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# Direct integral decomposition for periodic function spaces and application to Bloch waves

• In this paper, we study a direct integral decomposition for the spaces $L^2(O)$ and $H^1(O)$ based on $(\xi,Y^*)-$periodic functions. Using this decomposition we can write the Green's operator (associated to the classical Stokes system in fluid mechanics) in terms of a family of self-adjoint compact operators which depend on the parameter $\xi$. As a consequence, we obtain the so-called Bloch waves associated to the Stokes system in the case of a periodic perforated domain.
Mathematics Subject Classification: Primary: 35B27, 46A35, 35Q30; Secondary: 34B27, 47A70.

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