Small-time global exact controllability of the Navier–Stokes equation with Navier slip-with-friction boundary conditions

  • Jean-Michel Coron

    Université Pierre et Marie Curie, Paris, France
  • Frédéric Marbach

    Université Pierre et Marie Curie, Paris, France
  • Franck Sueur

    Université de Bordeaux, Talence, France
Small-time global exact controllability of the Navier–Stokes equation with Navier slip-with-friction boundary conditions cover

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Abstract

In this work, we investigate the small-time global exact controllability of the Navier–Stokes equation, both towards the null equilibrium state and towards weak trajectories.We consider a viscous incompressible fluid evolving within a smooth bounded domain, either in 2D or in 3D. The controls are only located on a small part of the boundary, intersecting all its connected components. On the remaining parts of the boundary, the fluid obeys a Navier slip-with-friction boundary condition. Even though viscous boundary layers appear near these uncontrolled boundaries, we prove that small-time global exact controllability holds. Our analysis relies on the controllability of the Euler equation combined with asymptotic boundary layer expansions. Choosing the boundary controls with care enables us to guarantee good dissipation properties for the residual boundary layers, which can then be exactly canceled using local techniques.

Cite this article

Jean-Michel Coron, Frédéric Marbach, Franck Sueur, Small-time global exact controllability of the Navier–Stokes equation with Navier slip-with-friction boundary conditions. J. Eur. Math. Soc. 22 (2020), no. 5, pp. 1625–1673

DOI 10.4171/JEMS/952