  We consider a coupled system consisting of the Navier-Stokes equations and a
porous medium type of Keller-Segel system that model the motion of swimming
bacteria living in fluid and consuming oxygen. We establish the global-in-time
existence of weak solutions for the Cauchy problem of the system in dimension
three. In addition, if the Stokes system, instead Navier-Stokes system, is
considered for the fluid equation, we prove that bounded weak solutions exist
globally in time.

Analysis of PDEs