Local solutions for nonhomogeneous Navier-Stokes equations with large flux

The local existence of solutions to nonhomogeneous Navier-Stokes equations in cylindrical domains with arbitrary large flux is demonstrated. The existence is proved by the method of successive approximations. To show the existence with the lowest possible regularity the special Besov spaces called the Sobolev-Slobodetskii spaces are used. The inflow and outflow are prescribed on the parts of the boundary which are perpendicular to the $x_3$-axis. Since the inflow and outflow are positive the crucial point of this paper is to verify that $x_3$-coordinate of velocity is also positive. Finally, we conclude the local existence such that the velocity belongs to $W_{\sigma} ^{2+{s},1+{s}/2}(\Omega^t)$, the gradient of pressure to $W_{\sigma} ^{{s},{s}/2}(\Omega^t)$ and the density to $W_{r,\infty}^{1,1}(\Omega^t)$, where ${s}\in(0,1)$, ${\sigma} >3/{s}$, $r>5/{s}$, $r>{\sigma} $.