Controlabilidade para alguns modelos da mecânica de fluidos.

Abstract

The aim of this thesis is the controllability of some fluid mechanic models. We are going to prove the existence of controls that drive the solution of our system from a prescribed initial state to a desired final state at a given positive time. The two first Chapters deal with the controllability of the Burgers-α and Leray-α models. The Leray-α model is a regularized variant of the Navier-Stokes system (α is a small positive parameter) that can also be viewed as a model for turbulent flows; the Burgers-α model can be viewed as a related toy model of Leray-α. We prove that the Leray-α and Burgers-α models are locally null controllable, with controls bounded independently of α. We also prove that, if the initial data are sufficiently small, the control of the Leray-α equations (resp. the Burgers-α equation) converge as α → 0+ to a null control of the Navier-Stokes equations (resp. the Burgers equation). The third Chapter deals with the boundary controllability of inviscid incompressible fluids for which thermal effects are important. They will be modeled through the so called Boussinesq approximation. In the zero heat diffusion case, by adapting and extending some ideas from J.-M. Coron [13, 15] and O. Glass [43, 44, 45], we establish the simultaneous global exact controllability of the velocity field and the temperature for 2D and 3D flows. When the heat diffusion coefficient is positive, we present some additional results concerning exact controllability for the velocity field and local null controllability of the temperature. The fourth Chapter is devoted to prove the local exact controllability to the trajectories for a coupled system of the Boussinesq kind, with a reduced number of controls. In the state system, the unknowns are the velocity field and pressure of the fluid (y, p), the temperature θ and an additional variable c that can be viewed as the concentration of a contaminant solute. We prove several results, that essentially show that it is sufficient to act locally in space on the equations satisfied by θ and c.