This paper considers the following chemotaxis-Stokes system: {nt+u⋅∇n=Δn−∇⋅(nc∇c),ct+u⋅∇c=Δc−nc,ut=Δu+∇P+n∇ϕ,∇⋅u=0,\documentclass[12pt]{minimal}
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\begin{document}$$ \left \{ \textstyle\begin{array}{l} n_{t}+u\cdot\nabla n=\Delta n-\nabla\cdot(\frac{n}{c}\nabla c), \\ c_{t}+u\cdot\nabla c=\Delta c-nc, \\ u_{t}=\Delta u+\nabla P+n\nabla\phi, \\ \nabla\cdot u=0, \end{array}\displaystyle \right . $$\end{document} in two-dimensional smoothly bounded domains, which can be seen as a model to describe the migration of aerobic bacteria swimming in an incompressible fluid. It is proved that the corresponding initial-boundary value problem possesses a global generalized solution for any sufficiently regular initial data (n0,c0,u0)\documentclass[12pt]{minimal}
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\begin{document}$(n_{0}, c_{0}, u_{0})$\end{document} satisfying n0≥0\documentclass[12pt]{minimal}
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\begin{document}$n_{0}\geq0$\end{document} and c0>0\documentclass[12pt]{minimal}
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\begin{document}$c_{0}>0$\end{document}. Moreover, the solution component c satisfies c(⋅,t)⇀⋆0\documentclass[12pt]{minimal}
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\begin{document}$c(\cdot,t)\overset{\star}{\rightharpoonup}0$\end{document} in L∞(Ω)\documentclass[12pt]{minimal}
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\begin{document}$L^{\infty}(\Omega )$\end{document} as t→∞\documentclass[12pt]{minimal}
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\begin{document}$t\rightarrow\infty$\end{document} and c(⋅,t)→0\documentclass[12pt]{minimal}
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\begin{document}$c(\cdot,t)\rightarrow0$\end{document} in Lp(Ω)\documentclass[12pt]{minimal}
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\begin{document}$L^{p}(\Omega)$\end{document} as t→∞\documentclass[12pt]{minimal}
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\begin{document}$t\rightarrow\infty$\end{document} for any p∈[1,∞)\documentclass[12pt]{minimal}
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\begin{document}$p\in[1,\infty)$\end{document}.
