Uniform stabilization of Boussinesq systems in critical $\mathbf{L}^q$-based Sobolev and Besov spaces by finite dimensional interior localized feedback controls

TL;DR

This paper develops a method to stabilize the d-dimensional Boussinesq system near an unstable equilibrium using minimal, localized feedback controls in low regularity Sobolev and Besov spaces, extending previous work on Navier-Stokes equations.

Contribution

It introduces explicit, finite-dimensional feedback controls for the Boussinesq system in critical low-regularity spaces, with controls localized inside the domain and minimal in number.

Findings

Achieved uniform stabilization near unstable equilibrium

Constructed feedback controls of minimal dimension

Extended previous Navier-Stokes stabilization results to Boussinesq systems

Abstract

We consider the d-dimensional Boussinesq system defined on a sufficiently smooth bounded domain, with homogeneous boundary conditions, and subject to external sources, assumed to cause instability. The initial conditions for both fluid and heat equations are taken of low regularity. We then seek to uniformly stabilize such Boussinesq system in the vicinity of an unstable equilibrium pair, in the critical setting of correspondingly low regularity spaces, by means of explicitly constructed, feedback controls, which are localized on an arbitrarily small interior subdomain. In addition, they will be minimal in number, and of reduced dimension: more precisely, they will be of dimension $(d-1)$ for the fluid component and of dimension $1$ for the heat component. The resulting space of well-posedness and stabilization is a suitable, tight Besov space for the fluid velocity component (close to $\mathbf{L}^3(\Omega$) for $ d = 3 $) and the space $L^q(\Omega$) for the thermal component, $ q > d $. Thus, this paper may be viewed as an extension of \cite{LPT.1}, where the same interior localized uniform stabilization outcome was achieved by use of finite dimensional feedback controls for the Navier-Stokes equations, in the same Besov setting.