Global Well-posedness of the Chemotaxis-Navier-Stokes Equations in two dimensions

We consider two dimensional Keller-Segel equations coupled with the Navier-Stokes equations modelled by Tuval et al.[32]. Assuming that the chemotactic sensitivity and oxygen consumption rate are nondecreasing and differentiable, we prove that there is no blow-up in a finite time for solutions with large initial data to chemotaxis-Navier-Stokes equations in two dimensions. In addition, temporal decays of solutions are shown, as time tends to infinity.
Comment: This paper has been withdrawn by the author due to a crucial error in the proof of Lemma 5