Uniform stabilization of Boussinesq systems in critical <inline-formula><tex-math id="M1">\begin{document}$ \mathbf{L}^q $\end{document}</tex-math></inline-formula>-based Sobolev and Besov spaces by finite dimensional interior localized feedback controls

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 L 3 (Ω) for d = 3) and the space L q (Ω) for the thermal component, q > d. Thus, this paper may be viewed as an extension of [63], 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.

Global boundary stabilization to trajectories of the deterministic and stochastic porous-media equation

“…For example: for the case d = 1 and β(z) = z m see [10,11]; the case β(z) = z 2 , which corresponds to the so-called Boussinesq equation (related to groundwater flow), for internal control see e.g. [12,17], while for boundary control see e.g. [22]; for boundary stabilization of the toy model related to the Navier-Stokes equation, the Burgers equation, see e.g.…”