Controllability of the Navier-Stokes Equation in a Rectangle with a Little Help of a Distributed Phantom Force

We consider the 2D incompressible Navier-Stokes equation in a rectangle with the usual no-slip boundary condition prescribed on the upper and lower boundaries. We prove that for any positive time, for any finite energy initial data, there exist controls on the left and right boundaries and a distributed force, which can be chosen arbitrarily small in any Sobolev norm in space, such that the corresponding solution is at rest at the given final time. Our work improves earlier results in Guerrero et al. (C R Math Acad Sci Paris 343(9):573-577, 2006), Guerrero et al. (J Math Pures Appl (9) 98(6):689-709, 2012) where the distributed force is small only in a negative Sobolev space. It is a further step towards an answer to Lions' question in Lions (Exact controllability for distributed systems. Some trends and some problems. In: Applied and industrial mathematics (Venice, 1989), vol. 56, Math. Appl., pp. 59-84. Kluwer Academic Publications, Dordrecht, 1991) about the small-time global exact boundary controllability of the Navier-Stokes equation with the no-slip boundary condition, for which no distributed force is allowed. Our analysis relies on the well-prepared dissipation method already used in Marbach (J Math Pures Appl (9) 102(2):364-384, 2014) for Burgers and in Coron et al. (J Eur Math Soc, 2016) for Navier-Stokes in the case of the Navier slip-with-friction boundary condition. In order to handle the larger boundary layers associated with the no-slip boundary condition, we perform a preliminary regularization into analytic functions with arbitrarily large analytic radius and prove a long-time nonlinear Cauchy-Kovalevskaya estimate relying only on horizontal analyticity, in the spirit of Chemin (Le systeme de Navier-Stokes incompressible soixante dix ans apres Jean Leray. In: Actes des Journees Mathematiques a la Memoire de Jean Leray, vol. 9, Seminar Congress, pp. 99-123. Societe Mathematique de France, Paris (2004), 2004), Zhang and Zhang (J Funct Anal 270(7):2591-2615, 2016).