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The Navier wall law at a boundary with random roughness. (English) Zbl 1176.35127
Summary: We consider the Navier-Stokes equation in a domain with irregular boundaries. The irregularity is modeled by a spatially homogeneous random process, with typical size \({\varepsilon\ll 1}\). In the parent paper [A. Basson and D. Gérard-Varet, Commun. Pure Appl. Math. 61, No. 7, 941–987 (2008; Zbl 1179.35207)], we derived a homogenized boundary condition of Navier type as \({\varepsilon\to 0}\). We show here that for a large class of boundaries, this Navier condition provides a \({O(\varepsilon^{3/2} |\ln \varepsilon|^{1/2})}\) approximation in \(L ^2\), instead of \({O(\varepsilon^{3/2})}\) for periodic irregularities. Our result relies on the study of an auxiliary boundary layer system. Decay properties of this boundary layer are deduced from a central limit theorem for dependent variables.

MSC:
35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
35A35 Theoretical approximation in context of PDEs
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