,u=0,\hfill & x\in \partial \Omega ,t>0,\hfill \\ n\left(x,0\right)=n_0\left(x\right),c\left(x,0\right)=c_0\left(x\right),u\left(x,0\right)=u_0\left(x\right),\hfill & x\in \Omega ,\hfill \end{array}$ (1)

其中，有界域 $\Omega \subset R^3$ 具有光滑边界，向量 $\nu$ 表示区域边界上的外法方向向量， $n=n\left(x,t\right)$ 和 $c=c\left(x,t\right)$ 分别表示细胞(一类大肠杆菌)密度和化学物质(氧气)浓度， $u=u\left(x,t\right)$ 表示流体(水)速度，P表示相关压力。

趋化灵敏度函数S和引力势 $\phi$ 是给定的函数，要求 $\phi \in C^2\left(\Omega \right)$ 并且 $S\left(x,n,c\right)\in C^2\left(\bar{\Omega }\times {\left[0,\infty \right)}^2;{ℝ}^{3\times 3}\right)$ 是一个矩阵值函数，满足对任意的 $\left(x,n,c\right)\in \bar{\Omega }\times \left[0,\infty \right)\times \left[0,\infty \right)$ ，有

$\left|S\left(x,n,c\right)\right|\le C_S{\left(1+n\right)}^{-\alpha },$ (2)

其中常数 $c_s>0$ 和任意的 $\alpha >0$ 。

一些研究已经表明，考虑了生物体自身重力对流体的影响，该模型或许更能够说明流体环境中生物趋化现象所观测到的结果。

为了得到我们的主要结果，假设初始条件满足与文献 [6] 中的相同假设：

$n_0\in C^0\left(\bar{\Omega }\right),c_0\in W^{1,\infty }\left(\Omega \right),u_0\in D\left(A^{\beta }\right),$ (3)

其中 $n_0,c_0$ 在 $\bar{\Omega }$ 上非负，参数 $\beta \in \left(\frac{3}{4},1\right)$ ，A表示斯托克斯算子在 $L^2\left(\Omega \right)$ 的螺线型子空间

$L_{\sigma }^2\left(\Omega \right):=\left\{\phi \in L^2\left(\Omega \right)|\nabla \cdot \phi =0\right\}$

中的投影。

在这些假设下，本文的主要结果是利用权函数的方法建立了正则化问题解的先验估计，参见论文第3部分，为进一步研究解的定性理论做好准备。

2. 正则化

为了克服非线性边界条件带来的困难，我们将首先做边界的正则化近似，相关处理过程也可以参考文献 [6] 。

为此，对任意的 $\epsilon \in \left(0,1\right)$ ，设 ${\left({\rho }_{\epsilon }\right)}_{\left(\epsilon \in \left(0,1\right)\right)}\subset C_0^{\infty }\left(\Omega \right)$ 是满足 $0\le {\rho }_{\epsilon }\le 1$ 的一列标准截断函数，当 $\epsilon \to 0$ 时， ${\rho }_{\epsilon }$ 逐点收敛于1，定义 $S_{\epsilon }\left(x,n,c\right)={\rho }_{\epsilon }\left(x\right)\cdot {\chi }_{\epsilon }\left(n\right)\cdot S\left(x,n,c\right),\left(x,n,c\right)\in \bar{\Omega }\times \left(0,\infty \right)\times \left(0,\infty \right)$ ，显然，在区域边界上有 $S_{\epsilon }=0$ 。

