  The issue of global well-posedness for the 3D inhomogenous incompressible
Navier-Stokes equations was first addressed by Kazhikov in 1974. In this
manuscript, we obtain its global well-posedness for the system with
density-dependent viscosity under the smallness assumption of initial velocity
in the critical space $\dot{B}_{p,1}^{-1+\frac 3p}$ with $p\in ]1, 9/2]$. To
the best of our knowledge, this is the first result about the global
well-posedness for which one does not assume any smallness condition on the
density when the initial density is far away from vacuum.
