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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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