In this paper, we study the initial-boundary value problem for the coupled chemotaxis(-Navier)-Stokes system with indirect signal production$\{\begin{array}{ccc}\hfill & {\partial }_{t}n+\mathit{u}\cdot \mathrm{\nabla }n=\mathrm{\Delta }n-\mathrm{\nabla }\cdot \left(n\mathrm{\nabla }c\right)+rn-\mu n^2,\hfill & \hfill (x,t)\in \mathrm{\Omega }\times (0,\infty ),\\ \hfill & {\partial }_{t}c+\mathit{u}\cdot \mathrm{\nabla }c=\mathrm{\Delta }c-c+v,\hfill & \hfill (x,t)\in \mathrm{\Omega }\times (0,\infty ),\\ \hfill & {\partial }_{t}v+\mathit{u}\cdot \mathrm{\nabla }v=\mathrm{\Delta }v-v+n,\hfill & \hfill (x,t)\in \mathrm{\Omega }\times (0,\infty ),\\ \hfill & {\partial }_t\mathit{u}+\kappa (\mathit{u}\cdot \mathrm{\nabla })\mathit{u}+\mathrm{\nabla }P=\mathrm{\Delta }\mathit{u}+n\mathrm{\nabla }\mathrm{\Phi },\mathrm{\nabla }\cdot \mathit{u}=0,\hfill & \hfill (x,t)\in \mathrm{\Omega }\times (0,\infty )\end{array}$ in a smooth bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^d(d=2,3)$, where $\kappa \in \{0,1\}$, $r\ge 0$ and $\mu >0$ are given constants. It is shown that when posed with no-flux/no-flux/no-flux/Dirichlet boundary condition and along with appropriate assumptions on regularity of the initial data $(n_0,c_0,v_0,{\mathit{u}}_0)$, the chemotaxis-Stokes system (i.e. $\kappa =0$) admits globally bounded classical solution in $\mathrm{\Omega }\subset {\mathbb{R}}^3$; the chemotaxis-Navier-Stokes system (i.e. $\kappa =1$) possesses globally bounded classical solution in $\mathrm{\Omega }\subset {\mathbb{R}}^2$.

本文研究具有间接信号产生的耦合趋化（Navier）-Stokes系统的初边值问题$\{\begin{array}{ccc}\hfill & {\partial }_{Ť}ñ+\mathit{ü}\cdot \mathrm{\nabla }ñ=\mathrm{\Delta }ñ-\mathrm{\nabla }\cdot （ñ\mathrm{\nabla }C）+\mathrm{[R}ñ-\mu {ñ}^{\mathrm{2个}}，\hfill & \hfill （X，Ť）\in \mathrm{\Omega }\times （0，\infty ），\\ \hfill & {\partial }_{Ť}C+\mathit{ü}\cdot \mathrm{\nabla }C=\mathrm{\Delta }C-C+v，\hfill & \hfill （X，Ť）\in \mathrm{\Omega }\times （0，\infty ），\\ \hfill & {\partial }_{Ť}v+\mathit{ü}\cdot \mathrm{\nabla }v=\mathrm{\Delta }v-v+ñ，\hfill & \hfill （X，Ť）\in \mathrm{\Omega }\times （0，\infty ），\\ \hfill & {\partial }_{Ť}\mathit{ü}+\kappa （\mathit{ü}\cdot \mathrm{\nabla }）\mathit{ü}+\mathrm{\nabla }P=\mathrm{\Delta }\mathit{ü}+ñ\mathrm{\nabla }\mathrm{\Phi }，\mathrm{\nabla }\cdot \mathit{ü}=0，\hfill & \hfill （X，Ť）\in \mathrm{\Omega }\times （0，\infty ）\end{array}$ 在光滑的有界域中 $\mathrm{\Omega }\subset {\mathbb{[R}}^d（d=\mathrm{2个}，3）$， 在哪里 $\kappa \in \{0，\mathrm{1个}\}$， $\mathrm{[R}\ge 0$ 和 $\mu >0$被赋予常数。结果表明，当存在无通量/无通量/无通量/ Dirichlet边界条件时，以及对初始数据规律性的适当假设时，$（{ñ}_0，C_0，v_0，{\mathit{ü}}_0）$，趋化性-斯托克斯系统（即 $\kappa =0$）承认在 $\mathrm{\Omega }\subset {\mathbb{[R}}^3$; 趋化性-Navier-Stokes系统（即$\kappa =\mathrm{1个}$）在 $\mathrm{\Omega }\subset {\mathbb{[R}}^{\mathrm{2个}}$。