Global solvability and boundedness to a coupled chemotaxis-fluid model with arbitrary porous medium diffusion

In this paper, we deal with the following coupled chemotaxis-fluid model{nt+u⋅∇n=Δnm−∇⋅(n(1+n)−α∇c)+γn−μn2,ct+u⋅∇c=Δc−c+n,ut=Δu−∇π+n∇φ,∇⋅u=0 in a bounded domain Ω⊂R3 with zero-flux boundary for n,c and no-slip boundary for u. It is shown that for any large initial datum, for any m>0, α>0, the problem admits a global weak solution, which is uniformly bounded. On the basis of this, the stability of the steady states also be discussed. The study of this paper improve the results in [15], in which, the global existence and boundedness of weak solutions are established for m>13, α>65−m.