Some Controllability Results forthe N-Dimensional Navier--Stokes and Boussinesq systems with N-1 scalar controls

In this paper we deal with some controllability problems for systems of the Navier– Stokes and Boussinesq kind with distributed controls supported in small sets. Our main aim is to control N-dimensional systems (N + 1 scalar unknowns in the case of the Navier–Stokes equations) with N − 1 scalar control functions. In a first step, we present some global Carleman estimates for suitable adjoint problems of linearized Navier–Stokes and Boussinesq systems. In this way, we obtain null controllability properties for these systems. Then, we deduce results concerning the local exact controllability to the trajectories. We also present (global) null controllability results for some (truncated) approximations of the Navier–Stokes equations. 1. Introduction and examples. Let Ω ⊂ R N (N = 2 or 3) be a bounded connected open set whose boundary ∂Ω is regular enough (for instance of class C 2). Let O ⊂ Ω be a (small) nonempty open subset and let T > 0. We will use the notation Q = Ω × (0, T) and Σ = ∂Ω × (0, T) and we will denote by n(x) the outward unit normal to Ω at the point x ∈ ∂Ω. On the other hand, we will denote by C, C 1 , C 2 ,. .. various positive constants (usually depending on Ω and O). We will be concerned with the following controlled Navier–Stokes and Boussinesq systems: