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Stochastic 3D Navier-Stokes equations in a thin domain and its $$\alpha$$-approximation. (English) Zbl 1143.76422
Summary: In the thin domain $$\mathcal O _\varepsilon = \mathbb T^2 \times (0,\varepsilon)$$, where $$\mathbb T^2$$ is a two-dimensional torus, we consider the 3D Navier-Stokes equations, perturbed by a white in time random force, and the Leray $$\alpha$$-approximation for this system. We study ergodic properties of these models and their connection with the corresponding 2D models in the limit $$\epsilon \rightarrow 0$$. In particular, under natural conditions concerning the noise we show that in some rigorous sense the 2D stationary measure $$\mu$$ comprises asymptotical in time statistical properties of solutions for the 3D Navier-Stokes equations in $$\mathcal O _\varepsilon$$, when $$\varepsilon \ll 1$$.

##### MSC:
 76D06 Statistical solutions of Navier-Stokes and related equations 35Q30 Navier-Stokes equations 76M35 Stochastic analysis applied to problems in fluid mechanics
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##### References:
 [1] Avrin, J.D., Large-eigenvalue global existence and regularity results for the navier – stokes equation, J. differential equations, 127, 365-390, (1996) · Zbl 0863.35075 [2] Bricmont, J.; Kupiainen, A.; Lefevere, R., Ergodicity of the 2D navier – stokes equations with random forcing, Comm. math. phys., 224, 65-81, (2001) · Zbl 0994.60066 [3] Chepyzhov, V.; Titi, E.; Vishik, M., On the convergence of solutions of the Leray-$$\alpha$$ model to the trajectory attractor of the 3D navier – stokes system, Discrete contin. dyn. syst., 17, 481-500, (2007) · Zbl 1146.35071 [4] Cheskidov, A.; Holm, D.; Olson, E.; Titi, E., On a Leray-$$\alpha$$ model of turbulence, Proc. R. soc. lond. ser.A, 461, 629-649, (2005) · Zbl 1145.76386 [5] Chueshov, I., On determining functionals for stochastic navier – stokes equations, Stoch. stoch. reports, 68, 45-64, (1999) · Zbl 0945.35093 [6] Chueshov, I.; Kuksin, S., Random kick-forced 3D navier – stokes equations in a thin domain, Arch. ration. mech. anal., 188, 117-153, (2008) · Zbl 1133.76015 [7] Chueshov, I.; Raugel, G.; Rekalo, A., Interface boundary value problem for the navier – stokes equations in thin two-layer domains, J. differential equations, 208, 449-493, (2005) · Zbl 1078.35084 [8] Constantin, P.; Foiaş, C., Navier-Stokes equations, (1988), University of Chicago Press Chicago · Zbl 0687.35071 [9] Cullen, M.J.P., A mathematical theory of large-scale atmosphere/Ocean flow, (2006), Imperial College Press London [10] Da Prato, G.; Zabszyk, J., Stochastic equations in infinite dimensions, (1992), Cambridge University Press Cambridge [11] E, W.; Mattingly, J.C.; Sinai, Ya.G., Gibbsian dynamics and ergodicity for the stochastically forced navier – stokes equation, Comm. math. phys., 224, 83-106, (2001) · Zbl 0994.60065 [12] Flandoli, F.; Maslowski, B., Ergodicity of the 2D navier – stokes equation under random perturbations, Comm. math. phys., 172, 119-141, (1995) · Zbl 0845.35080 [13] Hairer, M.; Mattingly, J., Ergodicity of the 2D navier – stokes equations with degenerate stochastic forcing, Ann. math., 164, 993-1032, (2006) · Zbl 1130.37038 [14] Iftimie, D., The 3D navier – stokes equations seen as a perturbation of the 2D navier – stokes equations, Bull. soc. math. France, 127, 473-517, (1999) · Zbl 0946.35059 [15] Iftimie, D.; Raugel, G., Some results on the navier – stokes equations in thin 3D domains, J. differential equations, 169, 281-331, (2001) · Zbl 0972.35085 [16] Ikeda, N.; Watanabe, S., Stochastic differential equations and diffusion processes, (1989), North-Holland Amsterdam · Zbl 0684.60040 [17] Komech, A.I.; Vishik, M.I., Strong solutions of 2D navier – stokes system and the corresponding Kolmogorov equations, Z. anal. anwendungen., 1, 3, 23-52, (1982), (in Russian) · Zbl 0512.60050 [18] Kuksin, S.B., Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions, (2006), European Mathematical Society Publishing House Zürich, see also mp_arc 06-178 [19] Kuksin, S.B.; Shirikyan, A., Stochastic dissipative PDE’s and Gibbs measures, Comm. math. phys., 213, 291-330, (2000) · Zbl 0974.60046 [20] Kuksin, S.B.; Shirikyan, A., Coupling approach to white-forced nonlinear PDE’s, J. math. pures appl., 81, 567-602, (2002) · Zbl 1021.37044 [21] Leray, J., Essai sur le mouvement d’un fluide visqueux emplissant l’espace, Acta math., 63, 193-248, (1934) · JFM 60.0726.05 [22] Moise, I.; Temam, R.; Ziane, M., Asymptotic analysis of the navier – stokes equations in thin domains, Topol. methods nonlinear anal., 10, 249-282, (1997) · Zbl 0957.35108 [23] Montgomery-Smith, S., Global regularity of the navier – stokes equation on thin three dimensional domains with periodic boundary conditions, Electronic J. diff. eqns., 11, 1-19, (1999) · Zbl 0923.35120 [24] Raugel, G.; Sell, G., Navier – stokes equations on thin domains. I: global attractors and global regularity of solutions, J. amer. math. soc., 6, 503-568, (1993) · Zbl 0787.34039 [25] Raugel, G.; Sell, G., Navier-Stokes equations on thin 3D domains. II. global regularity of spatially periodic solutions, (), 205-247 · Zbl 0804.35106 [26] Shirikyan, A., Ergodicity for a class of Markov processes and applications to randomly forced PDE’s, Russian J. math. phys., 12, 81-96, (2005) · Zbl 1132.60317 [27] Temam, R.; Ziane, M., Navier – stokes equations in three-dimensional thin domains with various boundary conditions, Adv. differential equations, 1, 499-546, (1996) · Zbl 0864.35083 [28] Vishik, M.I.; Komech, A.I.; Fursikov, A.V., Some mathematical problems of statistical hydromechanics, Russian math. surveys, 34, 5, 149-234, (1979) · Zbl 0503.76045 [29] Vishik, M.I.; Fursikov, A.V., Mathematical problems of statistical hydromechanics, (1988), Kluwer Academic Publishers Dordrecht · Zbl 0688.35077
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