A new maximality argument for a coupled fluid-structure interaction, with implications for a divergence-free finite element method

We study the well-posedness of a total linearization, with respect to a perturbation of the external forcing, of a free-boundary nonlinear elasticity-incompressible fluid interaction. The total linearization for the coupling modeled by the Navier-Stokes equations and the nonlinear equations of elastodynamics was obtained recently in [L. Bociu and J.-P. Zolésio, Evol. Equ. Control Theory, 2 (2013), pp. 55-79]. The equations and the free boundary were linearized together, and the result turned out to be quite different from the usual coupling of classical linear models. New additional terms are present on the common interface, some of them involving boundary curvatures and boundary acceleration. These terms play an important role in the final linearized system and cannot be neglected; their presence also introduces new challenges in the well-posedness analysis, which proceeds to establish that the evolution operator associated to the linearized system can be represented as a bounded perturbation of a maximal dissipative semigroup generator. Introduction.A typical fluid-structure interaction system is composed of a solid either surrounding a fluid flow (arteries, pipelines, reservoir tanks, etc.) or surrounded by a fluid or gas (blood cells, marine animals, bridge supports, aircraft, automobiles, etc.). This model falls in the category of free boundary problems, since the motion of the common boundary between the solid and the fluid is one of the unknowns in the coupled system. The ubiquity of this interaction type has led to a rapidly growing research interest in this model. Its mathematical solvability [5,9,15,17,22,21,26,32,33,34,37,38,49,55,52], numerical approximations [23,29,39,54,53,44,46], and stability [14,27,28,31] have been intensively studied.Mathematically, this interaction is described by a partial differential equation (PDE) system that couples the parabolic (fluid) and hyperbolic (elasticity) phases, where the key issue is that the traces of the elastic component at the energy level are not defined via the standard trace theory. The loss of regularity induced by the hyperbolic component left the basic question of existence open until recently.The first major development appeared in [16], where the authors established local existence and regularity of solutions for the coupling of Navier-Stokes equations and a linear elasticity model, describing a body submerged in a fluid, under the assumptions of smooth initial data; specifically, the initial fluid velocity w 0 has Sobolev regularity H 5 , and the initial data for the elastic component (ϕ 0 , ϕ 1 ) belongs to H 3 × H 2 . Moreover, due to the incompressibility condition on the fluid, the uniqueness result

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“…• For a C 1 compact boundary-less manifold M let {·, ·} 1 2 ,M denote the duality pairing between the Sobolev spaces H −1/2 (M) and H 1/2 (M).…”

Section: Norms and Duality Pairingsmentioning

confidence: 99%

Well-Posedness Analysis for a Linearization of a Fluid-Elasticity Interaction

Bociu¹,

Toundykov²,

Zolésio³

2015

SIAM J. Math. Anal.

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“…In Section 2 we introduce the functional and semigroup setup for the PDE problem. A PDE result pertaining to the fluid pressure, obtained in [6] and recorded here as Lemma 2.1, allows for a clear (and yet nontrivial) definition of the linear operator which underlies the fluid-structure dynamics. Section 3 is devoted to the spectral analysis for the dynamics operator, this being of intrinsic interest, as well as a prerequisite step for the proof of our main result, that is Theorem 1.5.…”

Section: Background and Further Remarksmentioning

confidence: 98%

“…In this section we recall from [6] the semigroup setup for the IBVP Problem (1.2)-(1.3). Although the C 0 -semigroup e A ρ t underlying the linear evolution (1.2)-(1.3) was introduced in [19], a more explicit definition of the dynamics operator is given, based upon a certain identification of the fluid pressure p as the solution to an appropriate elliptic problem; see Lemma 2.1 below.…”

Section: Semigroup Frameworkmentioning

confidence: 99%

“…The reader is referred to the monograph [16,Chapters 5,6,11,12] and to the recent paper [17] for well-posedness and asymptotic behavior results/techniques provided for other flow and flowplate interactions.…”

Section: Background and Further Remarksmentioning

confidence: 99%

“…e.g.,[6, Proposition 1]; in attaining the second estimate we have also used (3.32)); appealing again to(3.28) and (3.24) we then have q n | ∂O −→ 0 strongly in H −1/2 (O) . (3.35) G. Avalos, F. Bucci / J.…”

mentioning

confidence: 95%

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

Avalos

Bucci

2015

Journal of Differential Equations

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In this work we investigate the uniform stability properties of solutions to a well-established partial differential equation (PDE) model for a fluid-structure interaction. The PDE system under consideration comprises a Stokes flow which evolves within a three-dimensional cavity; moreover, a Kirchhoff plate equation is invoked to describe the displacements along a (fixed) portion -say, -of the cavity wall. Contact between the respective fluid and structure dynamics occurs on the boundary interface . The main result in the paper is as follows: the solutions to the composite PDE system, corresponding to smooth initial data, decay at the rate of O(1/t). Our method of proof hinges upon the appropriate invocation of a relatively recent resolvent criterion for polynomial decays of C 0 -semigroups. While the characterization provided by said criterion originates in the context of operator theory and functional analysis, the work entailed here is wholly within the realm of PDE.

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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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A new maximality argument for a coupled fluid-structure interaction, with implications for a divergence-free finite element method

Cited by 21 publications

References 10 publications

Well-Posedness Analysis for a Linearization of a Fluid-Elasticity Interaction

Well-Posedness Analysis for a Linearization of a Fluid-Elasticity Interaction

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

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

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