In this paper, we consider the following Keller–Segel(–Navier)–
Stokes system
where
Ω⊂RN (N=2,3) is a bounded domain with smooth boundary
∂Ω, κ∈R and
χ(c) is assumed to generalize the prototype
It is proved that i) for
κ≠0 and
N=2 or
κ=0 and
N∈{2,3}, the corresponding initial–boundary problem admits a unique global classical solution which is bounded; ii) for
κ≠0 and
N=3, the corresponding initial–boundary problem possesses at least one global weak solution.