  In this paper, we investigate a chemotaxis-fluid system involving both the
effect of potential force on cells and the effect of chemotactic force on
fluid: \begin{equation*} \left\{ \begin{split} \partial_t n +
\mathbf{u}\cdot\nabla n & = \Delta n - \nabla\cdot\left(\chi(c)n\nabla c\right)
+ \nabla\cdot(n\nabla\phi), \\ \partial_t c + \mathbf{u}\cdot\nabla c &= \Delta
c - nf(c), \\ \partial_t\mathbf{u} + \kappa(\mathbf{u}\cdot\nabla)\mathbf{u} +
\nabla P & = \Delta\mathbf{u} - n\nabla\phi + \chi(c)n\nabla c, \\
\nabla\cdot\mathbf{u} &= 0 \end{split} \right. \end{equation*} in
$\mathbb{R}^d\times(0,T)\, (d=2,3)$. One of the novelties and difficulties here
is that the coupling in this model is stronger and more nonlinear than the
most-studied chemotaxis-fluid model. We will first establish several
extensibility criteria of classical solutions, which ensure us to extend the
local solutions to global ones in the three dimensional chemotaxis-Stokes case
and in the two dimensional chemotaxis-Navier-Stokes version under suitable
smallness assumption on $\|c_0\|_{L^{\infty}}$ with the help of a new entropy
functional inequality. Some further decay estimates are also obtained under
some suitable growth restriction on the potential $\nabla \phi$ at infinity. As
a byproduct of the entropy functional inequality, we also establish the
global-in-time existence of weak solutions to the three dimensional
chemotaxis-Navier-Stokes system. To the best of our knowledge, this seems to be
the first work addressing the global well-posedness and decay property of
solutions to the Cauchy problem of self-consistent chemotaxis-fluid system.
