Decay estimate of solutions to the coupled chemotaxis-fluid equations in R3

We are concerned with a model arising from biology, which is coupled system of the chemotaxis equations and the viscous incompressible fluid equations through transport and external forcing. We study the large time behavior of solutions near a constant states to the chemotaxis-Navier-Stokes system in R3. Appealing to a pure energy method, we first obtain a global existence theorem by assuming that the H3 norm of the initial data is small, but the higher order derivatives can be arbitrary large. If the initial data belongs to homogeneous Sobolev norms Ḣ−s(0≤s<32) or homogeneous Besov norms Ḃ2,∞−s(0