Lptheory for Boussinesq system with Dirichlet boundary conditions

Acevedo, Paul; Amrouche, Chérif; Conca, Carlos

doi:10.1080/00036811.2018.1530762

Cited by 6 publications

(2 citation statements)

References 25 publications

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“…For any f , g ∈ L q (Ω), L q (Ω), there exists a solution (generally not unique) (y e , θ e , π e ) ∈ (W 2,q (Ω) ∩ W 1,q 0 (Ω)) × (W 2,q (Ω) ∩ W 1,q 0 (Ω)) × (W 1,q (Ω)/R). See [3], [4], [5] for q = 2. In the Hilbert space setting, see [19], [27], [74], [90], [44].…”

Section: Stationary Boussinesq Equationsmentioning

confidence: 99%

Finite dimensional boundary uniform stabilization of the Boussinesq system in Besov spaces by critical use of Carleman estimate-based inverse theory

Lasiecka,

Priyasad,

Triggiani

2022

Preprint

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We consider the d-dimensional Boussinesq system defined on a sufficiently smooth bounded domain, and subject to a pair {v, u} of controls localized on { Γ, ω}. Here, v is a scalar Dirichlet boundary control for the thermal equation, acting on an arbitrary small connected portion Γ of the boundary Γ = ∂Ω. Instead, u is a d-dimensional internal control for the fluid equation acting on an arbitrary small collar ω supported by Γ (Fig 1). 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 an explicitly constructed, finite dimensional feedback control pair {v, u} localized on { Γ, ω}. In addition, they will be minimal in number, and of reduced dimension: more precisely, u will be of dimension (d − 1), to include necessarily its d th component, and v will be of dimension 1. 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 a corresponding Besov space for the thermal component, q > d. Unique continuation inverse theorems for suitably over determined adjoint static problems play a critical role in the constructive solution. Their proof rests on Carleman-type estimates, a topic pioneered by M. V. Klibanov since the early 80's, after the 1939-breakthrough publication [17].

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Section: Stationary Boussinesq Equationsmentioning

confidence: 99%

Finite dimensional boundary uniform stabilization of the Boussinesq system in Besov spaces by critical use of Carleman estimate-based inverse theory

Lasiecka,

Priyasad,

Triggiani

2022

Preprint

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“…5) obtained from Φ * ij by omitting the d-componentϕ * (d) ijof the vector Φ * ij . Next,we construct the following matrix U i of size i × K, K = sup{ i…”

mentioning

confidence: 99%

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

Lasiecka¹,

Priyasad²,

Triggiani³

2020

Discrete &Amp; Continuous Dynamical Systems - B

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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.

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Unique Continuation Properties of Static Over-determined Eigenproblems: The Ignition Key for Uniform Stabilization of Dynamic Fluids by Feedback Controllers

Triggiani

2023

Advances in Mathematical Fluid Mechanics

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L^ptheory for Boussinesq system with Dirichlet boundary conditions

Cited by 6 publications

References 25 publications

Finite dimensional boundary uniform stabilization of the Boussinesq system in Besov spaces by critical use of Carleman estimate-based inverse theory

Finite dimensional boundary uniform stabilization of the Boussinesq system in Besov spaces by critical use of Carleman estimate-based inverse theory

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

Unique Continuation Properties of Static Over-determined Eigenproblems: The Ignition Key for Uniform Stabilization of Dynamic Fluids by Feedback Controllers

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