Jm. Coron et Av. Fursikov, GLOBAL EXACT CONTROLLABILITY OF THE 2D NAVIER-STOKES EQUATIONS ON A MANIFOLD WITHOUT BOUNDARY, Russian journal of mathematical physics, 4(4), 1996, pp. 429-448
Global exact controllability in arbitrary small time is established fo
r the Navier-Stokes equations of incompressible incompressible fluids
on any connected orientable compact Riemannian surface without boundar
y when the control is defined on an arbitrary nonempty open subset of
the surface.