Uniform local null control of the Leray-$\alpha$ model

This paper deals with the distributed and boundary controllability of the so called Leray-$\alpha$ model. This is a regularized variant of the Navier-Stokes system ($\alpha$ is a small positive parameter) that can also be viewed as a model for turbulent flows. We prove that the Leray-$\alpha$ equations are locally null controllable, with controls bounded independently of $\alpha$. We also prove that, if the initial data are sufficiently small, the controls converge as $\alpha \to 0^+$ to a null control of the Navier-Stokes equations. We also discuss some other related questions, such as global null controllability, local and global exact controllability to the trajectories, etc.