A coupled Keller–Segel–Stokes model: global existence for small initial data and blow-up delay

We study a system consisting of the elliptic-parabolic Keller-Segel equations coupled to Stokes equations by transport and gravitational forcing. We show global-in-time existence of solutions for small initial mass in 2D. In 3D we establish global existence assuming that the initial L 3/2 -norm is small. Moreover, we give numerical evidence that for this extension of the Keller-Segel system in 2D, solutions exist with mass above 8, which is the critical mass for the system without fluid.