We study global existence and uniqueness of solutions to instationary inhomogeneous Navier-Stokes equations on bounded domains of $\Bbb{R}^n, n\geq 3$, with initial velocity in $B^0_{q,\infty}(\Omega)$, $q\geq n$, and piecewise constant initial density.
To this end, first, existence for momentum equations with prescribed density is obtained based on maximal $L^\infty_\gamma$-regularity of the Stokes operator in little Nicolskii space $b^{s}_{q,\infty}(\Omega)$, $s\in\Bbb{R}$, exploited in [J. Math. Fluid Mech. 18(2016), 102-131] and existence for divergence problem in $b^{-s}_{q,\infty}(\Omega)$, s>0. Then, we obtain an existence result for transport equations in the space of pointwise multipliers for $b^{-s}_{q,\infty}(\Omega)$, 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.