In this paper, we consider the initial-boundary value problem of Keller–Segel–Navier–Stokes system ntκ+uκ·∇nκ=Δnκ-∇·(nκ(1+nκ)-α∇cκ),x∈Ω,t>0,ctκ+uκ·∇cκ=Δcκ-cκ+nκ,x∈Ω,t>0,utκ+κ(uκ·∇)uκ=Δuκ-∇Pκ+nκ∇ϕ,x∈Ω,t>0,∇·uκ=0,x∈Ω,t>0\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left\{ \begin{aligned}&n_t^{\kappa } +\mathbf{u}^{\kappa } \cdot \nabla n^{\kappa } =\varDelta n^{\kappa } - \nabla \cdot \big (n^{\kappa }(1+n^{\kappa })^{-\alpha }\nabla c^{\kappa } \big ),&\qquad x\in \varOmega ,\,t>0, \\&c_t^{\kappa } +\mathbf{u}^{\kappa } \cdot \nabla c^{\kappa } =\varDelta c^{\kappa } -c^{\kappa } +n^{\kappa },&\qquad x\in \varOmega ,\,t>0,\\&\mathbf{u}^{\kappa }_t+\kappa ({\mathbf {u}}^{\kappa } \cdot \nabla )\mathbf{u}^{\kappa } =\varDelta \mathbf{u}^{\kappa } -\nabla P^{\kappa } + n^{\kappa }\nabla \phi ,&\qquad x\in \varOmega ,\,t>0,\\&\nabla \cdot \mathbf{u}^{\kappa } =0,&\qquad x\in \varOmega ,\,t>0\\ \end{aligned} \right. \end{aligned}$$\end{document}with no-flux boundary conditions for nκ\documentclass[12pt]{minimal}
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\begin{document}$$n^{\kappa }$$\end{document} and cκ\documentclass[12pt]{minimal}
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\begin{document}$$c^{\kappa }$$\end{document} and a no-slip boundary condition for uκ\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf{u}^{\kappa }$$\end{document}, and with sufficiently regular initial data in a bounded domain Ω⊂R3\documentclass[12pt]{minimal}
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\begin{document}$$\varOmega \subset {\mathbb {R}}^3\,$$\end{document} with smooth boundary. Our result reveals that the solution (nκ,cκ,uκ)\documentclass[12pt]{minimal}
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\begin{document}$$(n^{\kappa },\,c^{\kappa },\,\mathbf{} u^{\kappa })$$\end{document} converges towards the solution of the corresponding Stokes variant (κ=0)\documentclass[12pt]{minimal}
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\begin{document}$$(\kappa =0)$$\end{document} with an exponential time decay rate as κ→0+\documentclass[12pt]{minimal}
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\begin{document}$$\kappa \rightarrow 0_{+}$$\end{document} provided that ‖n0‖L32(Ω),\documentclass[12pt]{minimal}
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\begin{document}$$\Vert n_0\Vert _{L^\frac{3}{2}(\varOmega )},$$\end{document}‖∇c0‖L3(Ω)\documentclass[12pt]{minimal}
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\begin{document}$$\Vert \nabla c_0\Vert _{L^3(\varOmega )}$$\end{document} and ‖u0‖L3(Ω)\documentclass[12pt]{minimal}
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\begin{document}$$\Vert {\mathbf {u}}_0\Vert _{L^3(\varOmega )}$$\end{document} are suitable small. Therefore, we can extend the obtained result on the Stokes limit in the two-dimensional case to the more complicated three-dimensional case.
