  The chemotaxis-Navier-Stokes system linking the chemotaxis equations \[ n_t +
u\cdot\nabla n = \Delta n - \nabla \cdot (n\chi(c)\nabla c) \] and \[ c_t +
u\cdot\nabla c = \Delta c-nf(c) \] to the incompressible Navier-Stokes
equations, \[ u_t + (u\cdot\nabla)u = \Delta u +\nabla P + n \nabla \Phi,
\qquad \nabla \cdot u = 0, \] is considered under homogeneous boundary
conditions of Neumann type for $n$ and $c$, and of Dirichlet type for $u$, in a
bounded convex domain $\Omega\subset R^3$ with smooth boundary, where $\Phi\in
W^{1,\infty}(\Omega)$, and where $f\in C^1([0,\infty))$ and $\chi\in
C^2([0,\infty))$ are nonnegative with $f(0)=0$. Problems of this type have been
used to describe the mutual interaction of populations of swimming aerobic
bacteria with the surrounding fluid. Up to now, however, global existence
results seem to be available only for certain simplified variants such as
e.g.the two-dimensional analogue, or the associated chemotaxis-Stokes system
obtained on neglecting the nonlinear convective term in the fluid equation. The
present work gives an affirmative answer to the question of global solvability
in the following sense: Under mild assumptions on the initial data, and under
modest structural assumptions on $f$ and $\chi$, inter alia allowing for the
prototypical case when \[ f(s)=s \quad {for all} s\ge 0 \qquad {and} \qquad
\chi \equiv const., \] the corresponding initial-boundary value problem is
shown to possess a globally defined weak solution.
