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Controllability of Boussinesq flows driven by finite-dimensional and physically localized forces

Manuel Rissel

TL;DR

This work proves that the planar Boussinesq system on $\mathbb{T}^2$ is approximately controllable using finite-dimensional, physically localized controls supported in a fixed region $\omega$. The authors build a generating Euler flow from observable families to induce a non-stationary transport mechanism that transfers energy from low to high frequencies, and then use Coron's return method to steer the temperature, with buoyancy driving the velocity. They construct universal finite-dimensional control spaces $\mathscr{F}_{\mathscr{v}}$ and $\mathscr{F}_{\mathscr{t}}$ independent of data such as $\nu,\tau$, and $\varepsilon$, enabling approximate steering of $(u,\theta)$ with $\|u(\cdot,\delta)-u_0\|$ and $\|\theta(\cdot,\delta)-\theta_1\|$ small for a short time, and extend the controllability to arbitrary time via the return mechanism. This yields the first known example of an incompressible fluid where approximate controllability is achieved by truly finite-dimensional, physically localized forces, with explicit constructions and a clear pipeline from transport controllability to nonlinear Boussinesq controllability.

Abstract

We show approximate controllability of Boussinesq flows in $\mathbb{T}^2 = \mathbb{R}^2 / 2π\mathbb{Z}^2$ driven by finite-dimensional controls that are supported in any fixed region $ω\subset \mathbb{T}^2$. This addresses a Boussinesq version of a question by Agrachev and provides the first known example of incompressible fluids with this property. In this context, we complement results obtained for the Navier--Stokes system by Agrachev--Sarychev (Comm. Math. Phys. 265, 2006), where the controls are finite-dimensional but not localized in physical space, and Nersesyan--Rissel (Comm. Pure Appl. Math. 78, 2025), where physically localized controls admit for special $ω$ a degenerate but not finite-dimensional structure. For our proof, we study controllability properties of tailored convection equations governed by time-periodic degenerately forced Euler flows that provide a twofold geometric mechanism: transport of information through $ω$ versus non-stationary mixing effects transferring energy from low-dimensional sources to higher frequencies. The temperature is then controlled by using Coron's return method, while the velocity is mainly driven by the buoyant force. When $ω$ contains two cuts of $\mathbb{T}^2$, our approach allows to effectively construct low-dimensional control spaces of dimensions that are independent of the choice of $ω$ within this class of control regions.

Controllability of Boussinesq flows driven by finite-dimensional and physically localized forces

TL;DR

This work proves that the planar Boussinesq system on is approximately controllable using finite-dimensional, physically localized controls supported in a fixed region . The authors build a generating Euler flow from observable families to induce a non-stationary transport mechanism that transfers energy from low to high frequencies, and then use Coron's return method to steer the temperature, with buoyancy driving the velocity. They construct universal finite-dimensional control spaces and independent of data such as , and , enabling approximate steering of with and small for a short time, and extend the controllability to arbitrary time via the return mechanism. This yields the first known example of an incompressible fluid where approximate controllability is achieved by truly finite-dimensional, physically localized forces, with explicit constructions and a clear pipeline from transport controllability to nonlinear Boussinesq controllability.

Abstract

We show approximate controllability of Boussinesq flows in driven by finite-dimensional controls that are supported in any fixed region . This addresses a Boussinesq version of a question by Agrachev and provides the first known example of incompressible fluids with this property. In this context, we complement results obtained for the Navier--Stokes system by Agrachev--Sarychev (Comm. Math. Phys. 265, 2006), where the controls are finite-dimensional but not localized in physical space, and Nersesyan--Rissel (Comm. Pure Appl. Math. 78, 2025), where physically localized controls admit for special a degenerate but not finite-dimensional structure. For our proof, we study controllability properties of tailored convection equations governed by time-periodic degenerately forced Euler flows that provide a twofold geometric mechanism: transport of information through versus non-stationary mixing effects transferring energy from low-dimensional sources to higher frequencies. The temperature is then controlled by using Coron's return method, while the velocity is mainly driven by the buoyant force. When contains two cuts of , our approach allows to effectively construct low-dimensional control spaces of dimensions that are independent of the choice of within this class of control regions.

Paper Structure

This paper contains 15 sections, 14 theorems, 175 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Description of the approach
  4. Explicit representations of the control spaces for a special case
  5. Literature
  6. Transport equations with generating drift
  7. Finite-dimensional and physically localized transport controls
  8. Finite-dimensional flushing profile
  9. Partition of unity
  10. Definition of the vector field $\overline{y}$
  11. Modified generating vector field
  12. Definition of the main convection profile
  13. Controllability of the main convection problem
  14. The nonlinear problem and conclusion of the main result
  15. Proof of the second main result

Key Result

Theorem 1.1

There are finite-dimensional spaces $\mathscr{F}_{\mathscr{v}} \subset C^{\infty}(\mathbb{T}^2; \mathbb{R}^2)$ and $\mathscr{F}_{\mathscr{t}} \subset C^{\infty}(\mathbb{T}^2; \mathbb{R})\cap H_{\operatorname{avg}}$ such that the following statement holds. For any given data there exists $\delta_0 > 0$ so that for each $\delta \in (0, \delta_0)$ there are $\xi \in L^2((0,\delta); \mathscr{F}_{\mat

Figures (2)

  • Figure 1: The behavior of $\overline{u}^{\star}$ and $g^{\star}$ from \ref{['equation:introlg']} on the time interval $[0, T^{\star}]$ is schematically denoted for a subdivision of $\mathbb{T}^2$ by letters from A to I and numbers from $1$ to $9$, respectively. It is then indicated how $G$ and $\overline{U}$ from \ref{['equation:introtrp']} are obtained with the help of $g^{\star}$ and $\overline{u}^{\star}$, and arrows indicate how information is propagated along $\overline{U}$ into the control zone (center square). In regions marked with "$-$", the flow or controls are inactive at the given time. For example, the values of $\overline{U}$ on the time interval $[t^1_a, t^1_b]$ marked with "A" in the center region of the domain correspond to the values of $\overline{u}^{\star}$ on the time interval $[0, T^{\star}]$ marked with "A" in the bottom-left region of the domain.
  • Figure 2: Illustration of the covering $(\mathcal{O}_i)_{i\in\{1,\dots,M\}}$ (three example squares printed) and important values of the cutoff functions $\mu$ and $\chi$. Within the inner dashed square, which includes the support of $\mu$, one has $\chi = 1$. Exterior to the outer dashed square, $\chi$ vanishes.

Theorems & Definitions (38)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • Example 3.1
  • Lemma 3.2: FursikovImanuvilov1999
  • ...and 28 more