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

Holder continuity of Keller-Segel equations of porous medium type coupled to fluid equations