Approximate controllability of the Navier–Stokes system in unbounded domains

The question of the approximate controllability for the 2- and the 3-dimensional Navier–Stokes system defined in the exterior of a bounded domain $\omega$ or in the entire space is studied. It is shown that one can find boundary controls or locally distributed controls (having support in a prescribed bounded domain) defined on the right-hand side of the system such that in prescribed time the solution of the Navier–Stokes system becomes arbitrarily close to an arbitrary prescribed divergence-free vector field.