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Small-time global exact controllability of the Navier-Stokes equation with Navier slip-with-friction boundary conditions. (English) Zbl 1447.93024

The small-time global exact null controllability problem for the Navier-Stokes equation was first suggested by J. L. Lions [in: Applied and industrial mathematics, Proc. Symp., Venice/Italy 1989, Math. Appl., D. Reidel Publ. Co. 56, . 59–84 (1991; Zbl 0735.93006)]. Continuing the study on this problem the present authors, in this interesting and elobarative work, have investigated the small-time global exact controllability of the Navier-Stokes equation for both towards the null equilibrium state and towards weak trajectories. They considered 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, the authors have proved that small-time global exact controllability holds. The analysis is based on the controllability of the Euler equation combined with asymptotic boundary layer expansions. Choosing the boundary controls with care enables to guarantee good dissipation properties for the residual boundary layers, which can be exactly canceled using local techniques. The authors provides an extensive list of references which covers known results and related previous works on the problem. Further extensions of this type of work in different directions are outlined at the end of the paper.

MSC:

93B05 Controllability
93C20 Control/observation systems governed by partial differential equations
35Q30 Navier-Stokes equations

Citations:

Zbl 0735.93006
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Full Text: DOI arXiv

References:

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