{
  "id": "2004.04358",
  "version": "v1",
  "published": "2020-04-09T04:01:12.000Z",
  "updated": "2020-04-09T04:01:12.000Z",
  "title": "Smooth controllability of the Navier-Stokes equation with Navier conditions. Application to Lagrangian controllability",
  "authors": [
    "J. Liao",
    "F. Sueur",
    "P. Zhang"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "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 controllability result in [J. M. Coron, F. Marbach and F. Sueur, Small-time global exact controllability of the Navier-Stokes equation with Navier slip-with-friction boundary conditions, J. Eur. Math. Soc., 22 (2020),1625-1673.] 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 [O. Glass and T. Horsin, Approximate Lagrangian controllability for the 2-D Euler equation. Application to the control of the shape of vortex patches, J. Math. Pures Appl. (9), 93 (2010), 61-90], [O. Glass and T. Horsin, Prescribing the motion of a set of particles in a three-dimensional perfect fluid, SIAM J. Control Optim., 50 (2012), 2726-2742], [T. Horsin and O. Kavian, Lagrangian controllability of inviscid incompressible fluids: a constructive approach, ESAIM Control Optim. Calc. Var., 23 (2017), 1179-1200] in the case of the Euler equations and the result [O. Glass and T. Horsin, Lagrangian controllability at low Reynolds number, ESAIM Control Optim. Calc. Var., 22 (2016), 1040-1053] in the case of the steady Stokes equations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-09T04:01:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "navier-stokes equation",
      "smooth controllability",
      "navier conditions",
      "approximate lagrangian controllability result",
      "global approximate lagrangian controllability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}