Rational rates of uniform decay for strong solutions to a fluid-structure PDE system

Avalos, George; Bucci, F.

doi:10.1016/j.jde.2015.01.037

Cited by 26 publications

(32 citation statements)

References 36 publications

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“…In addition to the manuscript , we should mention the papers and , which also deal with the problem of obtaining rational decay rates for coupled PDE systems of hyperbolic‐parabolic type, in multi‐dimensions, and which also obtain their polynomial stability objective by the agency of the resolvent criterion theorem 3 of .…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

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Uniform stability for solutions of a structural acoustics PDE model with no added dissipative feedback

Avalos

Geredeli

2016

Math Methods in App Sciences

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Communicated by P. SacksA rate of rational decay is obtained for smooth solutions of a PDE model, which has been used in the literature to describe structural acoustic flows. This structural acoustics model is composed of two distinct PDE systems: (i) a wave equation, to model the interior acoustic flow within the given cavity and (ii) a structurally damped elastic equation, to describe timeevolving displacements along the flexible portion 0 of the cavity walls. Moreover, the extent of damping in this elastic component is quantified by parameter Á 2 OE0, 1. The coupling between these two distinct dynamics occurs across the boundary interface 0 . Our main result is the derivation of uniform decay rates for classical solutions of this particular structural acoustic PDE, decay rates that are obtained without incorporating any additional boundary dissipative feedback mechanisms. In particular, in the case that full Kelvin-Voight damping is present in fourth-order elastic dynamics, that is, the structural acoustics system as it appears in the literature, solutions that correspond to smooth initial data decay at a rate of O t 1 6 Á. By way of deriving these stability results, necessary a priori inequalities for a certain static structural acoustics PDE model are generated here; these inequalities ultimately allow for an application of a recently derived resolvent criterion for rational decay.

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Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

“…Applying the boundary conditions in (23) and (24) to these relations, we then have (As usual @z0 @ D rz 0 , where .x/ is the unit tangential vector). Combining the two yields now the relation (34).…”

Section: Proof Of (I)mentioning

confidence: 99%

Uniform stability for solutions of a structural acoustics PDE model with no added dissipative feedback

Avalos

Geredeli

2016

Math Methods in App Sciences

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“…The one reference which pertains to exponential stability of the flow-structure system (4)-(6), in the special case U ≡ 0, is the work [12] (This paper also resolves wellposedness of [structurally] nonlinear versions of (4)- (6) in the case of zero ambient state.) However, the time dependent multiplier method which is adopted in [12], by way of establishing exponential decay, is inapplicable in the present case of generally nonzero vector field U which satisfies (15).…”

Section: Literaturementioning

confidence: 96%

“…Such a uniform bound on the resolvent, as it acts on the imaginary axis, ultimately allows for the wellknown resolvent characterization for exponential stability of bounded semigroups; see Theorem 23 in the Appendix, as well as [23] and [33]. (Such a static methodology was previouslyused in [8], [5] and [6] in the context of determining uniform decay rates for other compressible fluid-structure interactions.) The work to estimate said resolvent is undertaken in Section 5 below.…”

Section: Literaturementioning

confidence: 99%

Exponential stability of a nondissipative, compressible flow–structure PDE model

Avalos

Geredeli

2019

J. Evol. Equ.

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In this work, a result of exponential stability is obtained for solutions of a compressible flow-structure partial differential equation (PDE) model which has recently appeared in the literature. In particular, a compressible flow PDE and its associated state equation for the associated pressure variable, each evolving within a three dimensional domain O, are coupled to a fourth order plate equation which holds on a flat portion Ω of the boundary ∂O. Moreover, since this coupled PDE model is the result of a linearization of the compressible Navier-Stokes equations about an arbitrary state, the flow PDE component contains a generally nonzero ambient flow profile U. By way of obtaining the aforesaid exponential stability, a "frequency domain" approach is adopted here, an approach which is predicated on obtaining a uniform estimate on the resolvent of the associated flow-structure semigroup generator.

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“…Eventually, we are interested in learning if and how the presence of the dissipating fluid dynamics affects the stability of the structure (as in, e.g., [24,18,28,27], [4]). In particular, for the linear compressible fluid-structure interaction PDE model, we are interested in strong/uniform stability properties of the associated C 0 -semigroup.…”

Section: Introductionmentioning

confidence: 99%

Semigroup well-posedness of a linearized, compressible fluid with an elastic boundary

Avalos¹,

Geredeli²,

Webster³

2018

Discrete &Amp; Continuous Dynamical Systems - B

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We address semigroup well-posedness of the fluid-structure interaction of a linearized compressible, viscous fluid and an elastic plate (in the absence of rotational inertia). Unlike existing work in the literature, we linearize the compressible Navier-Stokes equations about an arbitrary state (assuming the fluid is barotropic), and so the fluid PDE component of the interaction will generally include a nontrivial ambient flow profile U. The appearance of this term introduces new challenges at the level of the stationary problem. In addition, the boundary of the fluid domain is unavoidably Lipschitz, and so the well-posedness argument takes into account the technical issues associated with obtaining necessary boundary trace and elliptic regularity estimates. Much of the previous work on flow-plate models was done via Galerkin-type constructions after obtaining good a priori estimates on solutions (specifically [18]-the work most pertinent to ours here); in contrast, we adopt here a Lumer-Phillips approach, with a view of associating solutions of the fluid-structure dynamics with a C0-semigroup e At t≥0 on the natural finite energy space of initial data. So, given this approach, the major challenge in our work becomes establishing of the maximality of the operator A which models the fluid-structure dynamics. In sum: our main result is semigroup well-posedness for the fully coupled fluid-structure dynamics, under the assumption that the ambient flow field U ∈ H 3 (O) has zero normal component trace on the boundary (a standard assumption with respect to the literature). In the final sections we address well-posedness of the system in the presence of the von Karman plate nonlinearity, as well as the stationary problem associated with the dynamics.

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Rational rates of uniform decay for strong solutions to a fluid-structure PDE system

Cited by 26 publications

References 36 publications

Uniform stability for solutions of a structural acoustics PDE model with no added dissipative feedback

Uniform stability for solutions of a structural acoustics PDE model with no added dissipative feedback

Exponential stability of a nondissipative, compressible flow–structure PDE model

Semigroup well-posedness of a linearized, compressible fluid with an elastic boundary

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