Global classical solutions in a self-consistent chemotaxis(-Navier)-Stokes system

The self-consistent chemotaxis-fluid system{n(t)+u<middle dot> del n= triangle n- del <middle dot>(n del c) +del <middle dot>(n del phi),x is an element of ohm, t >0,c(t)+u<middle dot> del c= triangle c-nc,x is an element of ohm, t >0,u(t)+kappa(u<middle dot> del)u+del P= triangle u-n del phi+n del c,x is an element of ohm, t >0,del <middle dot>u= 0,x is an element of ohm, t >0,is considered under no-flux boundary conditions forn, cand the Dirichlet boundary condi-tion foruon a bounded smooth domain ohm subset of RN(N= 2,3),kappa is an element of {0,1}. The existence ofglobal bounded classical solutions is proved under a smallness assumption on & Vert;c0 & Vert;L infinity(ohm).Both the effect of gravity (potential force) on cells and the effect of the chemotacticforce on fluid are considered here, and thus the coupling is stronger than the most stud-ied chemotaxis-fluid systems. The literature on self-consistent chemotaxis-fluid systems ofthis type so far concentrates on the nonlinear cell diffusion as an additional dissipativemechanism. To the best of our knowledge, this is the first result on the boundedness ofa self-consistent chemotaxis-fluid system with linear cell diffusion