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 Lp-spaces, the existence of self-similar solutions provided the initial data are small homogeneous functions.
- Asymptotic stability
- Continuous dependence for the initial data
- Existence of global mild solutions
- Keller-Segel system
- Navier-Stokes equations
- Self-similar solutions
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