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

We show approximate controllability of Boussinesq flows in ${\mathbb{T}}^2={\mathbb{R}}^2\mathrm{/}2\pi {\mathbb{Z}}^2$ driven by finite-dimensional controls that are supported in any fixed region $\omega \subset {\mathbb{T}}^2$. 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 $\omega$ 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 $\omega$ 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 $\omega$ 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 $\omega$ within this class of control regions.