我们接下来考虑以下正则化问题

$\{\begin{array}{ll}{n}_{\epsilon t}+u_{\epsilon }\cdot \nabla n_{\epsilon }=\Delta n_{\epsilon }-\nabla \cdot \left(n_{\epsilon }{S}_{\epsilon }\left(x,n_{\epsilon },c_{\epsilon }\right)\cdot \nabla c_{\epsilon }\right)+\nabla \cdot \left(n_{\epsilon }\nabla \varphi \right),\hfill & x\in \Omega ,t>0,\hfill \\ c_{\epsilon t}+u_{\epsilon }\cdot \nabla c_{\epsilon }=\Delta c_{\epsilon }-n_{\epsilon }{c}_{\epsilon },\hfill & x\in \Omega ,t>0,\hfill \\ u_{\epsilon t}+\nabla P_{\epsilon }=\Delta u_{\epsilon }-n_{\epsilon }\nabla \varphi +n_{\epsilon }{S}_{\epsilon }\left(x,n_{\epsilon },c_{\epsilon }\right)\nabla c_{\epsilon },\hfill & x\in \Omega ,t>0,\hfill \\ \nabla \cdot u_{\epsilon }=0,\hfill & x\in \Omega ,t>0,\hfill \\ \nabla n_{\epsilon }\cdot \nu =\nabla c_{\epsilon }\cdot \nu =0,u_{\epsilon }=0,\hfill & x\in \partial \Omega ,t>0,\hfill \\ n_{\epsilon }\left(x,0\right)=n_0\left(x\right),c_{\epsilon }\left(x,0\right)=c_0\left(x\right),u_{\epsilon }\left(x,0\right)=u_0\left(x\right),x\in \Omega \hfill & x\in \Omega ,\hfill \end{array}$ (4)

从文献 [7] 的引理2.1可以知道，对于任意的 $\epsilon \in \left(0,1\right)$ ，上述正则化问题具有经典解。

另一方面，利用质量守恒定律，很容易得到下面的引理。

引理 [6] 假设 $n_0,c_0,u_0$ 满足条件(3)，对任意的 $t>0$ ，方程(4)的解满足

${\displaystyle {\int }_{\Omega }{n}_{\epsilon }\left(x,t\right)\text{d}x}={\displaystyle {\int }_{\Omega }{n}_0\left(x\right)\text{d}x}$ 和 ${\Vert c_{\epsilon }\left(\cdot ,t\right)\Vert }_{L^{\infty }\left(\Omega \right)}\le {\Vert c_0\Vert }_{L^{\infty }\left(\Omega \right)}$ 。

3. $n_{\epsilon }$ 和 $\nabla c_{\epsilon }$ 的先验估计

本节我们在 $L^p\left(\Omega \right)$ 中得到 $n_{\epsilon }$ 的先验估计，在 $L^q\left(\Omega \right)$ 中得到 $\nabla c_{\epsilon }$ 的先验估计，该方法基于对 ${\displaystyle {\int }_{\Omega }{n}_{\epsilon }^p\phi \left(c_{\epsilon }\right)}$ 和 ${\displaystyle {\int }_{\Omega }{\left|\nabla c\right|}_{\epsilon }^q\phi \left(c_{\epsilon }\right)}$ 的加权处理，通过适当选择 $\phi$ ，推导出一些关于 ${\displaystyle {\int }_{\Omega }{n}_{\epsilon }^p\phi \left(c_{\epsilon }\right)}$ 时间演变的信息。

定理 3.1设 $p>1$ 和 $\left(n_{\epsilon },c_{\epsilon },u_{\epsilon },P_{\epsilon }\right)$ 是(4)的全局经典解。然后对于任何的非负非递减函数 $\phi \left(s\right)\in C^2\left(\left[0,+\infty \right)\right)$ ，对任意的 $t>0$ ，我们有

$\begin{array}{l}\frac{\text{d}}{\text{d}t}{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^p\phi \left(c_{\epsilon }\right)}+\frac{p\left(p-1\right)}{4}{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p-2}\phi \left(c_{\epsilon }\right){\left|\nabla n_{\epsilon }\right|}^2}+{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^p{\left|\nabla c_{\epsilon }\right|}^2}\left({\phi }^{″}\left(c_{\epsilon }\right)-\frac{p^2+15p}{4\left(p-1\right)}\frac{{{\phi }^{\prime }}^2\left(c_{\epsilon }\right)}{\phi \left(c_{\epsilon }\right)}\right)\\ \le p\left(p-1\right)C_S^2{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p-2\alpha }\phi \left(c_{\epsilon }\right){\left|\nabla c_{\epsilon }\right|}^2}+p{C}_S{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p-\alpha }{\phi }^{\prime }\left(c_{\epsilon }\right){\left|\nabla c_{\epsilon }\right|}^2}+p^2{\Vert \nabla \varphi \Vert }_{L^{\alpha }}^2{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^p\phi (\; c\; \epsilon \; )}\end{array}$

