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This work is motivated by a longstanding interest in the long time behavior of flow‐structure interaction (FSI) PDE dynamics. We consider a linearized compressible flow structure interaction (FSI) PDE model with a view of analyzing the stability properties of both the compressible flow and plate solution components. In our earlier work, we gave an answer in the affirmative to question of uniform stability for finite energy solutions of said compressible flow‐structure system, by means of a “frequency domain” approach. However, the frequency domain method of proof in that work is not “robust” (insofar as we can see), when one wishes to study longtime behavior of solutions of compressible flow‐structure PDE models, which track the appearance of the ambient state onto the boundary interface. Nor is a frequency domain approach in this earlier work availing when one wishes to consider the dynamics, in long time, of solutions to physically relevant nonlinear versions of the compressible flow‐structure PDE system under present consideration (e.g., the Navier–Stokes nonlinearity in the PDE flow component or a nonlinearity of Berger/Von Karman type in the plate equation). Accordingly, in the present work, we operate in the time domain by way of obtaining the necessary energy estimates, which culminate in an alternative proof for the uniform stability of finite energy compressible flow‐structure solutions. Since there is a need to avoid steady states in our stability analysis, as a prerequisite result, we also show here that zero is an eigenvalue for the generators of flow‐structure systems, whether the material derivative term be absent or present. Moreover, we provide a clean characterization of the (one dimensional) zero eigenspace, with or without material derivative, under an appropriate assumption on the underlying ambient vector field.
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Semigroup wellposedness and asymptotic stability of a compressible Oseen–structure interaction via a pointwise resolvent criterion
Abstract 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 isuniformly 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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- Award ID(s):
- 1907823
- PAR ID:
- 10443034
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Mathematische Nachrichten
- Volume:
- 296
- Issue:
- 3
- ISSN:
- 0025-584X
- Page Range / eLocation ID:
- p. 1135-1155
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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