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arXiv · 0812.4340

Very weak estimates for a rough Poisson-Dirichlet problem with natural vertical boundary conditions

Vuk Milisic
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Abstract

This work is a continuation of [E. Bonnetier, D.Bresch, V. Milisic, submitted]; it deals with rough boundaries in the simplified context of a Poisson equation. We impose Dirichlet boundary conditions on the periodic microscopic perturbation of a flat edge on one side and natural homogeneous Neumann boundary conditions are applied on the inlet/outlet of the domain. To prevent oscillations on the Neumann-like boundaries, we introduce a microscopic vertical corrector defined in a rough quarter-plane. In [E. Bonnetier, D.Bresch, V. Milisic, submitted] we studied a priori estimates in this setting; here we fully develop very weak estimates a la Necas [J. Necas. Les méthodes directes en théorie des équations elliptiques] in the weighted Sobolev spaces on an unbounded domain. We obtain optimal estimates which improve those derived in [E. Bonnetier, D.Bresch, V. Milisic, submitted]. We validate these results numerically, proving first order results for boundary layer approximation including the vertical correctors and a little less for the averaged wall-law introduced in the literature [W. Jäger and A. Mikelic. J. Diff. Equa., N. Neus, M. Neus, A. Mikelic, Appl. Anal. 2006].

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Vuk Milisic. 2008-12-23. Very weak estimates for a rough Poisson-Dirichlet problem with natural vertical boundary conditions. https://arxiv.org/abs/0812.4340

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