Methods and Applications of Analysis

Very Weak Estimates for a Rough Poisson-Dirichlet Problem with Natural Vertical Boundary Conditions

Vuk Milišić

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This work is a continuation of [3]; 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 [3] we studied a priori estimates in this setting; here we fully develop very weak estimates à la Nečas [17] in the weighted Sobolev spaces on an unbounded domain. We obtain optimal estimates which improve those derived in [3]. 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 [13, 18].

Article information

Methods Appl. Anal., Volume 16, Number 2 (2009), 157-186.

First available in Project Euclid: 2 November 2009

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Zentralblatt MATH identifier

Primary: 76D05: Navier-Stokes equations [See also 35Q30] 35B27: Homogenization; equations in media with periodic structure [See also 74Qxx, 76M50] 76Mxx: Basic methods in fluid mechanics [See also 65-XX] 65Mxx: Partial differential equations, initial value and time-dependent initial- boundary value problems

Wall-laws rough boundary Laplace equation multi-scale modelling boundary layers error estimates natural boundary conditions vertical boundary correctors


Milišić, Vuk. Very Weak Estimates for a Rough Poisson-Dirichlet Problem with Natural Vertical Boundary Conditions. Methods Appl. Anal. 16 (2009), no. 2, 157--186.

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