Decay estimates of the coupled chemotaxis-fluid equations in R3

In this paper, we are concerned with a Chemotaxis-Navier-Stokes model, arising from biology, which is a coupled system of the chemotaxis equations and the viscous incompressible fluid equations with transport and external force. The optimal convergence rates of classical solutions to the Chemotaxis-Navier-Stokes system for small initial perturbation around constant states are obtained by pure energy method under the assumption the initial data belong to H˙−s∩HN, N⩾3 (0⩽s<3/2). The H˙−s (0⩽s<3/2) negative Sobolev norms are shown to be preserved along time evolution. Compared to the result in [5], we obtain the optimal decay rates of the higher-order spatial derivatives of the solutions.