On Bloch waves for the Stokes equations

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January 2007, 7(1): 1-28. doi: 10.3934/dcdsb.2007.7.1

On Bloch waves for the Stokes equations

Grégoire Allaire ^1, , Carlos Conca ^2, , Luis Friz ^3, and Jaime H. Ortega ^3,

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, Chile
3.	Departamento de Ciencias Básicas, Facultad de Ciencias, Universidad del Bío-Bío, Avenida Andrés Bello s/n, Casilla 447, Chillán, Chile, Chile

Received January 2006 Revised August 2006 Published October 2006

Abstract
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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.

Keywords: Spectral Theory, Stokes Equation, Bloch waves, Homogenization..

Mathematics Subject Classification: 35P99, 35Q30, 47A75, 49J20, 93C20, 93B6.

Citation: Grégoire Allaire, Carlos Conca, Luis Friz, Jaime H. Ortega. On Bloch waves for the Stokes equations. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 1-28. doi: 10.3934/dcdsb.2007.7.1

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