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We consider the 2D incompressible Navier-Stokes equation in a rectangle withthe usual no-slip boundary condition prescribed on the upper and lowerboundaries. We prove that for any positive time, for any finite energy initialdata, there exist controls on the left and right boundaries and a distributedforce, which can be chosen arbitrarily small in any Sobolev norm in space, suchthat the corresponding solution is at rest at the given final time.  Our work improves earlier results where the distributed force is small onlyin a negative Sobolev space. It is a further step towards an answer toJacques-Louis Lions' question about the small-time global exact boundarycontrollability of the Navier-Stokes equation with the no-slip boundarycondition, for which no distributed force is allowed.  Our analysis relies on the well-prepared dissipation method already used forBurgers and for Navier-Stokes in the case of the Navier slip-with-frictionboundary condition. In order to handle the larger boundary layers associatedwith the no-slip boundary condition, we perform a preliminary regularizationinto analytic functions with arbitrarily large analytic radius and prove along-time nonlinear Cauchy-Kovalevskaya estimate relying only on horizontalanalyticity.

Controllability of the Navier-Stokes equation in a rectangle with a little help of a distributed phantom force