  In this paper, we are concerned with the global wellposedness of 2-D
density-dependent incompressible Navier-Stokes equations with variable
viscosity, in a critical functional frame- work which is invariant by the
scaling of the equations and under a non-linear smallness condition on
fluctuation of the initial density which has to be doubly exponential small
compared with the size of the initial velocity. In the second part of the
paper, we apply our methods combined with the techniques of R. Danchin and P.
B. Mucha to prove the global existence of solutions to inhomogeneous
Navier-Stokes system with piecewise constant initial density which has small
jump at the interface and is away from vacuum. In particular, this latter
result removes the smallness condition for the initial velocity in a
corresponding theorem of R. Danchin and P. B. Mucha.
