CORON, Jean-Michel; MARBACH, Frédéric; SUEUR, Franck; ZHANG, Ping

doi:10.1007/s40818-019-0073-4

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CORON, Jean-Michel
Laboratoire Jacques-Louis Lions [LJLL (UMR_7598)]
Control And GEometry [CaGE]

MARBACH, Frédéric
École normale supérieure - Rennes [ENS Rennes]
Institut de Recherche Mathématique de Rennes [IRMAR]

SUEUR, Franck
Institut de Mathématiques de Bordeaux [IMB]

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Ce document a été publié dans

Annals of PDE. 2019-11, vol. 5, n° 17

Springer

Résumé en anglais

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 where the distributed force is small only in a negative Sobolev space. It is a further step towards an answer to Jacques-Louis Lions' question 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 for Burgers and 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 nonlin-ear Cauchy-Kovalevskaya estimate relying only on horizontal analyticity.< Réduire

URI

https://oskar-bordeaux.fr/handle/20.500.12278/192739

DOI

http://dx.doi.org/10.1007/s40818-019-0073-4

Project ANR

Bords, oscillations et couches limites dans les systèmes différentiels - ANR-16-CE40-0027
Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation - ANR-11-LABX-0020

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Unités de recherche

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251