We deal with the 3D Navier–Stokes equation in a smooth simply connected bounded domain, with controls on a non-empty open part of the boundary and a Navier slip-with-friction boundary condition on the remaining, uncontrolled, part of the boundary. We extend the small-time global exact null controllability result in Coron et al. (J Eur Math Soc 22:1625–1673, 2020) from Leray weak solutions to the case of smooth solutions. Our strategy relies on a refinement of the method of well-prepared dissipation of the viscous boundary layers which appear near the uncontrolled part of the boundary, which allows to handle the multi-scale features in a finer topology. As a byproduct of our analysis we also obtain a small-time global approximate Lagrangian controllability result, extending to the case of the Navier–Stokes equations the recent results (Glass and Horsin in J Math Pures Appl (9) 93:61–90, 2010; Glass and Horsin in SIAM J Control Optim 50: 2726–2742, 2012; Horsin and Kavian in ESAIM Control Optim Calc Var 23:1179–1200, 2017) in the case of the Euler equations and the result (Glass and Horsin in ESAIM Control Optim Calc Var 22:1040–1053, 2016) in the case of the steady Stokes equations.

Publication:

Archive for Rational Mechanics and Analysis volume 243, pages869–941 (2022)

Published: 07 January 2022

Author:

J. Liao

Institute of Applied Physics and Computational Mathematics, Beijing 100094, China.

Academy of Mathematics and Systems Science, The Chinese Academy of Sciences, Beijing 100190, China. e-mail: liaojiajiang15@mails.ucas.ac.cn

F. Sueur

Institut de Mathématiques de Bordeaux, Université Bordeaux, 33405 Talence Cedex, France. e-mail: Franck.Sueur@math.u-bordeaux.fr

Institut Universitaire de France, 1 rue Descartes, 75231 Paris Cedex 05, France.

P. Zhang

Academy of Mathematics and Systems Science and Hua Loo-Keng Key Laboratory of Mathematics, The Chinese Academy of Sciences, Beijing 100190, China. e-mail: zp@amss.ac.cn

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China.