On asymptotically almost periodic mild solutions of chemotaxis-fluid systems on bounded domains

We study the well-posedness and exponential stability of almost periodic (AP-) and asymptotically almost periodic (AAP-) mild solutions for the chemotaxis-Navier-Stokes systems on a bounded domain Ω ⊂ ℝd (d ⩾ 2) with the smooth boundary. Our strategy is: we first prove the well-posedness of mild solutions for the corresponding linear systems by using the dispersive and smoothing estimates of the Newmann heat semigroup. Then, we establish the well-posedness of AP- and AAP-mild solutions for the linear systems. Next, we combine the well-posed results obtained for linear systems and fixed-point arguments to establish the well-posedness of AP- and AAP-mild solutions for the semilinear systems. Finally, the exponential decay of these solutions is proven by using the Gronwall inequality.