  We consider a coupled system consisting of a degenerate porous medium type of
Keller-Segel system and Stokes system modeling the motion of swimming bacteria
living in fluid and consuming oxygen. We establish the global existence of weak
solutions and H\"older continuous solutions in dimension three, under the
assumption that the power of degeneracy is above a certain number depending on
given parameter values. To show H\"older continuity of weak solutions, we
consider a single degenerate porous medium equation with lower order terms, and
via a unified method of proof, we obtain H\"older regularity, which is of
independent interest.
