In this note, we discuss the problem of large deviations for the stochastic 2D Navier–Stokes equations. We show that the occupation measures of the trajectories of the system satisfy a large deviations principle, provided that the noise acts on all Fourier modes. In the case when the noise is more degenerate and acts on all the determining modes, we obtain an LDP of local type. The proofs use the methods introduced in [13, 20] based on a Kifer-type sufficient condition for LDP and a multiplicative ergodic theorem.
@article{SLSEDP_2016-2017____A17_0,
author = {Nersesyan, Vahagn},
title = {Large deviations results for the stochastic {Navier{\textendash}Stokes} equations},
journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
note = {talk:17},
pages = {1--10},
publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2016-2017},
doi = {10.5802/slsedp.112},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/slsedp.112/}
}
TY - JOUR AU - Nersesyan, Vahagn TI - Large deviations results for the stochastic Navier–Stokes equations JO - Séminaire Laurent Schwartz — EDP et applications N1 - talk:17 PY - 2016-2017 SP - 1 EP - 10 PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique UR - http://archive.numdam.org/articles/10.5802/slsedp.112/ DO - 10.5802/slsedp.112 LA - en ID - SLSEDP_2016-2017____A17_0 ER -
%0 Journal Article %A Nersesyan, Vahagn %T Large deviations results for the stochastic Navier–Stokes equations %J Séminaire Laurent Schwartz — EDP et applications %Z talk:17 %D 2016-2017 %P 1-10 %I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique %U http://archive.numdam.org/articles/10.5802/slsedp.112/ %R 10.5802/slsedp.112 %G en %F SLSEDP_2016-2017____A17_0
Nersesyan, Vahagn. Large deviations results for the stochastic Navier–Stokes equations. Séminaire Laurent Schwartz — EDP et applications (2016-2017), Exposé no. 17, 10 p. doi : 10.5802/slsedp.112. http://archive.numdam.org/articles/10.5802/slsedp.112/
[1] A. A. Agrachev and A. V. Sarychev. Navier–Stokes equations: controllability by means of low modes forcing. J. Math. Fluid Mech., 7(1):108–152, 2005. | DOI | MR | Zbl
[2] J. Bricmont, A. Kupiainen, and R. Lefevere. Exponential mixing of the 2D stochastic Navier–Stokes dynamics. Comm. Math. Phys., 230(1):87–132, 2002. | DOI | Zbl
[3] A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications. Springer–Verlag, Berlin, 2000. | Zbl
[4] J.-D. Deuschel and D. W. Stroock. Large Deviations. Academic Press, Boston, 1989. | Zbl
[5] M. D. Donsker and S. R. S. Varadhan. Asymptotic evaluation of certain Markov process expectations for large time, I–II. Comm. Pure Appl. Math., 28:1–47, 279–301, 1975. | DOI | MR | Zbl
[6] W. E, J. C. Mattingly, and Ya. Sinai. Gibbsian dynamics and ergodicity for the stochastically forced Navier–Stokes equation. Comm. Math. Phys., 224(1):83–106, 2001. | DOI | MR | Zbl
[7] F. Flandoli and B. Maslowski. Ergodicity of the 2D Navier–Stokes equation under random perturbations. Comm. Math. Phys., 172(1):119–141, 1995. | DOI | MR | Zbl
[8] M. I. Freidlin and A. D. Wentzell. Random Perturbations of Dynamical Systems. Springer, New York–Berlin, 1984. | Zbl
[9] M. Gourcy. A large deviation principle for 2D stochastic Navier–Stokes equation. Stochastic Process. Appl., 117(7):904–927, 2007. | DOI | MR | Zbl
[10] M. Gourcy. Large deviation principle of occupation measure for a stochastic Burgers equation. Ann. Inst. H. Poincaré Probab. Statist., 43(4):375–408, 2007. | DOI | Numdam | MR | Zbl
[11] M. Hairer and J. C. Mattingly. Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing. Ann. of Math. (2), 164(3):993–1032, 2006. | DOI | MR | Zbl
[12] V. Jakšić, V. Nersesyan, C.-A. Pillet, and A. Shirikyan. Large deviations from a stationary measure for a class of dissipative PDEs with random kicks. Comm. Pure Appl. Math., 68(12):2108–2143, 2015. | DOI | MR | Zbl
[13] V. Jakšić, V. Nersesyan, C.-A. Pillet, and A. Shirikyan. Large deviations and mixing for dissipative PDE’s with unbounded random kicks. Accepted for publication in Nonlinearity, 2017. | DOI | MR | Zbl
[14] I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus. Springer-Verlag, New York, 1991. | Zbl
[15] Y. Kifer. Large deviations in dynamical systems and stochastic processes. Trans. Amer. Math. Soc., 321(2):505–524, 1990. | DOI | MR | Zbl
[16] S. Kuksin and A. Shirikyan. Stochastic dissipative PDEs and Gibbs measures. Comm. Math. Phys., 213(2):291–330, 2000. | DOI | MR | Zbl
[17] S. Kuksin and A. Shirikyan. Coupling approach to white-forced nonlinear PDEs. J. Math. Pures Appl. (9), 81(6):567–602, 2002. | DOI | MR | Zbl
[18] S. Kuksin and A. Shirikyan. Mathematics of Two-Dimensional Turbulence. Cambridge University Press, Cambridge, 2012. | DOI | Zbl
[19] J.-L. Lions. Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Dunod, 1969. | Zbl
[20] D. Martirosyan and V. Nersesyan. Local large deviations principle for occupation measures of the stochastic damped nonlinear wave equation. Accepted for publication in Annales de l’IHP: PS, 2017. | DOI | MR | Zbl
[21] V. Nersesyan. Large deviations for the Navier–Stokes equations driven by a white-in-time noise. Preprint, 2017. | DOI | MR
[22] C. Odasso. Exponential mixing for stochastic PDEs: the non-additive case. Probab. Theory Related Fields, 140(1-2):41–82, 2008. | DOI | MR | Zbl
[23] L. Wu. Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems. Stochastic Process., 91(2):205–238, 2001. | DOI | MR | Zbl
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