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

We show approximate controllability of Boussinesq flows in T² = R² / 2πZ² driven by finite-dimensional controls that are supported in any fixed region ω⊂ T². This addresses a Boussinesq version of a question by Agrachev and provides a first example of fluid PDEs controllable in that sense. In particular, we add in this context to 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 the torus and a closed square of side-length L, our approach yields explicit control spaces for the velocity and temperature of dimensions 2 + 182π/L² + 82π/L⁴ and 2 + 82π/L², respectively.