Global existence in critical Besov spaces for the coupled chemotaxis-fluid equations

In this paper, we are concerned with a system coupling the parabolic-parabolic Keller-Segel system to the viscous incompressible Navier-Stokes equations in spatial dimensions three, and prove the global existence of solutions with small initial data in critical Besov spaces with minimal regularity order by using the elementary paradifferential calculus and a special type of iteration scheme.