Wellposedness of the Keller-Segel Navier-Stokes equations in the critical Besov spaces

Abstract

We consider the Keller-Segel model coupled with the incompressibleNavier-Stokes equations in dimension three. We prove the local intime existence of the solution for large initial data and the global intime existence of the solution for small initial data plus somesmallness condition on the gravitational potential in the critical Besovspaces, which are new results for the model.