In this article, we show that the Navier-Stokes system with variable
density and viscosity is locally well-posed in the Besov
space
$$
\dot B^{\frac{N}{p}}_{p\,1}(\R^N)\times\big(\dot
B^{\frac{N}{p}-1}_{p\,1}(\R^N)\big)^N,
$$
for $1 < p\leq N$ when the
initial density approaches a strictly positive constant. This result
generalizes the work by R. Danchin for the case where the viscosity
is constant and $p=2$ (see [Danchin, R.: Density-dependent incompressible viscous fluids in critical
spaces. Proc. Roy. Soc. Edinburgh Sect. A 133 (2003), 1311-1334.]). Moreover, we prove existence
and uniqueness in the Sobolev space\arriba{2}
$$
H^{\frac{N}{2}+\alpha}(\R^N)\times\big(H^{\frac{N}{2}-1+\alpha}(\R^N)\big)^N
$$
for $\alpha>0,$ generalizing R. Danchin's result for the case where
viscosity is constant (see [Danchin, R.: Local and global well-posedness results for flows of inhomogeneous
viscous fluids. Adv. Differential Equations 9 (2004), 353-386.]).