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Global controllability of Boussinesq channel flows only through the temperature

Manuel Rissel

Abstract

We show the global approximate controllability of the Boussinesq system with viscosity and diffusion in a planar periodic channel by using only a temperature control supported in a thin strip. At the walls, a slip boundary condition is chosen for the fluid and the normal derivative of the temperature is assumed to vanish. This contributes a first global controllability result of such type for the Boussinesq system in the presence of non-periodic boundary conditions. We resort to a small-time scaling argument to control the vorticity through a large initial temperature. Moreover, relying on the special choice of the domain, we employ J.-M. Coron's return method in order to steer the temperature without significantly impacting the vorticity.

Global controllability of Boussinesq channel flows only through the temperature

Abstract

We show the global approximate controllability of the Boussinesq system with viscosity and diffusion in a planar periodic channel by using only a temperature control supported in a thin strip. At the walls, a slip boundary condition is chosen for the fluid and the normal derivative of the temperature is assumed to vanish. This contributes a first global controllability result of such type for the Boussinesq system in the presence of non-periodic boundary conditions. We resort to a small-time scaling argument to control the vorticity through a large initial temperature. Moreover, relying on the special choice of the domain, we employ J.-M. Coron's return method in order to steer the temperature without significantly impacting the vorticity.

Paper Structure

This paper contains 17 sections, 5 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Proof of the main result
  4. Remarks on the Boussinesq system
  5. Steering the vorticity through the initial temperature
  6. Step 1. Ansatz.
  7. Step 2. Estimates.
  8. Controlling the temperature
  9. Step 1. Controlling a family of linear problems.
  10. Step 2. Controls for the nonlinear problem.
  11. Step 3. Asymptotic expansions.
  12. Step 4. Estimates.
  13. Conclusion
  14. Step 1. Smoothing.
  15. Step 2. Reaching a special temperature state.
  16. ...and 2 more sections

Key Result

Theorem 1.1

The system equation:Boussinesq is globally approximately controllable by using only a temperature control. That is, for arbitrary there exists a control $\eta \in {\rm C}^{\infty}(\mathscr{C}\times[0,T];\mathbb{R})$ with $\operatorname{supp}(\eta) \subset \omegaup\times(0,T)$ such that the unique solution $(\bm{u}, \theta)$ to equation:Boussinesq satisfies

Theorems & Definitions (15)

  • Theorem 1.1: Main result
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Theorem 2.4
  • ...and 5 more