Global controllability of the bousiinesq system with Navier-slip-with-friction and Robin boundary conditions

Abstract

In this paper, we deal with the global exact controllability to the trajectories of theBoussinesq system posed in 2D or 3D smooth bounded domains. The velocity field of the fluid mustsatisfy a Navier-slip-with-friction boundary condition, and a Robin boundary condition is imposedto the temperature. We assume that one can act on the velocity and the temperature on a small partof the boundary. For the proof, we first transform the boundary control problem into a distributedcontrol problem. Then, we prove a global approximate controllability result by adapting the strategyof Coron, Marbach, and Sueur [J. Eur. Math. Soc.(JEMS), 22 (2020), pp. 1625--1673]; this relieson the controllability properties of the inviscid Boussinesq system and the analysis of appropriateasymptotic boundary layer expansions. Finally, we conclude with a local controllability result; as inmany other cases, this can be established as a consequence of the null controllability of a linearizedsystem through a fixed-point argument. Our contribution can be viewed as an extension of theresults in [J. Eur. Math. Soc.(JEMS), 22 (2020), pp. 1625--1673], where thermal effects were notconsidered. Thus, we prove that the ideas behind the controllability properties of the Euler systemand the well-prepared dissipation technique can be adapted to the present situation. Furthermore, wecover all the classical boundary conditions for the temperature, that is, those of the Robin, Neumann,and Dirichlet kinds.