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

Irena Lasiecka; Buddhika Priyasad; Roberto Triggiani

doi:10.3934/dcdsb.2020187

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2020, Volume 25, Issue 10: 4071-4117. Doi: 10.3934/dcdsb.2020187

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

1.
Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA
2.
IBS, Polish Academy of Sciences, Warsaw, Poland
3.
Institute for Mathematics and Scientific Computing, University of Graz, A-8010 Graz, Austria

Received: August 2019

Revised: March 2020

Early access: June 2020

Published: October 2020

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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 [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.

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Mathematics Subject Classification: 34H05, 35B35, 35K40, 93C20.

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