October  2020, 25(10): 4071-4117. doi: 10.3934/dcdsb.2020187

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 Published  October 2020 Early access  June 2020

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.

Citation: Irena Lasiecka, Buddhika Priyasad, Roberto Triggiani. Uniform stabilization of Boussinesq systems in critical $ \mathbf{L}^q $-based Sobolev and Besov spaces by finite dimensional interior localized feedback controls. Discrete and Continuous Dynamical Systems - B, 2020, 25 (10) : 4071-4117. doi: 10.3934/dcdsb.2020187
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show all references

References:
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H. Amann, Linear and Quasilinear Parabolic Problems: Volume I: Abstract Linear Theory, Monographs in Mathematics, 89. Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-3-0348-9221-6.

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H. Amann, On the strong solvability of the Navier-Stokes equations, J. Math. Fluid Mech., 2 (2000), 16-98.  doi: 10.1007/s000210050018.

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P. Acevedo Tapia, $L^p$-Theory for the Boussinesq System, Ph.D theis, Universidad de Chille, Faculatad de Ciencias Fisicas y Mathematicas, Departamento de Ingeniearia Mathematica, Santiago de Chille, 2015.

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P. AcevedoC. Amrouche and C. Conca, Boussinesq system with non-homogeneous boundary conditions, Appl. Math. Lett., 53 (2016), 39-44.  doi: 10.1016/j.aml.2015.09.015.

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P. AcervedoC. Amrouche and C. Conca, $L^p$ theory for Boussinesq system with Dirichlet boundary conditions, Appl. Anal., 98 (2019), 272-294.  doi: 10.1080/00036811.2018.1530762.

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M. Badra, Abstract settings for stabilization of nonlinear parabolic systems with Riccati-based strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control, Discrete and Continuous Dynamical Systems, 32 (2012), 1169-1208.  doi: 10.3934/dcds.2012.32.1169.

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R. W. Brockett, Asymptotic and feedback stabilization, Differential Geometric Control Theory, Birkhäuser, Basel, (1983), 181–191.

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V. Barbu and R. Triggiani, Internal stabilization of Navier-Stokes equations with finite-dimensional controllers, Indiana Univ. Math. J., 53 (2004), 1443-1494.  doi: 10.1512/iumj.2004.53.2445.

[12]

V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations, Mem. Amer. Math. Soc., 181 (2006), x+128 pp. doi: 10.1090/memo/0852.

[13]

V. BarbuI. Lasiecka and R. Triggiani, Abstract settings fr tangential boundary stabilization of Naiver Stokes equations by high-and low-gain feedback controllers, Nonlinear Analysis, 64 (2006), 2704-2746.  doi: 10.1016/j.na.2005.09.012.

[14]

V. BarbuI. Lasiecka and R. Triggiani, Local exponential stabilization strategies of the Navier Stokes equations, $d = 2, 3$, via feedback stabilization of its linearization, Optimal of Coupled Systems of Partial Differential Equations, Internat. Ser. Numer. Math., Birkhäuser, Basel, 155 (2007), 13-46.  doi: 10.1007/978-3-7643-7721-2_2.

[15]

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[16]

J. A. BurnsX. M. He and W. W. Hu, Feedback stabilization of a thermal fluid system with mixed boundary control, Computers and Mathematics with Applications, 71 (2016), 2170-2191.  doi: 10.1016/j.camwa.2016.01.011.

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