Suppression of blow-up in multi-species Patlak-Keller-Segel-Navier-Stokes system via the Poiseuille flow in a finite channel

In this paper, we consider the multi-species parabolic-elliptic Patlak-Keller-Segel system coupled with the Navier-Stokes equations near the 2-D Poiseuille flow $(\ A(1-y^2), 0\ )$ in a finite channel $\Omega=\mathbb{T}\times\mathbb{I}$ with $ \mathbb{I}=(-1,1)$. Furthermore, the Navier-slip boundary condition is imposed on the perturbation of velocity $u$. We show that if the Poiseuille flow is sufficiently strong ($A$ is large enough), the solutions to the system are global in time without any smallness restriction on the initial cell mass.