In this paper, we study the existence and uniqueness of periodic mild solutions for the Patlak-Keller-Segel-Navier-Stokes system in the whole space \({\mathbb {R}}^d\) (where \(d \geqslant 4\) ). First, we employ dispersive estimates, along with linear and bilinear estimates, to establish the existence of bounded mild solutions to the corresponding linear system. Next, we prove a Massera-type principle that guarantees the existence of periodic solutions for the linear system. By combining these linear results with fixed-point arguments, we establish the well-posedness of periodic mild solutions for the semilinear system. Furthermore, we demonstrate the polynomial stability of these solutions.
