Journal of the European Mathematical Society


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Published online first: 2020-02-11
DOI: 10.4171/JEMS/952

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

Jean-Michel Coron[1], Frédéric Marbach[2] and Franck Sueur[3]

(1) Université Pierre et Marie Curie, Paris, France
(2) Université Pierre et Marie Curie, Paris, France
(3) Université de Bordeaux, Talence, France

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.

Keywords: Controllability, Navier–Stokes, boundary layers

Coron Jean-Michel, Marbach Frédéric, Sueur Franck: Small-time global exact controllability of the Navier–Stokes equation with Navier slip-with-friction boundary conditions. J. Eur. Math. Soc. Electronically published on February 11, 2020. doi: 10.4171/JEMS/952 (to appear in print)