Homogenization of stokes system using bloch waves

Grégoire Allaire; Tuhin Ghosh; Muthusamy Vanninathan; Grégoire Allaire; Tuhin Ghosh; Muthusamy Vanninathan

doi:10.3934/nhm.2017022

Networks and Heterogeneous Media

2017, Volume 12, Issue 4: 525-550. doi: 10.3934/nhm.2017022

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Homogenization of stokes system using bloch waves

1.
Centre de Mathématiques Appliquées, Ecole Polytechnique, CNRS, Université Paris-Saclay, Palaiseau, France
2.
Centre for Applicable Matematics, Tata Institute of Fundamental Research, Bangalore, India
3.
Current address: Mathematics Department, IIT-Bombay, Mumbai, India

Received: 01 September 2016 Revised: 01 April 2017
Primary: 35P99, 35Q30, 47A75; Secondary: 49J20, 93C20, 93B60

In this work, we study the Bloch wave homogenization for the Stokes system with periodic viscosity coefficient. In particular, we obtain the spectral interpretation of the homogenized tensor. The presence of the incompressibility constraint in the model raises new issues linking the homogenized tensor and the Bloch spectral data. The main difficulty is a lack of smoothness for the bottom of the Bloch spectrum, a phenomenon which is not present in the case of the elasticity system. This issue is solved in the present work, completing the homogenization process of the Stokes system via the Bloch wave method.
- Spectral theory,
- Bloch waves,
- Stokes equation,
- homogenization,
- periodic media
Citation: Grégoire Allaire, Tuhin Ghosh, Muthusamy Vanninathan. Homogenization of stokes system using bloch waves[J]. Networks and Heterogeneous Media, 2017, 12(4): 525-550. doi: 10.3934/nhm.2017022

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Abstract

In this work, we study the Bloch wave homogenization for the Stokes system with periodic viscosity coefficient. In particular, we obtain the spectral interpretation of the homogenized tensor. The presence of the incompressibility constraint in the model raises new issues linking the homogenized tensor and the Bloch spectral data. The main difficulty is a lack of smoothness for the bottom of the Bloch spectrum, a phenomenon which is not present in the case of the elasticity system. This issue is solved in the present work, completing the homogenization process of the Stokes system via the Bloch wave method.

References

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