Temporal decay in negative Besov spaces for the 3D coupled chemotaxis-fluid equations

In this paper, we investigate the temporal decay for the three-dimensional coupled chemotaxis-fluid equations with low regularity assumptions on initial data in homogeneous Besov spaces. By using the Fourier splitting argument, we establish two weighted energy inequalities, which show that certain weighted negative Besov norms of solutions are preserved along time evolution. Combining such scaled energy estimates and the interpolation in Besov norms, we obtain the optimal decay rates of global solutions by solving an ordinary differential inequality.