Global Controllability of the Boussinesq 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 the Boussi-
nesq system posed in 2D or 3D smooth bounded domains. The velocity field of the fluid must
satisfy a Navier slip-with-friction boundary condition and a Robin boundary condition is im-
posed to the temperature. We assume that one can act on the velocity and the temperature on
a small part of the boundary. For the proof, we first transform the boundary control problem
into a distributed control problem. Then, we prove a global approximate controllability result
by adapting the strategy of Coron et al [J. Eur. Math. Soc., 22 (2020), pp. 1625–1673]; this
relies on the controllability properties of the inviscid Boussinesq system and the analysis of
appropriate asymptotic boundary layer expansions. Finally, we conclude with a local control-
lability result; as in many other cases, this can be established as a consequence of the null
controllability of a linearized system through a fixed-point argument. Our contribution can be
viewed as an extension of the results in [J. Eur. Math. Soc., 22 (2020), pp. 1625–1673], where
thermal effects were not considered. Thus, we prove that the ideas behind the controllability
properties of the Euler system and the well-prepared dissipation technique can be adapted
to the present situation. Furthermore, we cover all the classical boundary conditions for the
temperature, that is, those of the Robin, Neumann and Dirichlet kinds.
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