Existence and uniqueness theorem on mild solutions to the Keller-Segel system coupled with the Navier-Stokes fluid

We consider the Keller-Segel system coupled with the Navier-Stokes fluid in the whole space, and prove the existence of global mild solutions with the small initial data in the scaling invariant space. Our method is based on the implicit function theorem which yields necessarily continuous dependence of solutions for the initial data. As a byproduct, we show the asymptotic stability of solutions as the time goes to infinity. Since we may deal with the initial data in the weak L^p-spaces, the existence of self-similar solutions provided the initial data are small homogeneous functions.