Abstract
Gaussian measures μ β,ν are associated to some stochastic 2D models of turbulence. They are Gibbs measures constructed by means of an invariant quantity of the system depending on some parameter β (related to the 2D nature of the fluid) and the viscosity ν. We prove the existence and the uniqueness of the global flow for the stochastic viscous system; moreover the measure μ β,ν is invariant for this flow and is the unique invariant measure. Finally, we prove that the deterministic inviscid equation has a μ β,ν-stationary solution (for any ν>0).
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Albeverio, S., Cruzeiro, A.B.: Global flows with invariant (Gibbs) measures for Euler and Navier-Stokes two-dimensional fluids. Commun. Math. Phys. 129(3), 431–444 (1990)
Albeverio, S., Ferrario, B.: Uniqueness of solutions of the stochastic Navier-Stokes equation with invariant measure given by the enstrophy. Ann. Probab. 32(2), 1632–1649 (2004)
Barbato, D., Barsanti, M., Bessaih, H., Flandoli, F.: Some rigorous results on a stochastic GOY model. J. Stat. Phys. 125(3), 677–716 (2006)
Bessaih, H., Ferrario, B.: Invariant Gibbs measures of the energy for shell models of turbulence; the inviscid and viscous cases. Nonlinearity 25, 1075–1097 (2012)
Bessaih, H., Millet, A.: Large deviation principle and inviscid shell models. Electron. J. Probab. 14(89), 2551–2579 (2009)
Boffetta, G., Ecke, R.E.: Two-dimensional turbulence. Annu. Rev. Fluid Mech. 44, 427–451 (2012)
Bouchet, F., Corvellec, M.: Invariant measures for the 2D Euler and Vlasov equations. J. Stat. Mech. 08, P08021 (2010)
Chojnowska-Michalik, A., Goldys, B.: Symmetric Ornstein-Uhlenbeck semigroups and their generators. Probab. Theory Relat. Fields 124(4), 459–486 (2002)
Chueshov, I., Millet, A.: Stochastic 2D hydrodynamical type systems: well posedness and large deviations. Appl. Math. Optim. 61(3), 379–420 (2010)
Constantin, P., Levant, B., Titi, E.S.: Analytic study of shell models of turbulence. Physica D 219(2), 120–141 (2006)
Da Prato, G.: Kolmogorov Equations for Stochastic PDEs. Advances Courses in Mathematics CRM Barcelona. Birkhäuser, Basel (2004)
Da Prato, G.: An Introduction to Infinite-Dimensional Analysis. Springer, Berlin (2006)
Da Prato, G., Debussche, A.: Two-dimensional Navier-Stokes equations driven by a space-time white noise. J. Funct. Anal. 196(1), 180–210 (2002)
Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications, vol. 44. Cambridge University Press, Cambridge (1992)
Da Prato, G., Zabczyk, J.: Ergodicity for Infinite Dimensional Systems. LMS Lecture Notes, vol. 229. Cambridge University Press, Cambridge (1996)
Debussche, A.: The 2D-Navier-Stokes equations perturbed by a delta correlated noise. In: Probabilistic Methods in Fluids, pp. 115–129. World Scientific, River Edge (2003)
Eberle, A.: Uniqueness and Non-uniqueness of Semigroups Generated by Singular Diffusion Operators. Lecture Notes in Mathematics, vol. 1718. Springer, Berlin (1999)
Ferrario, B.: A note on a result of Liptser-Shiryaev. Stoc. Anal. Appl. (2012). doi:10.1080/07362994.2012.727139. Eprint arXiv:1005.0237v2
Ferrario, B.: Uniqueness and absolute continuity for semilinear SPDE’s. In: The Proceedings of Ascona 2011. Progress in Probability. Birkhäuser, Basel (2012, to appear)
Flandoli, F.: An introduction to 3D stochastic fluid dynamics. In: Da Prato, G., Röckner, M. (eds.) SPDE in Hydrodynamic: Recent Progress and Prospects, pp. 51–150. Lectures Given at the C.I.M.E. Summer School Held in Cetraro, August 29–September 3, 2005. Lecture Notes in Math., vol. 1942. Springer, Berlin; Fondazione C.I.M.E., Florence (2008)
Gledzer, E.B.: System of hydrodynamic type admitting two quadratic integrals of motion. Dokl. Akad. Nauk SSSR 209, 1046–1048 (1973). Engl. transl.: Sov. Phys. Dokl. 18, 216–217 (1973)
Kuksin, S.B.: The Eulerian limit for 2D statistical hydrodynamics. J. Stat. Phys. 115(1–2), 469–492 (2004)
Kuo, H.H.: Gaussian Measures in Banach Spaces. Lecture Notes in Mathematics, vol. 463. Springer, Berlin (1975)
L’vov, V.S., Podivilov, E., Pomyalov, A., Procaccia, I., Vandembroucq, D.: Improved shell model of turbulence. Phys. Rev. E 58, 1811–1822 (1998)
Ohkitani, K., Yamada, M.: Lyapunov spectrum of a chaotic model of three-dimensional turbulence. J. Phys. Soc. Jpn. 56, 4210–4213 (1987)
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, New York (1983)
Temam, R.: Navier-Stokes Equations. Theory and Numerical Analysis. Studies in Mathematics and Its Applications, vol. 2. North-Holland, Amsterdam (1984)
Acknowledgements
We are very grateful to an anonymous referee that helped in making better the exposition of this paper. The work of H. Bessaih was partially supported by the NSF grant No. DMS 0608494, and by GNAMPA-INDAM project. We would like to also acknowledge the hospitality of EPFL Lausanne where some parts of this paper have been refined while the authors were visiting the Bernoulli Center.
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Bessaih, H., Ferrario, B. Invariant Measures of Gaussian Type for 2D Turbulence. J Stat Phys 149, 259–283 (2012). https://doi.org/10.1007/s10955-012-0601-z
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DOI: https://doi.org/10.1007/s10955-012-0601-z