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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. [ABSTRACT FROM AUTHOR]

The present paper is about a famous extension of the Prandtl equation, the so-called Interactive Boundary Layer (IBL) model. This model has been used intensively in the numerics of steady boundary layer flows and compares favorably to the Prandtl model, especially past separation. We consider here the unsteady version of the IBL model and study its linear well-posedness, namely, the linear stability of shear flow solutions to high frequency perturbations. We show that the IBL model exhibits strong unrealistic instabilities, which are in particular distinct from Tollmien-- Schlichting waves. We also exhibit similar instabilities for a Prescribed Displacement Thickness (PDT) model, which is one of the building blocks of numerical implementations of the IBL model. [ABSTRACT FROM AUTHOR]