Global Existence, Unconditional Uniqueness, and Stability for the Patlak–Keller–Segel–Navier–Stokes Systems on the Whole Space

In this paper, we study the existence, unconditional uniqueness, and polynomial stability of mild solutions to the Patlak–Keller–Segel–Navier–Stokes systems on the whole space 𝕉^d (where d 4). We work in the framework of weak L^p spaces, i. e. , L^p, (𝕉^d). First, we use dispersive estimates together with linear and bilinear estimates to prove the existence of bounded mild solutions to the corresponding linear systems. Then, by fixed point arguments, we obtain the well-posedness of mild solutions to the semilinear systems. Moreover, we prove the unconditional uniqueness in the space L^p (𝕉^d). Finally, we establish polynomial stability for mild solutions by using the Yamazaki estimates.