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The Stokes problem in a periodic layer. (English) Zbl 1227.35048
The authors consider the steady Stokes problem $$-\Delta v+\nabla p=f$$, $$-\nabla \cdot v=f_\nabla$$, posed in a domain $$\Omega$$ of $$\mathbb R^3$$ which is unbounded in two directions and bounded in the third one and which has a periodic structure near infinity. Outside a finite ball, the domain $$\Omega$$ has a bounded 3D pattern $$S$$ which is infinitely repeated along two directions. $$S$$ may contain some hole. The boundary conditions $$v=g$$ are imposed on $$\partial\Omega$$.
The main result of the paper proves that the solution $$(v,p)\in W_\beta^{l+1}(\Omega)^3\times W_\beta^{l,1}(\Omega)$$ of the Stokes problem has an asymptotic representation, under some hypotheses on the data of this problem. This asymptotic representation involves the solutions of Stokes problems in the reference cell $$S$$. The proof of this asymptotic representation is essentially a consequence of the properties of the solutions of the cell problems.
##### MSC:
 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 35Q30 Navier-Stokes equations 76D07 Stokes and related (Oseen, etc.) flows 76M50 Homogenization applied to problems in fluid mechanics
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