This paper reconsiders the Keller–Segel–Stokes system with indirect signal production nt+u·∇n=Δn-χ∇·(n∇v)+rn-μnα,vt+u·∇v=Δv-v+w,wt+u·∇w=Δw-w+n,ut=Δu-∇P+n∇ϕ,∇·u=0(⋆)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} n_{t}+u\cdot \nabla n=\Delta n-\chi \nabla \cdot (n\nabla v)+rn-\mu n^\alpha ,\\ v_{t}+u\cdot \nabla v=\Delta v-v+w,\\ w_{t}+u\cdot \nabla w=\Delta w-w+n,\\ u_t=\Delta u-\nabla P+n\nabla \phi ,\qquad \nabla \cdot u=0 \end{array}\right. }\quad (\star ) \end{aligned}$$\end{document}in a bounded domain Ω⊂R3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Omega \subset \mathbb {R}^3$$\end{document} with smooth boundary, where χ>0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\chi >0$$\end{document}, r∈R\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$r\in \mathbb {R}$$\end{document}, μ>0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mu >0$$\end{document}, α>1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\alpha >1$$\end{document} and ϕ∈W2,∞(Ω)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\phi \in W^{2,\infty }(\Omega )$$\end{document}. In the case α=2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\alpha =2$$\end{document} of quadratic degradation, the global boundedness of classical solution for the no-flux/no-flux/no-flux/Dirichlet initial-boundary value problem of (⋆)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(\star )$$\end{document} has been established in Dai and Liu (J Differ Equ 314:201–250, 2022). In some situations α∈(53,2)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\alpha \in (\frac{5}{3},2)$$\end{document} of subquadratic degradation, the present study reveals that for all reasonably regular initial data, the associated initial-boundary value problem possesses a globally bounded classical solution. This result significantly improves the above mentioned one. It is worth noting that in previously known results on the corresponding system with direct signal production, the blow-up of solution can be prevented by either suitably large quadratic degradation coefficient or sublinear signal production. In comparison with these results for the case of direct signal production, the present result mathematically quantizes the regularizing effect of the indirect signal production mechanism on the Keller–Segel–Stokes system in the sense that the well-known destabilizing action of chemotactic cross-diffusion can be completely excluded by certain subquadratic degradations with arbitrarily small coefficient.
