Remarks on local controllability for the Boussinesq system with Navier boundary condition

Cristhian Montoya

doi:10.5802/crmath.29

Abstract

This note deals with the local exact controllability to a particular class of trajectories for the Boussinesq system with nonlinear Navier–slip boundary conditions and internal controls having vanishing components. Briefly speaking, in two dimensions, the local exact controllability property is obtained using only one control in the heat equation, whereas two scalar controls are required in three dimensions.

Highlights

The interaction of incompressible fluids with a diffusion process can be modeled by a coupled system between the Navier–Stokes and heat equations, usually called Boussinesq system
Nonlinear Navier–type boundary conditions for the fluid flow and homogeneous Neumann conditions for the diffusion equation are considered in order to study the local exact controllability for the Boussinesq system with few scalar controls
Guerrero [10] shows the local exact controllability to the trajectories of the Boussinesq system with Dirichlet boundary conditions, the same author proven in [11] the local exact controllability to the trajectories for the Navier–Stokes with Navier–slip boundary conditions

Summary

Introduction

The interaction of incompressible fluids with a diffusion process can be modeled by a coupled system between the Navier–Stokes and heat equations, usually called Boussinesq system. Guerrero [10] shows the local exact controllability to the trajectories of the Boussinesq system with Dirichlet boundary conditions, the same author proven in [11] the local exact controllability to the trajectories for the Navier–Stokes with Navier–slip boundary conditions In both papers N + 1 distributed scalar controls supported in small sets are considered. Concerning the N -dimensional Boussinesq system with Dirichlet conditions, in [6] the authors proved that the local exact controllability to the trajectories can be achieved with N − 1 scalar controls, under certain geometric assumption on the control domain. One of the main result in this Note concerns the local controllability to a particular class of trajectories of (1) This result is presented in the following theorem.

A new Carleman inequality

Local controllability for the Boussinesq system