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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

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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