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On Bloch waves for the Stokes equations

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


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