Existence of incompressible and immiscible flows in critical function spaces on bounded domains

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.