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Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise

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  • Sritharan, S.S.
  • Sundar, P.

Abstract

A Wentzell-Freidlin type large deviation principle is established for the two-dimensional Navier-Stokes equations perturbed by a multiplicative noise in both bounded and unbounded domains. The large deviation principle is equivalent to the Laplace principle in our function space setting. Hence, the weak convergence approach is employed to obtain the Laplace principle for solutions of stochastic Navier-Stokes equations. The existence and uniqueness of a strong solution to (a) stochastic Navier-Stokes equations with a small multiplicative noise, and (b) Navier-Stokes equations with an additional Lipschitz continuous drift term are proved for unbounded domains which may be of independent interest.

Suggested Citation

  • Sritharan, S.S. & Sundar, P., 2006. "Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise," Stochastic Processes and their Applications, Elsevier, vol. 116(11), pages 1636-1659, November.
    • Handle: RePEc:eee:spapps:v:116:y:2006:i:11:p:1636-1659
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    Cited by:

    1. Swie[combining cedilla]ch, Andrzej, 2009. "A PDE approach to large deviations in Hilbert spaces," Stochastic Processes and their Applications, Elsevier, vol. 119(4), pages 1081-1123, April.
    2. Liu, Wei & Röckner, Michael & Zhu, Xiang-Chan, 2013. "Large deviation principles for the stochastic quasi-geostrophic equations," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 3299-3327.
    3. Salins, M., 2021. "Systems of small-noise stochastic reaction–diffusion equations satisfy a large deviations principle that is uniform over all initial data," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 159-194.
    4. Budhiraja, Amarjit & Chen, Jiang & Dupuis, Paul, 2013. "Large deviations for stochastic partial differential equations driven by a Poisson random measure," Stochastic Processes and their Applications, Elsevier, vol. 123(2), pages 523-560.
    5. Maroulas, Vasileios & Xiong, Jie, 2013. "Large deviations for optimal filtering with fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 123(6), pages 2340-2352.
    6. Cai, Yujie & Huang, Jianhui & Maroulas, Vasileios, 2015. "Large deviations of mean-field stochastic differential equations with jumps," Statistics & Probability Letters, Elsevier, vol. 96(C), pages 1-9.
    7. Ankit Kumar & Manil T. Mohan, 2023. "Large Deviation Principle for Occupation Measures of Stochastic Generalized Burgers–Huxley Equation," Journal of Theoretical Probability, Springer, vol. 36(1), pages 661-709, March.
    8. Sundar, P. & Yin, Hong, 2009. "Existence and uniqueness of solutions to the backward 2D stochastic Navier-Stokes equations," Stochastic Processes and their Applications, Elsevier, vol. 119(4), pages 1216-1234, April.
    9. Ganguly, Arnab, 2018. "Large deviation principle for stochastic integrals and stochastic differential equations driven by infinite-dimensional semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 128(7), pages 2179-2227.
    10. Wei Wang & Jianliang Zhai & Tusheng Zhang, 2022. "Stochastic Two-Dimensional Navier–Stokes Equations on Time-Dependent Domains," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2916-2939, December.
    11. Duan, Jinqiao & Millet, Annie, 2009. "Large deviations for the Boussinesq equations under random influences," Stochastic Processes and their Applications, Elsevier, vol. 119(6), pages 2052-2081, June.
    12. Mohan, Manil T., 2020. "Well posedness, large deviations and ergodicity of the stochastic 2D Oldroyd model of order one," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 4513-4562.
    13. Cipriano, Fernanda & Torrecilla, Iván, 2015. "Inviscid limit for 2D stochastic Navier–Stokes equations," Stochastic Processes and their Applications, Elsevier, vol. 125(6), pages 2405-2426.
    14. Xiuwei Yin & Jiang-Lun Wu & Guangjun Shen, 2022. "Well-Posedness for Stochastic Fractional Navier–Stokes Equation in the Critical Fourier–Besov Space," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2940-2959, December.

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