Existence and decay of global smooth solutions to the coupled chemotaxis–fluid model

In this paper, we consider a coupled chemotaxis–fluid model in R3R3, which describes the so-called “chemotaxis Boycott effect” arising from the interplay of chemotaxis and diffusion of nutrients or signaling chemicals in bacterial suspensions. It is shown that the Cauchy problem has a unique global-in-time solution (n,c,u)(x,t)(n,c,u)(x,t) on R3×(0,∞)R3×(0,∞), provided the invariant initial norm ‖(u0,∇c0)‖L3+‖n0‖L3/2‖(u0,∇c0)‖L3+‖n0‖L3/2 is suitably small, or the diffusion coefficients of cells, substrate and fluid (i.e. λ, ν, μ  ) are large enough. We also show that the invariant norm ‖(u,∇c)(t)‖L33+‖n(t)‖L3/23/2 is monotone decreasing in t   for all t≥0t≥0.