Controllability of Navier-Stokes equations by few low modes forcing

Investigated are 2D and 3D Navier-Stokes equations with periodic boundary conditions, controlled by the low-frequency in spatial variables external force. Using principles of geometric control theory, global controllability is established for finite-dimensional Galerkin's approximations of Navier-Stokes equations. In the case of two spatial variables also obtained is surjectivity of finite-dimensional projections of sets of attainability for initial Navier-Stokes equation. The latter result uses the continuity property, which has independent significance. Demonstrated is continuous dependence of the 2D Navier-Stokes equation solution on external force for the case, when the force space is characterized with weak relaxation topology.