  In this work, we investigate the small-time global exact controllability of
the Navier-Stokes equation, both towards the null equilibrium state and towards
weak trajectories. We consider a viscous incompressible fluid evolving within a
smooth bounded domain, either in 2D or in 3D. The controls are only located on
a small part of the boundary, intersecting all its connected components. On the
remaining parts of the boundary, the fluid obeys a Navier slip-with-friction
boundary condition. Even though viscous boundary layers appear near these
uncontrolled boundaries, we prove that small-time global exact controllability
holds. Our analysis relies on the controllability of the Euler equation
combined with asymptotic boundary layer expansions. Choosing the boundary
controls with care enables us to guarantee good dissipation properties for the
residual boundary layers, which can then be exactly canceled using local
techniques.
