Navier-Stokes equations in aperture domains: Global existence with bounded flux and time-periodic solutions

We consider the Navier-Stokes equations in an aperture domain of the three-dimensional Euclidean space. We are interested in proving the existence of regular solutions corresponding to small initial data and flux through the aperture. The flux is assumed to be smooth and bounded on (0, +). As a consequence, we prove the existence of a time-periodic solution corresponding to a time-periodic flux through the aperture. Finally, we compare our solution with a solution belonging to a wider class, showing that, if such a solution does exist, then the two solutions coincide.