On Bloch waves for the Stokes equations
Grégoire Allaire, Carlos Conca, Luis Friz, Jaime H. Ortega · Discrete and Continuous Dynamical Systems - B · 2007
In this work, we study the Bloch wave decomposition for the Stokesequations in a periodic media in $\R^d$. We prove that, because of theincompressibility constraint, the Bloch eigenvalues, as functions ofthe Bloch frequency $\xi$, are not continuous at theorigin. Nevertheless, when $\xi$ goes to zero in a fixed direction, weexhibit a new limit spectral problem for which the eigenvalues aredirectionally differentiable. Finally, we present an analogous studyfor the Bloch wave decomposition for a periodic perforated domain.