On Bloch waves for the Stokes equations

Grégoire Allaire; Carlos Conca; Luis Friz; Jaime H. Ortega

doi:10.3934/dcdsb.2007.7.1

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2007, Volume 7, Issue 1: 1-28. Doi: 10.3934/dcdsb.2007.7.1

This issue Previous Article Next Article Well-posedness of the modified Crank-Nicholson difference schemes in Bochner spaces

On Bloch waves for the Stokes equations

1.
Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 PALAISEAU Cedex
2.
Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 170-3, Correo 3, Santiago
3.
Departamento de Ciencias Básicas, Facultad de Ciencias, Universidad del Bío-Bío, Avenida Andrés Bello s/n, Casilla 447, Chillán

Received: December 31, 2005

Revised: July 31, 2006

Published: December 2006

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Abstract

In this work, we study the Bloch wave decomposition for the Stokes equations in a periodic media in $\R^d$. We prove that, because of the incompressibility constraint, the Bloch eigenvalues, as functions of the Bloch frequency $\xi$, are not continuous at the origin. Nevertheless, when $\xi$ goes to zero in a fixed direction, we exhibit a new limit spectral problem for which the eigenvalues are directionally differentiable. Finally, we present an analogous study for the Bloch wave decomposition for a periodic perforated domain.

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Mathematics Subject Classification: 35P99, 35Q30, 47A75, 49J20, 93C20, 93B60.

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