A class of parabolic-elliptic Keller–Segel–Stokes systems generalizing the prototype [graphic not available: see fulltext] is considered under boundary conditions of homogeneous Neumann type for n (the density of the cell population) and c (the chemical concentration), and Dirichlet type for u (the velocity field), in a bounded domain Ω⊆R2\documentclass[12pt]{minimal}
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\begin{document}$$\Omega \subseteq {\mathbb {R}}^2$$\end{document} with smooth boundary, where CS>0\documentclass[12pt]{minimal}
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\begin{document}$$C_S>0$$\end{document} and ϕ\documentclass[12pt]{minimal}
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\begin{document}$$\phi $$\end{document} is a given sufficiently smooth function. The model is proposed to describe chemotaxis-fluid interaction in cases when the evolution of the chemoattractant is essentially dominated by production through cells. Moreover, the chemical diffuses much faster than the cells move. It is shown that under the condition that α>0,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \alpha >0, \end{aligned}$$\end{document}for any sufficiently smooth initial data (n0,u0)\documentclass[12pt]{minimal}
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\begin{document}$$(n_0,u_0 )$$\end{document} satisfying some compatibility conditions, the associated initial-boundary-value problem (KSF) possesses a global bounded classical solution. In comparison to the result for the corresponding fluid-free system, it is easy to see that the restriction on α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} here is optimal. Building on this boundedness property, it can finally even be proved that the corresponding solution of the system decays to (n¯0,n¯0,0)\documentclass[12pt]{minimal}
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\begin{document}$$({\bar{n}}_0,{\bar{n}}_0,0)$$\end{document} exponentially if CS\documentclass[12pt]{minimal}
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\begin{document}$$C_S$$\end{document} is smaller, where n¯0=1|Ω|∫Ωn0\documentclass[12pt]{minimal}
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\begin{document}$${\bar{n}}_0=\frac{1}{|\Omega |}\int _{\Omega }n_0$$\end{document}. Our main tool is consideration of the energy functional ∫Ωn1+α,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \int _{\Omega }n^{1+\alpha }, \end{aligned}$$\end{document}which is a new energy-like functional.