证明对(4)的前两个方程式进行直接计算，对任意的 $t>0$ ，可以得到

$\begin{array}{c}\frac{\text{d}}{\text{d}t}{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^p\phi \left(c_{\epsilon }\right)}={\displaystyle {\int }_{\Omega }p{n}_{\epsilon }^{p-1}\phi \left(c_{\epsilon }\right)}\left(\Delta n_{\epsilon }-\nabla \cdot \left(n_{\epsilon }{S}_{\epsilon }\left(x,n_{\epsilon },c_{\epsilon }\right)\cdot \nabla c_{\epsilon }\right)-u_{\epsilon }\cdot \nabla n_{\epsilon }+\nabla \cdot \left(n_{\epsilon }\nabla \varphi \right)\right)\\ +{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^p{\phi }^{\prime }\left(c_{\epsilon }\right)}\left(\Delta c_{\epsilon }-n_{\epsilon }{c}_{\epsilon }-u_{\epsilon }\cdot \nabla c_{\epsilon }\right)\\ =:I_1+I_2,\end{array}$

经过若干分部积分，对任意的 $t>0$ ，有

$\begin{array}{c}{I}_1=-p\left(p-1\right){\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p-2}\phi \left(c_{\epsilon }\right){\left|\nabla n_{\epsilon }\right|}^2}-p{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p-1}{\phi }^{\prime }\left(c_{\epsilon }\right)\nabla n_{\epsilon }\cdot \nabla c_{\epsilon }}-p{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p-1}\phi \left(c_{\epsilon }\right)u_{\epsilon }\cdot \nabla n_{\epsilon }}\\ +p\left(p-1\right){\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p-1}\phi \left(c_{\epsilon }\right)\nabla n_{\epsilon }\cdot \left(S_{\epsilon }\left(x,n_{\epsilon },c_{\epsilon }\right)\cdot \nabla c_{\epsilon }\right)}+p{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^p{\phi }^{\prime }\left(c_{\epsilon }\right)\nabla c_{\epsilon }\cdot \left(S_{\epsilon }\left(x,n_{\epsilon },c_{\epsilon }\right)\cdot \nabla c_{\epsilon }\right)}\\ -p\left(p-1\right){\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p-1}\phi \left(c_{\epsilon }\right)\nabla n_{\epsilon }\cdot \nabla \varphi }-p{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^p{\phi }^{\prime }\left(c_{\epsilon }\right)\nabla c_{\epsilon }\cdot \nabla \varphi }\end{array}$

和

$I_2=-p{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p-1}{\phi }^{\prime }\left(c_{\epsilon }\right)\nabla n_{\epsilon }\cdot \nabla c_{\epsilon }}-{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^p{\phi }^{″}\left(c_{\epsilon }\right){\left|\nabla c_{\epsilon }\right|}^2}-{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p+1}{c}_{\epsilon }{\phi }^{\prime }\left(c_{\epsilon }\right)}-{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^p{\phi }^{\prime }\left(c_{\epsilon }\right)u_{\epsilon }\cdot \nabla c_{\epsilon }}$ .

注意到，

$\begin{array}{l}-p{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p-1}\phi \left(c_{\epsilon }\right)u_{\epsilon }\cdot \nabla n_{\epsilon }}-{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^p{\phi }^{\prime }\left(c_{\epsilon }\right)u_{\epsilon }\cdot \nabla c_{\epsilon }}\\ =-{\displaystyle {\int }_{\Omega }\phi \left(c_{\epsilon }\right)u_{\epsilon }\cdot \nabla n_{\epsilon }^p}-{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^p{u}_{\epsilon }\cdot \nabla \phi \left(c_{\epsilon }\right)}={\displaystyle {\int }_{\Omega }{n}_{\epsilon }^p\phi \left(c_{\epsilon }\right)\left(\nabla \cdot u_{\epsilon }\right)}=0,\end{array}$

