We consider an initial-value problem for the incompressible chemotaxis-Stokes equations generalizing the porous-medium-type diffusion model with tensor-valued sensitivity $\left\{\begin{array}{ccc}\hfill & n_t+u\cdot \nabla n=\Delta n^m-\nabla \cdot (nS(x,n,c)\cdot \nabla c),\hfill & \hfill x\in \Omega ,t>0,\\ \hfill & c_t+u\cdot \nabla c=\Delta c-nc,\hfill & \hfill x\in \Omega ,t>0,\\ \hfill & u_t+\nabla P=\Delta u+n\nabla \varphi ,\hfill & \hfill x\in \Omega ,t>0,\\ \hfill & \nabla \cdot u=0,\hfill & \hfill x\in \Omega ,t>0\end{array}\right.\left(CNF\right)$ in a bounded domain $\Omega \subseteq {\mathbb{R}}^3$ with a smooth boundary, where $\varphi \in W^{2,\infty }\left(\Omega \right)$ is a given gravitational potential function. Problems of this type have been used to describe the mutual interaction between swimming aerobic bacteria populations and the surrounding fluid. The main feature is that the chemotactic sensitivity $S$ is a given parameter matrix on $\Omega \times {[0,\infty )}^2$, whose Frobenius norm satisfies $\left|S(x,n,c)\right|\le S_0\left(c\right)$ for all $(x,n,c)\in \bar{\Omega }\times [0,\infty )\times [0,\infty )$ with $S_0\left(c\right)$ nondecreasing on $[0,\infty )$. Based on a new energy-type argument combined with a new estimation technique, we show that if $m>\frac{11}{4}-\sqrt{3}$, an associated initial–boundary value problem admits at least one globally bounded weak solution. The above mentioned results significantly improved and extended previous results of several authors.

我们考虑不可压缩趋化性-斯托克斯方程的初值问题，该方程推广了具有张量值灵敏度的多孔介质型扩散模型$\left\{\begin{array}{ccc}\hfill & n_{吨}+你\cdot \nabla n=\Delta n^{米}-\nabla \cdot (n\mathrm{小号}(X,n,C)\cdot \nabla C),\hfill & \hfill X\in \Omega ,吨>0,\\ \hfill & C_{吨}+你\cdot \nabla C=\Delta C-nC,\hfill & \hfill X\in \Omega ,吨>0,\\ \hfill & {你}_{吨}+\nabla 磷=\Delta 你+n\nabla \phi ,\hfill & \hfill X\in \Omega ,吨>0,\\ \hfill & \nabla \cdot 你=0,\hfill & \hfill X\in \Omega ,吨>0\end{array}\right.\left(CñF\right)$在有界域中$\Omega \subseteq {\mathbb{R}}^3$具有平滑边界，其中$\phi \in W^{2,\infty }\left(\Omega \right)$是给定的引力势函数。这种类型的问题已被用于描述游泳的需氧细菌种群与周围流体之间的相互作用。主要特点是趋化敏感性$\mathrm{小号}$是一个给定的参数矩阵$\Omega \times {[0,\infty )}^2$，其 Frobenius 范数满足$\left|\mathrm{小号}(X,n,C)\right|\le {\mathrm{小号}}_0\left(C\right)$对所有人$(X,n,C)\in \bar{\Omega }\times [0,\infty )\times [0,\infty )$和${\mathrm{小号}}_0\left(C\right)$不减$[0,\infty )$. 基于一种新的能量类型论证与一种新的估计技术相结合，我们表明，如果$米>\frac{11}{4}-\sqrt{3}$，相关的初始边界值问题至少有一个全局有界弱解。上述结果显着改进和扩展了几位作者以前的结果。