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

hal.structure.identifierLaboratoire Jacques-Louis Lions [LJLL (UMR_7598)]
hal.structure.identifierControl And GEometry [CaGE ]
dc.contributor.authorCORON, Jean-Michel
hal.structure.identifierÉcole normale supérieure - Rennes [ENS Rennes]
hal.structure.identifierInstitut de Recherche Mathématique de Rennes [IRMAR]
dc.contributor.authorMARBACH, Frédéric
hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorSUEUR, Franck
hal.structure.identifierAcademy of Mathematics and Systems Science [Beijing]
dc.contributor.authorZHANG, Ping
dc.date.accessioned2024-04-04T02:59:23Z
dc.date.available2024-04-04T02:59:23Z
dc.date.created2018
dc.date.issued2019-11
dc.identifier.issn2524-5317
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/192739
dc.description.abstractEnWe 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.
dc.description.sponsorshipBords, oscillations et couches limites dans les systèmes différentiels - ANR-16-CE40-0027
dc.description.sponsorshipCentre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation - ANR-11-LABX-0020
dc.language.isoen
dc.publisherSpringer
dc.title.enControllability of the Navier-Stokes equation in a rectangle with a little help of a distributed phantom force
dc.typeArticle de revue
dc.identifier.doi10.1007/s40818-019-0073-4
dc.subject.halMathématiques [math]/Optimisation et contrôle [math.OC]
dc.subject.halMathématiques [math]/Equations aux dérivées partielles [math.AP]
dc.identifier.arxiv1801.01860
bordeaux.journalAnnals of PDE
bordeaux.volume5
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.issue17
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
bordeaux.peerReviewedoui
hal.identifierhal-01676663
hal.version1
hal.popularnon
hal.audienceInternationale
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-01676663v1
bordeaux.COinSctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=Annals%20of%20PDE&rft.date=2019-11&rft.volume=5&rft.issue=17&rft.eissn=2524-5317&rft.issn=2524-5317&rft.au=CORON,%20Jean-Michel&MARBACH,%20Fr%C3%A9d%C3%A9ric&SUEUR,%20Franck&ZHANG,%20Ping&rft.genre=article


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