Global controllability of Boussinesq channel flows only through the temperature

TL;DR

This paper proves the first global approximate controllability of Boussinesq flows in a periodic channel using only temperature control in a thin strip, combining vorticity control and Coron's return method.

Contribution

It introduces a novel controllability result for Boussinesq systems with non-periodic boundary conditions using only temperature control.

Findings

Achieved global approximate controllability of Boussinesq flows with temperature control.

Developed a small-time scaling argument for vorticity control.

Applied Coron's return method to steer temperature without affecting vorticity.

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.