The coupled PDE system arising in fluid/structure interaction. I. Explicit semigroup generator and its spectral properties

Avalos, George; Triggiani, Roberto

doi:10.1090/conm/440/08475

Cited by 81 publications

(161 citation statements)

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“…Semigroup well-posedness for problem (1.5) [7,8] Paper [8] shows that the Stokes-Lamé system (1.5a-g) admits a semigroup formulation on the space H (see also [7] for a simplified model): In fact, let us define the following "Dirichlet" and "Neumann" maps D s and N f : …”

Section: Fluid-structure Physical and Pde Modelmentioning

confidence: 99%

“…In [8], it was shown that the pressure term in (1.5a) can be eliminated via the expression (1.14) where the G i are as defined above (see also [7] for a simplified model). In consequence, the fluid-structure PDE system (1.5a-g) can be abstractly expressed as (1.15) where the pressure has been eliminated, and where…”

Section: Fluid-structure Physical and Pde Modelmentioning

confidence: 99%

“…with the domain of definition D(A) ⊂ H. An explicit description of the domain of A is given in [7,8]; [w 0 , w 1 , u 0 ] ∈ D(A), means that there exists a harmonic "pressure" function π ∈ L 2 ( f ) such that [w 0 , w 1 , u 0 , π] satisfies: (a) for η = 0:…”

Section: Fluid-structure Physical and Pde Modelmentioning

confidence: 99%

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Boundary feedback stabilization of a coupled parabolic–hyperbolic Stokes–Lamé PDE system

Avalos

Triggiani

2009

J. Evol. Equ.

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We consider a parabolic-hyperbolic coupled system of two partial differential equations (PDEs), which governs fluid-structure interactions, and which features a suitable boundary dissipation term at the interface between the two media. The coupled system consists of Stokes flow coupled to the Lamé system of dynamic elasticity, with the respective dynamics being coupled on a boundary interface, where dissipation is introduced. Such a system is semigroup well-posed on the natural finite energy space (Avalos and Triggiani in Discr Contin Dynam Sys, to appear). Here we prove that, moreover, such semigroup is uniformly (exponentially) stable in the corresponding operator norm, with no geometrical conditions imposed on the boundary interface. This result complements the strong stability properties of the undamped case (Avalos and Triggiani in Discr Contin Dynam Sys, to appear).

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Section: Fluid-structure Physical and Pde Modelmentioning

confidence: 99%

Section: Fluid-structure Physical and Pde Modelmentioning

confidence: 99%

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Boundary feedback stabilization of a coupled parabolic–hyperbolic Stokes–Lamé PDE system

Avalos

Triggiani

2009

J. Evol. Equ.

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“…The relation in (5.10) proves dissipativity of A. The proof of maximality follows now the same arguments as in [30,33].…”

Section: Generation Of the Semigroupmentioning

confidence: 69%

“…The literature already contains two treatments leading to the abstract model corresponding to this PDE system. One approachintroduced in [30] and pursued in [31,32] -eliminates the pressure by introducing two suitable Green's maps of appropriate elliptic problems in ! f ; one with Dirichlet datum on !…”

Section: Specialization Of Abstract Model To the Fluidstructure Intermentioning

confidence: 99%

Min-max Game Theory for Elastic and Visco-Elastic Fluid Structure Interactions

Lasiecka¹

2013

TOAMJ

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Abstract:We present the salient features of a min-max game theory developed in the context of coupled PDE's with an interface. Canonical applications include linear fluid-structure interaction problem modeled by Oseen's equations coupled with elastic waves. We shall consider two models for the structures: elastic and visco-elastic. Control and disturbance are allowed to act at the interface between the two media. The sought-after saddle solutions are expressed in a pointwise feedback form, which involves a Riccati operator; that is, an operator satisfying a suitable non-standard Riccati differential equation. Motivations, applications as well as a brief historical account are also provided.

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Semigroup wellposedness and asymptotic stability of a compressible Oseen–structure interaction via a pointwise resolvent criterion

Geredeli

2023

Mathematische Nachrichten

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In this study, we consider the Oseen structure of the linearization of a compressible fluid-structure interaction (FSI) system for which the interaction interface is under the effect of material derivative term. The flow linearization is taken with respect to an arbitrary, variable ambient vector field. This process produces extra "convective derivative" and "material derivative" terms, which render the coupled system highly nondissipative. We show first a new well-posedness result for the full incorporation of both Oseen terms, which provides a uniformly bounded semigroup via dissipativity and perturbation arguments. In addition, we analyze the long time dynamics in the sense of asymptotic (strong) stability in an invariant subspace (one-dimensional less) of the entire state space, where the continuous semigroup is uniformly bounded. For this, we appeal to the pointwise resolvent condition introduced in Chill and Tomilov [Stability of operator semigroups: ideas and results, perspectives in operator theory Banach center publications, 75 (2007), Institute of Mathematics Polish Academy of Sciences, Warszawa, 71-109], which avoids an immensely technical and challenging spectral analysis and provides a short and relatively easy-to-follow proof.

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The coupled PDE system arising in fluid/structure interaction. I. Explicit semigroup generator and its spectral properties

Cited by 81 publications

References 0 publications

Boundary feedback stabilization of a coupled parabolic–hyperbolic Stokes–Lamé PDE system

Boundary feedback stabilization of a coupled parabolic–hyperbolic Stokes–Lamé PDE system

Min-max Game Theory for Elastic and Visco-Elastic Fluid Structure Interactions

Semigroup wellposedness and asymptotic stability of a compressible Oseen–structure interaction via a pointwise resolvent criterion

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