  We consider the Cauchy problem for the Keller-Segel system of consumption
type coupled with the incompressible Euler equations in $\mathbb{R}^2$. This
coupled system describes a biological phenomenon in which aerobic bacteria
living in slightly viscous fluids (such as water) move towards a higher oxygen
concentration to survive. We firstly prove the local existence of smooth
solutions for arbitrary smooth initial data. Then we show that these smooth
solutions can be extended globally if the initial density of oxygen is
sufficiently small. The main ingredient in the proof is the $W^{1,q}$-energy
estimate $(q>2)$ motivated by the partially inviscid two-dimensional Boussinesq
system in \cite{C06}. Our result improves the well-known global well-posedness
of the two-dimensional Keller-Segel system of consumption type coupled with the
incompressible Navier-Stokes equations.

Analysis of PDEs