Decay Rates of Interactive Hyperbolic-Parabolic PDE Models with Thermal Effects on the Interface

Lasiecka, Irena; Lebiedzik, Catherine

doi:10.1007/s002450010010

Cited by 23 publications

(36 citation statements)

References 23 publications

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“…The proof of existence is similar to the one in [11] (see also [8]) and uses the standard Galerkin's method, the proof of uniqueness and continuity uses analyticity of thermoelastic semigroup in (13)- (16) and results of Sect. 3.…”

Section: Existence Uniqueness and Continuitymentioning

confidence: 99%

“…Hybrid PDE systems that arise from structural acoustic models under thermal effects were studied by I. Lasiecka and C. Lebiedzik in [16,17,18]. In this case the undamped wave equation of type (7) with U = 0 is defined on a bounded three-dimensional domain O and the thermoelastic plate equation is defined on an interface Ω, the flat part of ∂O.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, in the case of a nonlinear thermoelastic system with free boundary conditions (see [18]) some additional mechanical damping acting on ∂Ω is assumed, but there is no need for such damping in the case of clamped or hinged boundary conditions (see [17]). Both cases α > 0 and α = 0 are considered in [16,17,18], but the proofs are quite different.…”

Section: Introductionmentioning

confidence: 99%

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Dynamics of a thermoelastic von Kármán plate in a subsonic gas flow

Ryzhkova

2006

Z. angew. Math. Phys.

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We discuss the problem of non-linear oscillations of a clamped thermoelastic plate in a subsonic gas flow. The dynamics of the plate is described by von Kármán system in the presence of thermal effects. No mechanical damping is assumed. To describe the influence of the gas flow we apply the linearized theory of potential flows. Our main result states that each weak solution of the problem considered tends to the set of the stationary points of the problem.A similar problem was considered in [27], but with rotational inertia accounted for, i.e. with the additional term −α∆utt, α > 0, and the same result on stabilization was obtained. There was introduced the decomposition of the solution such that the one term tends to zero and the other is compact in special ("local energy") topology. This decomposition enables us to prove the main result. But the case of rotational inertia neglected (α = 0) appears more difficult. Low a priori smoothness of ut in the case α = 0 prevents us to construct such a decomposition. In order to prove additional smoothness of ut we use analyticity of the corresponding thermoelastic semigroup proved in [25].The isothermal variant of this problem with additional mechanical damping term −ǫ∆ut, ǫ > 0 was considered in [13] and stabilization to the set of stationary solutions to the problem was proved. The problem, considered in the present work can also be regarded as an extension of the result of [18] to the case when gas occupies an unbounded domain. (2000). 35B40, 35D05, 35G25. Mathematics Subject Classification

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Section: Existence Uniqueness and Continuitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Dynamics of a thermoelastic von Kármán plate in a subsonic gas flow

Ryzhkova

2006

Z. angew. Math. Phys.

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“…In particular, hybrid PDE systems that arise from structural acoustic models were studied by I. Lasiecka and C. Lebiedzik in [5][6][7]. In this case the undamped wave equation of type (7) with U = 0 is defined on a bounded three-dimensional domain O and the thermoelastic plate equation is defined on an interface Ω, the flat part of ∂O.…”

Section: Introductionmentioning

confidence: 99%

Stabilization of von Kármán plate in the presence of thermal effects in a subsonic potential flow of gas

Ryzhkova¹

2004

Journal of Mathematical Analysis and Applications

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We discuss the problem of nonlinear oscillations of a clamped plate in the presence of thermal effects in a subsonic gas flow. The dynamics of the plate is described by von Kármán system in the presence of thermal effects, in which rotational inertia is accounted for. To describe influence of the gas flow we apply the linearized theory of potential flows. Our main result states that each weak solution of the problem considered tends to the set of the stationary points of the problem.

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“…A 2-dimensional thermoelastic plate with clamped boundary conditions and subject to interior point control -the model of the present paper -is an ideal wall of a structural acoustic chamber, subject to piezo-ceramic control action, for the purpose of noise reduction [1,2,3,6,15,22,4,5] to quote a few references.…”

Section: The Pathology Of the Critical Case N =mentioning

confidence: 99%

The Optimal Interior Regularity for the Critical Case of a Clamped Thermoelastic System with Point Control Revisited

Lebiedzik

Triggiani

2011

Modern Aspects of the Theory of Partial Differential Equations

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Abstract. In the case of clamped thermoelastic systems with interior point control defined on a bounded domain Ω, the critical case is n = dim Ω = 2. Indeed, an optimal interior regularity theory was obtained in [Triggiani, Discrete Contin. Dyn. Syst., 2007] for n = 1 and n = 3. However, in this reference, an ' -loss' of interior regularity has occurred due to a peculiar pathology: the incompatibility of the B.C. of the spaces H 3/2 0 (Ω) and H 3/2 00 (Ω). This problem for n = 2 was rectified in [Triggiani, J. Differential Equations, 2008]: this establishes the sought-after interior regularity of the thermoelastic problem through a technical analysis based on sharp boundary (trace) regularity theory of Kirchhoff and wave equations. As an additional bonus, a sharp boundary regularity of the elastic displacement is also obtained. In the present paper, we revisit that problem using a technique developed by these authors to circumvent the pathology of the incompatible boundary conditions. This yields a more direct proof of the optimal interior regularity (but not of the boundary regularity).

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Decay Rates of Interactive Hyperbolic-Parabolic PDE Models with Thermal Effects on the Interface

Cited by 23 publications

References 23 publications

Dynamics of a thermoelastic von Kármán plate in a subsonic gas flow

Dynamics of a thermoelastic von Kármán plate in a subsonic gas flow

Stabilization of von Kármán plate in the presence of thermal effects in a subsonic potential flow of gas

The Optimal Interior Regularity for the Critical Case of a Clamped Thermoelastic System with Point Control Revisited

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