  In this paper, we investigate an initial-boundary value problem for a
chemotaxis-fluid system in a general bounded regular domain $\Omega \subset
\mathbb{R}^N$ ($N\in\{2,3\}$), not necessarily being convex. Thanks to the
elementary lemma given by Mizoguchi & Souplet [10], we can derive a new type of
entropy-energy estimate, which enables us to prove the following: (1) for
$N=2$, there exists a unique global classical solution to the full
chemotaxis-Navier-Stokes system, which converges to a constant steady state
$(n_\infty, 0,0)$ as $t\to+\infty$, and (2) for $N=3$, the existence of a
global weak solution to the simplified chemotaxis-Stokes system. Our results
generalize the recent work due to Winkler [15,16], in which the domain $\Omega$
is essentially assumed to be convex.
