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We deal with the 3D Navier-Stokes equation in a smooth simply connectedbounded domain, with controls on a non-empty open part of the boundary and aNavier slip-with-friction boundary condition on the remaining, uncontrolled,part of the boundary. We extend the small-time global exact controllabilityresult in [J. M. Coron, F. Marbach and F. Sueur, Small-time global exactcontrollability of the Navier-Stokes equation with Navier slip-with-frictionboundary conditions, J. Eur. Math. Soc., 22 (2020),1625-1673.] from Leray weaksolutions to the case of smooth solutions.  Our strategy relies on a refinement of the method of well-prepareddissipation of the viscous boundary layers which appear near the uncontrolledpart of the boundary, which allows to handle the multi-scale features in afiner topology.  As a byproduct of our analysis we also obtain a small-time global approximateLagrangian controllability result, extending to the case of the Navier-Stokesequations the recent results [O. Glass and T. Horsin, Approximate Lagrangiancontrollability for the 2-D Euler equation. Application to the control of theshape of vortex patches, J. Math. Pures Appl. (9), 93 (2010), 61-90], [O. Glassand T. Horsin, Prescribing the motion of a set of particles in athree-dimensional perfect fluid, SIAM J. Control Optim., 50 (2012), 2726-2742],[T. Horsin and O. Kavian, Lagrangian controllability of inviscid incompressiblefluids: a constructive approach, ESAIM Control Optim. Calc. Var., 23 (2017),1179-1200] in the case of the Euler equations and the result [O. Glass and T.Horsin, Lagrangian controllability at low Reynolds number, ESAIM Control Optim.Calc. Var., 22 (2016), 1040-1053] in the case of the steady Stokes equations.

Smooth controllability of the Navier-Stokes equation with Navier conditions. Application to Lagrangian controllability