  We study global existence and uniqueness of solutions to instationary
inhomogeneous Navier-Stokes equations on bounded domains of $\R^n, n\geq 3$,
with initial velocity in $B^0_{q,\infty}(\Om)$, $q\geq n$, and piecewise
constant initial density. \par To this end, first, existence for momentum
equations with prescribed density is obtained based on maximal
$L^\infty_\ga$-regularity of the Stokes operator in little Nicolskii space
$b^{s}_{q,\infty}(\Om)$, $s\in\R$, exploited in \cite{RiZh14} and existence for
divergence problem in $b^{-s}_{q,\infty}(\Om)$, $s>0$. Then, we obtain an
existence result for transport equations in the space of pointwise multipliers
for $b^{-s}_{q,\infty}(\Om)$, $s>0$. Finally, the existence of the
inhomogeneous Navier-Stokes equations is proved via an iterate scheme while the
proof of uniqueness is done via a Lagrangian approach based on the prior
results on momentum equations and transport equation.
