In a bounded planar domain Ω\documentclass[12pt]{minimal}
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\begin{document}$\varOmega $\end{document} with smooth boundary, the initial-boundary value problem of homogeneous Neumann type for the Keller-Segel-fluid system
{nt+∇⋅(nu)=Δn−∇⋅(n∇c),x∈Ω,t>0,0=Δc−c+n,x∈Ω,t>0,\documentclass[12pt]{minimal}
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\begin{document} $$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad }l} n_{t} + \nabla \cdot (nu) = \Delta n - \nabla \cdot (n\nabla c), & x\in \varOmega , \ t>0, \\ 0 = \Delta c -c+n, & x\in \varOmega , \ t>0, \end{array}\displaystyle \right . \end{aligned}$$ \end{document} is considered, where u\documentclass[12pt]{minimal}
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\begin{document}$u$\end{document} is a given sufficiently smooth velocity field on Ω‾×[0,∞)\documentclass[12pt]{minimal}
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\begin{document}$\overline {\varOmega }\times [0,\infty )$\end{document} that is tangential on ∂Ω\documentclass[12pt]{minimal}
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\begin{document}$\partial \varOmega $\end{document} but not necessarily solenoidal.
