Homogenization and long time asymptotic of a fluid-structure interaction problem

Grégoire Allaire; Alessandro Ferriero

doi:10.3934/dcdsb.2008.9.199

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2008, Volume 9, Issue 2: 199-220. Doi: 10.3934/dcdsb.2008.9.199

This issue Previous Article Next Article Lie symmetries, qualitative analysis and exact solutions of nonlinear Schrödinger equations with inhomogeneous nonlinearities

Homogenization and long time asymptotic of a fluid-structure interaction problem

Grégoire Allaire^1, and
Alessandro Ferriero^2,

1.
Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 PALAISEAU Cedex
2.
Section de Mathématiques, EPFL, 1015 Lausanne

Received: January 31, 2007

Revised: October 31, 2007

Published: February 2008

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Abstract

We study the homogenization of an unsteady fluid-structure interaction problem with a scaling corresponding to a long time asymptotic regime. We consider oscillating initial data which are Bloch wave packets corresponding to tubes vibrating in opposition of phase. We prove that the initial displacements follow the rays of geometric optics and that the envelope function evolves according to a Schr ̈odinger equation which can be interpreted as an effect of dispersion.

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Mathematics Subject Classification: Primary: 35B27, 74Q10.

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