经过整理，对任意的 $t>0$ ，有

$\begin{array}{c}\frac{\text{d}}{\text{d}t}{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^p\phi \left(c_{\epsilon }\right)}=-p\left(p-1\right){\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p-2}\phi \left(c_{\epsilon }\right){\left|\nabla n_{\epsilon }\right|}^2}-2p{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p-1}{\phi }^{\prime }\left(c_{\epsilon }\right)\nabla n_{\epsilon }\cdot \nabla c_{\epsilon }}-{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^p{\phi }^{\prime }\left(c_{\epsilon }\right){\left|\nabla c_{\epsilon }\right|}^2}\\ +p{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^p{\phi }^{\prime }\left(c_{\epsilon }\right)\nabla c_{\epsilon }\cdot \left(S_{\epsilon }\left(x,n_{\epsilon },c_{\epsilon }\right)\cdot \nabla c_{\epsilon }\right)}+p\left(p-1\right){\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p-1}\phi \left(c_{\epsilon }\right)\nabla n_{\epsilon }\cdot \left(S_{\epsilon }\left(x,n_{\epsilon },c_{\epsilon }\right)\cdot \nabla c_{\epsilon }\right)}\\ -p\left(p-1\right){\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p-1}\phi \left(c_{\epsilon }\right)\nabla n_{\epsilon }\cdot \nabla \varphi }-p{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^p{\phi }^{\prime }\left(c_{\epsilon }\right)\nabla c_{\epsilon }\cdot \nabla \varphi }-{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p+1}{c}_{\epsilon }{\phi }^{\prime }(\; c\; \epsilon \; )}\end{array}$

结合条件(2)和 ${\phi }^{\prime }\left(c_{\epsilon }\right)$ 的非负性，对任意的 $t>0$ ，我们可以继续估计

$\begin{array}{l}\frac{\text{d}}{\text{d}t}{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^p\phi \left(c_{\epsilon }\right)}+p\left(p-1\right){\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p-2}\phi \left(c_{\epsilon }\right){\left|\nabla n_{\epsilon }\right|}^2}+{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^p{\phi }^{″}\left(c_{\epsilon }\right){\left|\nabla c_{\epsilon }\right|}^2}\\ =p\left(p-1\right){\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p-1}\phi \left(c_{\epsilon }\right)\nabla n_{\epsilon }\cdot \left(S_{\epsilon }\left(x,n_{\epsilon },c_{\epsilon }\right)\cdot \nabla c_{\epsilon }\right)}+p{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^p{\phi }^{\prime }\left(c_{\epsilon }\right)\nabla c_{\epsilon }\cdot \left(S_{\epsilon }\left(x,n_{\epsilon },c_{\epsilon }\right)\cdot \nabla c_{\epsilon }\right)}\\ -2p{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p-1}{\phi }^{\prime }\left(c_{\epsilon }\right)\nabla n_{\epsilon }\cdot \nabla c_{\epsilon }}-p\left(p-1\right){\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p-1}\phi \left(c_{\epsilon }\right)\nabla n_{\epsilon }\cdot \nabla \varphi }-p{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^p{\phi }^{\prime }\left(c_{\epsilon }\right)\nabla c_{\epsilon }\cdot \nabla \varphi }-{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p+1}{c}_{\epsilon }{\phi }^{\prime }\left(c_{\epsilon }\right)}\\ \le p\left(p-1\right)C_S{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p-\alpha -1}\phi \left(c_{\epsilon }\right)\left|\nabla n_{\epsilon }\right|\cdot \left|\nabla c_{\epsilon }\right|}+p{C}_S{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p-\alpha }{\phi }^{\prime }\left(c_{\epsilon }\right){\left|\nabla c_{\epsilon }\right|}^2}+2p{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p-1}{\phi }^{\prime }\left(c_{\epsilon }\right)\left|\nabla n_{\epsilon }\right|\cdot \left|\nabla c_{\epsilon }\right|}\\ +p\left(p-1\right){\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p-1}\phi \left(c_{\epsilon }\right)\left|\nabla n_{\epsilon }\right|\cdot \left|\nabla \varphi \right|}+p{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^p{\phi }^{\prime }\left(c_{\epsilon }\right)}\left|\nabla c_{\epsilon }\right|\cdot \left|\nabla \varphi \right|.\end{array}$

