  In 2004, Dombrowski et al. showed that suspensions of aerobic bacteria often
develop flows from the interplay of chemotaxis and buoyancy, which is described
as the chemotaxis-Navier-Stokes model, and they observed self-concentration
occurs as a turbulence by exhibiting transient, reconstituting, high-speed
jets, which entrains nearby fluid to produce paired, oppositely signed
vortices. In order to investigate the properties of these vortices (singular
points), one approach is to follow the partial regularity theory of
Caffarelli-Kohn-Nirenberg to study the singularity properties of suitable weak
solutions. In this paper, we established the existence of suitable weak
solution for the three dimensional chemotaxis-Navier-Stokes equations, where
the main difficulty is to establish appropriate local energy inequalities.
