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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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##### References:
 [1] Achdou, Y., Le Tallec, P., Valentin, F., Pironneau, O.: Constructing wall laws with domain decomposition or asymptotic expansion techniques. In: Symposium on Advances in Computational Mechanics, Vol. 3 (Austin, TX, 1997) Comput. Methods Appl. Mech. Engrg. 151, 1–2, 215–232 (1998) · Zbl 0920.76063 [2] Achdou, Y., Mohammadi, B., Pironneau, O., Valentin, F.: Domain decomposition & wall laws. In: Recent developments in domain decomposition methods and flow problems (Kyoto, 1996; Anacapri, 1996), Vol. 11 of GAKUTO Internat. Ser. Math. Sci. Appl. Tokyo: Gakkōtosho 1998, pp. 1–14 · Zbl 0931.76077 [3] Achdou Y., Pironneau O., Valentin F.: Effective boundary conditions for laminar flows over periodic rough boundaries. J. Comput. Phys. 147(1), 187–218 (1998) · Zbl 0917.76013 · doi:10.1006/jcph.1998.6088 [4] Amirat Y., Bresch D., Lemoine J., Simon J.: Effect of rugosity on a flow governed by stationary Navier-Stokes equations. Quart. Appl. Math. 59(4), 769–785 (2001) · Zbl 1019.76014 [5] Avellaneda M., Lin F.-H.: Compactness methods in the theory of homogenization. Comm. Pure Appl. Math. 40(6), 803–847 (1987) · Zbl 0632.35018 · doi:10.1002/cpa.3160400607 [6] Avellaneda M., Lin F.-H.: L p bounds on singular integrals in homogenization. Comm. Pure Appl. Math. 44(8–9), 897–910 (1991) · Zbl 0761.42008 · doi:10.1002/cpa.3160440805 [7] Baladi, V.: Decay of correlations. In: Smooth ergodic theory and its applications (Seattle, WA, 1999), Vol. 69 of Proc. Sympos. Pure Math. Providence, RI: Amer. Math. Soc. 2001, pp. 297–325 · Zbl 0993.37003 [8] Basson A., Gérard-Varet D.: Wall laws for fluid flows at a boundary with random roughness. Comm. Pure Applied Math. 61(7), 941–987 (2008) · Zbl 1179.35207 · doi:10.1002/cpa.20237 [9] Bechert D., Bartenwerfer M.: The viscous flow on surfaces with longitudinal ribs. J. Fluid Mech. 206(1), 105–129 (1989) · doi:10.1017/S0022112089002247 [10] Bresch D., Gérard-Varet D.: Roughness-induced effects on the quasi-geostrophic model. Comm. Math. Phys. 253(1), 81–119 (2005) · Zbl 1149.76680 · doi:10.1007/s00220-004-1173-9 [11] Bresch, D., Milisic, V.: Higher order boundary layer correctors and wall laws derivation: a unified approach. http://arXiv.org./list/math/0611083 , 2006 [12] De Bouard, A., Craig, W., Daz-Espinosa, O., Guyenne, P., Sulem, C.: Long wave expansions for water waves over random topography. http://arXiv.org./list/math/07100389 , 2007 · Zbl 1228.76029 [13] Durrett R.: Probability: theory and examples. Second ed. Belmont, CA: Duxbury Press, 1996 · Zbl 1202.60001 [14] Galdi, G.P.: An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I, Vol. 38 of Springer Tracts in Natural Philosophy. New York: Springer-Verlag, 1994 [15] Gérard-Varet, D.: Highly rotating fluids in rough domains. J. Math. Pures Appl. (9) 82, 11, 1453–1498 (2003) · Zbl 1033.76008 [16] Giaquinta M.: Multiple integrals in the calculus of variations and nonlinear elliptic systems, vol. 105 of Annals of Mathematics Studies. Princeton, NJ: Princeton University Press, 1983 · Zbl 0516.49003 [17] Giaquinta M., Modica G.: Nonlinear systems of the type of the stationary Navier-Stokes system. J. Reine Angew. Math. 330, 173–214 (1982) · Zbl 0492.35018 [18] Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Classics in Mathematics. Berlin: Springer-Verlag, 2001 (Reprint of the 1998 edition) · Zbl 1042.35002 [19] Jäger W., Mikelić A.: On the roughness-induced effective boundary conditions for an incompressible viscous flow. J. Differ. Eq. 170(1), 96–122 (2001) · Zbl 1009.76017 · doi:10.1006/jdeq.2000.3814 [20] Jäger W., Mikelić A.: Couette flows over a rough boundary and drag reduction. Comm. Math. Phys. 232(3), 429–455 (2003) · Zbl 1062.76012 [21] Jäger, W., Mikelić, A., Neuss, N.: Asymptotic analysis of the laminar viscous flow over a porous bed. SIAM J. Sci. Comput. 22, 6, 2006–2028 (2000) (electronic) · Zbl 0980.35124 [22] Jikov V.V., Kozlov S.M., Oleĭnik O.A.: Homogenization of differential operators and integral functionals. Berlin: Springer-Verlag, 1994. (Translated from the Russian by G. A. Yosifian [G. A. Iosif’ yan]) [23] Luchini P.: Asymptotic analysis of laminar boundary-layer flow over finely grooved surfaces. European J. Mech. B Fluids 14(2), 169–195 (1995) · Zbl 0835.76024 [24] Varadhan S., Zygouras N.: Behavior of the solution of a random semilinear heat equation. Comm. Pure Applied Math. 61(9), 1298–1329 (2008) · Zbl 1152.60077 · doi:10.1002/cpa.20256 [25] Varadhan, S.R.S.: Stochastic processes. Notes based on a course given at New York University during the year 1967/68. New York: Courant Institute of Mathematical Sciences New York University, 1968
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