Generation of bounded semigroups in nonlinear subsonic flow–structure interactions with boundary dissipation

Lasiecka, Irena; Webster, Justin T.

doi:10.1002/mma.1518

Cited by 17 publications

(15 citation statements)

References 26 publications

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“…In this case, however, asymptotic smoothness is arrived at in a straightforward way, which additionally exploits the compactness of the von Karman nonlinearity (with respect to the energy identity) in the case where u t in H 1 0 (Ω). The mathematical hurdles arising in the γ = 0 case (where u t ∈ L 2 (Ω)) begin at the outset with well-posedness of weak solutions; indeed, many well-posedness and long-time behavior analysis [36,30,16] are dramatically complicated when γ = 0. Thus, it is no wonder that, to date, the nonrotational von Karman plate with delayed terms has not been considered.…”

Section: Statement Of Main Resultsmentioning

confidence: 99%

Attractors for Delayed, Nonrotational von Karman Plates with Applications to Flow-Structure Interactions Without any Damping

Čhuešhov¹,

Lasiecka

Webster

2014

Communications in Partial Differential Equations

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This paper is devoted to a long time behavior analysis associated with flow structure interactions at subsonic and supersonic velocities. It turns out that an intrinsic component of that analysis is the study of attracting sets corresponding to von Karman plate equations with delayed terms and without rotational terms. The presence of delay terms in the dynamical system leads to the loss of gradient structure while the absence of rotational terms in von Karman plates leads to the loss of compactness of the orbits. Both these features make the analysis of long time behavior rather subtle rendering the established tools in the theory of PDE dynamical systems not applicable. It is our goal to develop methodology that is capable of handling this class of problems.Key terms: nonlinear plate, PDE with delay, long-time behavior of solutions, dynamical systems, global attractors, flow-structure interaction, MSC 2010: 35L20, 74F10, 35Q74. known from experiment (and also confirmed by numerics), that the potential flow (particularly at the supersonic speeds) has the ability of inducing a certain amount of stability in the moving structure. This is the case even when the structure itself does not possess mechanical damping mechanisms. If one writes down the equations for the interactive system, along with the standard energy balance, this dissipative effect is not exhibited at all; quantities are conserved and not dissipated. Thus, there must be some "hidden" mechanism which produces this dissipation. It is our goal to shed some light on this phenomenon. As it turns out, the decoupling technique introduced in [5, 6], which reduces the analysis of full flow-structure interaction to that of a certain delayed plate model, allows us to observe certain stabilizing effects of the flow. These occur in the form of non-conservative forces acting upon the structure as the "downwash" of the flow. This idea was already applied to Berger plate models [7,17] in the proof of existence of attractors corresponding to the associated reduced plate problem with a delayed term.In fact, well-posedness and long-time behavior analyses of nonlinear plate PDEs with delays have been treated in [8] (see also [14]): first, in the case of the von Karman model with rotational inertia, and secondly, in [7,17], in the case of the Berger model with a small intensity of delayed term (this corresponds to a large speed U of the flow of gas -hypersonic). These expositions flesh out the existence and properties of global attractors for the general plate with delay in the presence of a 'natural' form of interior damping, and then apply this general result to the specific delayed (aeroelastic) force given in the full flow-plate coupling.It should be noted that the presence of rotational inertia parameter, while drastically improving the topological properties of the model, is neither natural nor desirable in the context of flow-structure interaction. First, the original model for flow structure interaction describes the interaction between the mid-surface of the pl...

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Section: Statement Of Main Resultsmentioning

confidence: 99%

Attractors for Delayed, Nonrotational von Karman Plates with Applications to Flow-Structure Interactions Without any Damping

Čhuešhov¹,

Lasiecka

Webster

2014

Communications in Partial Differential Equations

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100

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“…d(x) c 0 > 0 for all x in Ω, since energy methods require the use of commutators to reconstruct the full energy in observability type estimates. More specifically, the use of geometrically constrained damping in the form of damping active in a collar of the boundary has arisen in the study of coupled dynamics [56,43,6].…”

Section: Motivation and Literaturementioning

confidence: 99%

Smooth attractors of finite dimension for von Karman evolutions with nonlinear frictional damping localized in a boundary layer

Geredeli

Lasiecka

Webster

2013

Journal of Differential Equations

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In this paper dynamic von Karman equations with localized interior damping supported in a boundary collar are considered. Hadamard well-posedness for von Karman plates with various types of nonlinear damping are well known, and the long-time behavior of nonlinear plates has been a topic of recent interest. Since the von Karman plate system is of "hyperbolic type" with critical nonlinearity (noncompact with respect to the phase space), this latter topic is particularly challenging in the case of geometrically constrained, nonlinear damping. In this paper we first show the existence of a compact global attractor for finite energy solutions, and we then prove that the attractor is both smooth and finite dimensional. Thus, the hyperbolic-like flow is stabilized asymptotically to a smooth and finite dimensional set.

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“…It is of interest to note that boundary control via moments are typically expressed through a hinged type condition. See [22] for details in the case of the plate alone, and [58] for a well-posedness analysis of these boundary conditions in the flow-plate model.…”

Section: Physical Configurations and Other Flow Boundary Conditionsmentioning

confidence: 99%

Flow-plate interactions: Well-posedness and long-time behavior

Čhuešhov¹,

Lasiecka²,

Webster³

2014

DCDS-S

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We consider flow-structure interactions modeled by a modified wave equation coupled at an interface with equations of nonlinear elasticity. Both subsonic and supersonic flow velocities are treated with Neumann type flow conditions, and a novel treatment of the so called Kutta-Joukowsky flow conditions are given in the subsonic case. The goal of the paper is threefold: (i) to provide an accurate review of recent results on existence, uniqueness, and stability of weak solutions, (ii) to present a construction of finite dimensional, attracting sets corresponding to the structural dynamics and discuss convergence of trajectories, and (iii) to state several open questions associated with the topic. This second task is based on a decoupling technique which reduces the analysis of the full flow-structure system to a PDE system with delay.

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Generation of bounded semigroups in nonlinear subsonic flow–structure interactions with boundary dissipation

Cited by 17 publications

References 26 publications

Attractors for Delayed, Nonrotational von Karman Plates with Applications to Flow-Structure Interactions Without any Damping

Attractors for Delayed, Nonrotational von Karman Plates with Applications to Flow-Structure Interactions Without any Damping

Smooth attractors of finite dimension for von Karman evolutions with nonlinear frictional damping localized in a boundary layer

Flow-plate interactions: Well-posedness and long-time behavior

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