另一方面，利用杨氏不等式对上式右边进行合适的放缩，对任意的 $t>0$ ，可以得到

$p\left(p-1\right)C_S{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p-\alpha -1}\phi \left(c_{\epsilon }\right)\left|\nabla n_{\epsilon }\right|\cdot \left|\nabla c_{\epsilon }\right|}\le \frac{p\left(p-1\right)}{4}{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p-2}\phi \left(c_{\epsilon }\right){\left|\nabla n_{\epsilon }\right|}^2}+p\left(p-1\right)C_S^2{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p-2\alpha }\phi \left(c_{\epsilon }\right){\left|\nabla c_{\epsilon }\right|}^2}$

$2p{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p-1}{\phi }^{\prime }\left(c_{\epsilon }\right)\left|\nabla n_{\epsilon }\right|\cdot \left|\nabla c_{\epsilon }\right|}\le \frac{p\left(p-1\right)}{4}{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p-2}\phi \left(c_{\epsilon }\right){\left|\nabla n_{\epsilon }\right|}^2}+\frac{4p}{p-1}{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^p\frac{{{\phi }^{\prime }}^2\left(c_{\epsilon }\right)}{\phi \left(c_{\epsilon }\right)}{\left|\nabla c_{\epsilon }\right|}^2}$

$p\left(p-1\right){\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p-1}\phi \left(c_{\epsilon }\right)\left|\nabla n_{\epsilon }\right|\cdot \left|\nabla \varphi \right|}\le \frac{p\left(p-1\right)}{4}{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p-2}\phi \left(c_{\epsilon }\right){\left|\nabla n_{\epsilon }\right|}^2}+p\left(p-1\right){\displaystyle {\int }_{\Omega }{n}_{\epsilon }^p\phi \left(c_{\epsilon }\right){\left|\nabla \varphi \right|}^2}$

$p{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^p{\phi }^{\prime }\left(c_{\epsilon }\right)\left|\nabla c_{\epsilon }\right|\cdot \left|\nabla \varphi \right|}\le \frac{1}{4}{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^p\frac{{{\phi }^{\prime }}^2\left(c_{\epsilon }\right)}{\phi \left(c_{\epsilon }\right)}{\left|\nabla c_{\epsilon }\right|}^2}+p{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^p\phi \left(c_{\epsilon }\right){\left|\nabla \varphi \right|}^2}$

代入方程经过整理很很容易得到定理3.1。

下面，建立对 $\nabla c_{\epsilon }$ 的估计。然而，利用权函数可以消除流体函数 $u_{\epsilon }$ 的影响，下一个定理的证明与上面的证明方法相似。

定理3.2 设 $p>1$ ， $1<q<2$ ， $\left(n_{\epsilon },c_{\epsilon },u_{\epsilon },P_{\epsilon }\right)$ 为(4)的全局经典解。取与定理3.1相同的辅助函数 $\phi \left(s\right)\in C^2\left(\left[0,+\infty \right)\right)$ ，对任意的 $t>0$ ，我们有

$\begin{array}{l}\frac{\text{d}}{\text{d}t}{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q}\phi \left(c_{\epsilon }\right)}+\frac{q}{2}{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q-2}\phi \left(c_{\epsilon }\right){\left|D^2{c}_{\epsilon }\right|}^2}+{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q+2}}\left({\phi }^{″}\left(c_{\epsilon }\right)-2q\frac{{{\phi }^{\prime }}^2\left(c_{\epsilon }\right)}{\phi \left(c_{\epsilon }\right)}\right)\\ \le \frac{1}{2}q{\left(2q-2\right)}^2{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q-2}\phi \left(c_{\epsilon }\right)}+\frac{1}{8}p\left(p-1\right){\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p-2}\phi \left(c_{\epsilon }\right){\left|\nabla n_{\epsilon }\right|}^2}\\ +{\displaystyle {\int }_{\Omega }\frac{32q^2}{p\left(p-1\right)}{n}_{\epsilon }^{2-p}\phi \left(c_{\epsilon }\right){\left|\nabla c_{\epsilon }\right|}^{4q-2}{c}_{\epsilon }^2}+q{\displaystyle {\int }_{\partial \Omega }{\left|\nabla c_{\epsilon }\right|}^{2q-2}\phi \left(c_{\epsilon }\right)\frac{\partial {\left|\nabla c_{\epsilon }\right|}^2}{\partial \upsilon }}\end{array}$

