Existence and uniqueness theorems for a class of equations of interaction type between a vibrating structure and a fluid

Marie-Thérèse Aimar (1) , Abdelkader Intissar (2)

1 I2M - Institut de Mathématiques de Marseille
2 UPP - Université Pascal Paoli

The problems of the interaction between a vibrating structure and a fluid have been studied by many authors, see for example the interesting article [7] and their references. The principal objective of this work is to investigate the solvability of some problems of interaction between structure and fluid by a mathematical method based upon the analytic semigroups and fractional powers of operators and which can be applied to wider range of physical situations. In this paper, we develop this method on a three-dimensional model of interaction between a vibrating structure and a light fluid occupying a bounded domain in IR3. This model was introduced in J. Sound Vibration 177 (1994) [3] by Filippi-Lagarrigue-Mattei for an one-dimensional clamped thin plate, extended by an in nite perfectly rigid baffle. Intissar and Jeribihave shown in J. Math. Anal. Appl. (2004) [4] the existence of a Riesz basis of generalized eigenvectors of this one-dimensional model. A two-dimensional model of the vibration and the acoustic radiation of a baffled rectangular plate in contact with a dense fluid was consideredby Mattei in J. Sound Vibration (1996) [9].

Mots clés: spectral analysis; analytic semigroups; fractional powers of operators; non-selfadjoint operators; Riesz basis; coupled partial differential equations; fluid-structure interaction

Domaines: Analyse fonctionnelle [math.FA]; Equations aux dérivées partielles [math.AP]

Marie-Thérèse Aimar, Abdelkader Intissar. Existence and uniqueness theorems for a class of equations of interaction type between a vibrating structure and a fluid. Mathematica Aeterna, 2012, 2 (1), pp.21-30. ⟨hal-01290377⟩