  We consider the global approximate controllability of the two-dimensional
incompressible Navier-Stokes system driven by a physically localized and
degenerate force. In other words, the fluid is regulated via four scalar
controls that depend only on time and appear as coefficients in an effectively
constructed driving force supported in a given subdomain. Our idea consists of
squeezing low mode controls into a small region, essentially by tracking their
actions along the characteristic curves of a linearized vorticity equation. In
this way, through explicit constructions and by connecting Coron's return
method with recent concepts from geometric control, the original problem for
the nonlinear Navier-Stokes system is reduced to one for a linear transport
equation steered by a global force. This article can be viewed as an attempt to
tackle a well-known open problem due to Agrachev.
