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Global controllability of Boussinesq flows by using only a temperature control

Vahagn Nersesyan, Manuel Rissel

TL;DR

This work establishes global approximate controllability for the 2D incompressible Boussinesq system on the torus using only a temperature control localized to a horizontal strip, removing smallness restrictions on data. The authors develop a multi-stage framework that separately handles vorticity and temperature via transport-based controls, leveraging buoyancy coupling and a hydrodynamic scaling to transition from linearized transport controllability to the nonlinear system. A key innovation is the use of finitely decomposable actuators built from fixed transported Fourier modes, enabling physically localized controls and, in one variant, explicit feedback terms. The results extend the scope of controllability for buoyancy-influenced flows, relax localization topologies compared to prior work, and provide a constructive scheme for implementing heat-based flow steering with potential practical implications.

Abstract

We show that buoyancy driven flows can be steered in an arbitrary time towards any state by applying as control only an external temperature profile in a subset of small measure. More specifically, we prove that the 2D incompressible Boussinesq system on the torus is globally approximately controllable via physically localized heating or cooling. In addition, our controls have an explicitly prescribed structure; even without such structural requirements, large data controllability results for Boussinesq flows driven merely by a physically localized temperature profile were so far unknown. The presented method exploits various connections between the model's underlying transport-, coupling-, and scaling mechanisms.

Global controllability of Boussinesq flows by using only a temperature control

TL;DR

This work establishes global approximate controllability for the 2D incompressible Boussinesq system on the torus using only a temperature control localized to a horizontal strip, removing smallness restrictions on data. The authors develop a multi-stage framework that separately handles vorticity and temperature via transport-based controls, leveraging buoyancy coupling and a hydrodynamic scaling to transition from linearized transport controllability to the nonlinear system. A key innovation is the use of finitely decomposable actuators built from fixed transported Fourier modes, enabling physically localized controls and, in one variant, explicit feedback terms. The results extend the scope of controllability for buoyancy-influenced flows, relax localization topologies compared to prior work, and provide a constructive scheme for implementing heat-based flow steering with potential practical implications.

Abstract

We show that buoyancy driven flows can be steered in an arbitrary time towards any state by applying as control only an external temperature profile in a subset of small measure. More specifically, we prove that the 2D incompressible Boussinesq system on the torus is globally approximately controllable via physically localized heating or cooling. In addition, our controls have an explicitly prescribed structure; even without such structural requirements, large data controllability results for Boussinesq flows driven merely by a physically localized temperature profile were so far unknown. The presented method exploits various connections between the model's underlying transport-, coupling-, and scaling mechanisms.
Paper Structure (38 sections, 14 theorems, 136 equations, 4 figures)

This paper contains 38 sections, 14 theorems, 136 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The main controllability problem
  3. Highly degenerate controllability problems
  4. Notations
  5. Main results
  6. Related literature and outline
  7. Outline of the paper.
  8. Controls for linear transport problems
  9. Preliminary constructions
  10. Partition of the torus
  11. Convection strategy
  12. A generating vector field
  13. Non-localized degenerate controls
  14. Localized degenerate controls
  15. Step 1. Definition of a localized control.
  16. ...and 23 more sections

Key Result

Theorem 1.2

Let the integer $r\geq 0$, viscosity $\nu > 0$, diffusivity $\tau > 0$, control time $T~>~0$, initial and target states $(\bm{u}_0, \bm{u}_T) \in \mathbf{H}^{r}\times\mathbf{H}^{r}$, $(\theta_0, \theta_T) \in {\rm H}^{r}\times{\rm H}^{r}$, external forces $(\bm{\Phi}_{\operatorname{ext}}, \psi_{\ope to the Boussinesq problem equation:BoussinesqVelocity satisfies

Figures (4)

  • Figure 1: The control region $\omegaup \subset \mathbb{T}^2$ is any open horizontal strip and $\bm{e}_{\operatorname{grav}}$ points vertically.
  • Figure 2: The order in which the building blocks for our control force are chosen.
  • Figure 3: Subdivision of the proofs for Theorems \ref{['theorem:main']} and \ref{['theorem:secondmain']}.
  • Figure 4: An exemplary open covering of $\mathbb{T}^2$ by $K = 6$ overlapping strips $\mathcal{O}_1, \dots, \mathcal{O}_6$, the boundaries of which are in an alternating way depicted as solid, dashed, and dotted lines. The overlapping region due to vertical periodicity is highlighted by a dotted pattern. The reference strip $\mathcal{O}$ contained inside the control region is displayed as a (red) filled rectangle.

Theorems & Definitions (27)

  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • proof
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Theorem 2.6
  • ...and 17 more