Decay estimates of the coupled chemotaxis–fluid equations in [formula omitted

Abstract

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.