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Small-time global exact controllability to the trajectories for the viscous Boussinesq system

F. W. Chaves-Silva, E. Fernández-Cara, K. Le Balc'h, J. L. F. Machado, D. A. Souza

Abstract

In this paper, we deal with the global exact controllability to the trajectories of the Boussinesq system. We consider 2D and 3D smooth bounded domains. The velocity field of the fluid must satisfy a Navier slip-with-friction boundary condition and a Robin boundary condition is imposed to the temperature. We assume that one can act on the velocity and the temperature on an arbitrary small part of the boundary. The proof relies on three main arguments. First, we transform the problem into a distributed controllability problem by using a domain extension procedure. Then, we prove a global approximate controllability result by following the strategy of Coron et al [J. Eur. Math. Soc., 22 (2020), pp. 1625-1673], which deals with the Navier-Stokes equations. This part relies on the controllability of the inviscid Boussinesq system and asymptotic boundary layer expansions. Finally, we conclude with a local controllability result that we establish with the help of a linearization argument and appropriate Carleman estimates.

Small-time global exact controllability to the trajectories for the viscous Boussinesq system

Abstract

In this paper, we deal with the global exact controllability to the trajectories of the Boussinesq system. We consider 2D and 3D smooth bounded domains. The velocity field of the fluid must satisfy a Navier slip-with-friction boundary condition and a Robin boundary condition is imposed to the temperature. We assume that one can act on the velocity and the temperature on an arbitrary small part of the boundary. The proof relies on three main arguments. First, we transform the problem into a distributed controllability problem by using a domain extension procedure. Then, we prove a global approximate controllability result by following the strategy of Coron et al [J. Eur. Math. Soc., 22 (2020), pp. 1625-1673], which deals with the Navier-Stokes equations. This part relies on the controllability of the inviscid Boussinesq system and asymptotic boundary layer expansions. Finally, we conclude with a local controllability result that we establish with the help of a linearization argument and appropriate Carleman estimates.

Paper Structure

This paper contains 31 sections, 19 theorems, 259 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Bibliographical comments
  4. Strategy of the proof
  5. Domain extension and smoothing effect
  6. Domain extension
  7. Smoothing effect of the uncontrolled Boussinesq system
  8. Approximate controllability problem
  9. Time scaling
  10. The special case of the slip boundary condition
  11. Ansatz with no correction term
  12. Inviscid flow
  13. Flushing
  14. Equations and estimates for the remainder
  15. The case of Navier slip-with-friction boundary conditions
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $T>0$ be a positive time, let $(u_0,\theta_0) \in L^2_c(\Omega)^n\times L^2(\Omega)$ be an initial state and let $(\overline{u}, \overline{\theta}) \in W_T(\Omega)$ be a weak trajectory of eq_bou. Then, there exists a weak controlled solution to eq_bou in $W_T(\Omega)$ that satisfies

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Proposition 2.1
  • Definition 2.1
  • Proposition 2.2
  • Lemma 2.1
  • Proposition 3.1
  • Lemma 3.1
  • ...and 15 more