证明首先，通过直接计算可以得出，对任意的 $t>0$ ，有

$\begin{array}{c}\frac{\text{d}}{\text{d}t}{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q}\phi \left(c_{\epsilon }\right)}={\displaystyle {\int }_{\Omega }2q{\left|\nabla c_{\epsilon }\right|}^{2q-1}\phi \left(c_{\epsilon }\right)\nabla \cdot \left(\Delta c_{\epsilon }-n_{\epsilon }{c}_{\epsilon }-u_{\epsilon }\cdot \nabla c_{\epsilon }\right)}\\ +{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q}{\phi }^{\prime }\left(c_{\epsilon }\right)\left(\Delta c_{\epsilon }-n_{\epsilon }{c}_{\epsilon }-u_{\epsilon }\cdot \nabla c_{\epsilon }\right)}\\ =:J_1+J_2\end{array}$

已知 $\nabla \xi \cdot \nabla \text{Δ}\xi =\frac{1}{2}\text{Δ}{\left|\nabla \xi \right|}^2-{\left|D^2\xi \right|}^2$ ，从而，对任意的 $t>0$ ，通过分部积分法可以得出

$\begin{array}{c}{J}_1=-q\left(2q-2\right){\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q-2}\phi \left(c_{\epsilon }\right)\left(D^2{c}_{\epsilon }\cdot \nabla c_{\epsilon }\right)}-q{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q+1}{\phi }^{\prime }\left(c_{\epsilon }\right)}-2q{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q-2}\phi \left(c_{\epsilon }\right){\left|D^2{c}_{\epsilon }\right|}^2}\\ -2q{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q}{n}_{\epsilon }\phi \left(c_{\epsilon }\right)}-2q{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q-2}{c}_{\epsilon }\phi \left(c_{\epsilon }\right)\left(\nabla n_{\epsilon }\cdot \nabla c_{\epsilon }\right)}\\ -2q{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q-1}\phi \left(c_{\epsilon }\right)\left(u_{\epsilon }\cdot D^2{c}_{\epsilon }\right)}+q{\displaystyle {\int }_{\partial \Omega }{\left|\nabla c_{\epsilon }\right|}^{2q-2}\phi \left(c_{\epsilon }\right)\frac{\partial {\left|\nabla c_{\epsilon }\right|}^2}{\partial v}}\end{array}$

和

$\begin{array}{c}{J}_2=-2q{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q-1}{\phi }^{\prime }\left(c_{\epsilon }\right)\left(D^2{c}_{\epsilon }\cdot \nabla c_{\epsilon }\right)}-{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q+2}{\phi }^{″}\left(c_{\epsilon }\right)}\\ -{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q}{n}_{\epsilon }{c}_{\epsilon }{\phi }^{\prime }\left(c_{\epsilon }\right)}-{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q}{\phi }^{\prime }\left(c_{\epsilon }\right)\left(u_{\epsilon }\cdot \nabla c_{\epsilon }\right)}\end{array}$

同时，注意到J₁和J₂右侧两项刚好满足

$-2q{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q-1}\phi \left(c_{\epsilon }\right)\left(u_{\epsilon }\cdot D^2{c}_{\epsilon }\right)}-{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q}{\phi }^{\prime }\left(c_{\epsilon }\right)\left(u_{\epsilon }\cdot \nabla c_{\epsilon }\right)}={\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q}\phi \left(c_{\epsilon }\right)\left(\nabla \cdot u_{\epsilon }\right)}=0$

