Global Existence and Temporal Decay in Keller-Segel Models Coupled to Fluid Equations

We consider a Keller-Segel model coupled to the incompressible Navier-Stokes equations in spatial dimensions two and three. We establish the local existence of regular solutions and present some blow-up criteria for both cases that equations of oxygen concentration is of parabolic or hyperbolic type. We also prove that solutions exist globally in time and upper bounds of temporal decays are obtained under the some smallness conditions of initial data.