Local and global existence of solutions of a Keller-Segel model coupled to the incompressible fluid equations

We consider a Keller-Segel model coupled to the incompressible fluid equations which describes the dynamics of swimming bacteria. We mainly take the incompressible Navier-Stokes equations for the fluid equation part. In this case, we first show the existence of unique local-in-time solutions for large data in scaling invariant Besov spaces. We then proceed to show that these solutions can be defined globally-in-time if some smallness conditions are imposed to initial data. We also show the existence of unique global-in-time self-similar solutions when initial data are sufficiently small in scaling invariant Besov spaces. But, these solutions do not exhibit (expected) temporal decay rates. So, we change the fluid part to the Stokes equations and we derive temporal decay rates of the bacteria density and the fluid velocity.