这对我们的最终结果非常重要。整理J₁和J₂，对任意的 $t>0$ ，有

$\begin{array}{l}\frac{\text{d}}{\text{d}t}{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q}\phi \left(c_{\epsilon }\right)}+2q{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q-2}\phi \left(c_{\epsilon }\right){\left|D^2{c}_{\epsilon }\right|}^2}+{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q+2}{\phi }^{″}\left(c_{\epsilon }\right)}\\ =-q\left(2q-2\right){\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q-2}\phi \left(c_{\epsilon }\right)\left(D^2{c}_{\epsilon }\cdot \nabla c_{\epsilon }\right)}-2q{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q-1}{\phi }^{\prime }\left(c_{\epsilon }\right)\left(D^2{c}_{\epsilon }\cdot \nabla c_{\epsilon }\right)}\\ -q{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q+1}{\phi }^{\prime }\left(c_{\epsilon }\right)}-2q{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q}{n}_{\epsilon }\phi \left(c_{\epsilon }\right)}-{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q}{n}_{\epsilon }{c}_{\epsilon }{\phi }^{\prime }\left(c_{\epsilon }\right)}\\ -2q{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q-2}{c}_{\epsilon }\phi \left(c_{\epsilon }\right)\left(\nabla n_{\epsilon }\cdot \nabla c_{\epsilon }\right)}+q{\displaystyle {\int }_{\partial \Omega }{\left|\nabla c_{\epsilon }\right|}^{2q-2}\phi \left(c_{\epsilon }\right)\frac{\partial {\left|\nabla c_{\epsilon }\right|}^2}{\partial \upsilon }}\\ \le q\left(2q-2\right){\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q-2}\phi \left(c_{\epsilon }\right)\left|D^2{c}_{\epsilon }\right|\cdot \left|\nabla c_{\epsilon }\right|}+2q{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q-1}{\phi }^{\prime }\left(c_{\epsilon }\right)\left|D^2{c}_{\epsilon }\right|\cdot \left|\nabla c_{\epsilon }\right|}\\ +2q{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q-2}{c}_{\epsilon }\phi \left(c_{\epsilon }\right)\left|\nabla n_{\epsilon }\right|\cdot \left|\nabla c_{\epsilon }\right|}+q{\displaystyle {\int }_{\partial \Omega }{\left|\nabla c_{\epsilon }\right|}^{2q-2}\phi \left(c_{\epsilon }\right)\frac{\partial {\left|\nabla c_{\epsilon }\right|}^2}{\partial \upsilon }}\end{array}$

推导中充分用到了 $\phi \left(c_{\epsilon }\right)$ 和 ${\phi }^{\prime }\left(c_{\epsilon }\right)$ 的非负性。对任意的 $t>0$ ，最后调用三角不等式和杨氏不等式有

$q\left(2q-2\right){\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q-2}\phi \left(c_{\epsilon }\right)\left|D^2{c}_{\epsilon }\right|\cdot \left|\nabla c_{\epsilon }\right|}\le \frac{q}{2}{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q-2}\phi \left(c_{\epsilon }\right){\left|D^2{c}_{\epsilon }\right|}^2}+\frac{1}{2}q{\left(2q-2\right)}^2{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q}\phi (\; c\; \epsilon \; )}$

$2q{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q-1}{\phi }^{\prime }\left(c_{\epsilon }\right)\left|D^2{c}_{\epsilon }\right|\cdot \left|\nabla c_{\epsilon }\right|}\le \frac{q}{2}{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q-2}\phi \left(c_{\epsilon }\right){\left|D^2{c}_{\epsilon }\right|}^2}+2q{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q+2}\frac{{{\phi }^{\prime }}^2\left(c_{\epsilon }\right)}{\phi (\; c\; \epsilon \; )}}$

$2q{\displaystyle {\int }_{\Omega }{\left|\nabla c_{\epsilon }\right|}^{2q-2}{c}_{\epsilon }\phi \left(c_{\epsilon }\right)\left|\nabla n_{\epsilon }\right|\cdot \left|\nabla c_{\epsilon }\right|}\le \frac{p\left(p-1\right)}{8}{\displaystyle {\int }_{\Omega }{n}_{\epsilon }^{p-2}\phi \left(c_{\epsilon }\right){\left|\nabla n_{\epsilon }\right|}^2}+{\displaystyle {\int }_{\Omega }\frac{32q^2}{p\left(p-1\right)}{n}_{\epsilon }^{2-p}\phi \left(c_{\epsilon }\right){\left|\nabla c_{\epsilon }\right|}^{4q-2}{c}_{\epsilon }^2}$

代入方程经过整理很很容易得到定理3.2。

4. 结论

本文基于考虑重力对细胞运动的影响和趋化本身对流体的交互作用下的趋化流体方程，利用权函数的方法建立正则化问题解的先验估计，克服了流体项在数学处理上的困难，为进一步研究解的定性理论做好了准备。

致谢

感谢审稿人的审阅及对文章的意见和建议。

基金项目

阿坝师范学院校级科研项目(ASC20-01)。