Bernoulli

  • Bernoulli
  • Volume 21, Number 4 (2015), 2351-2392.

Large deviations for 2-D stochastic Navier–Stokes equations driven by multiplicative Lévy noises

Jianliang Zhai and Tusheng Zhang

Full-text: Open access

Abstract

In this paper, we establish a large deviation principle for two-dimensional stochastic Navier–Stokes equations driven by multiplicative Lévy noises. The weak convergence method introduced by Budhiraja, Dupuis and Maroulas [ Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011) 725–747] plays a key role.

Article information

Source
Bernoulli, Volume 21, Number 4 (2015), 2351-2392.

Dates
Received: December 2013
Revised: May 2014
First available in Project Euclid: 5 August 2015

Permanent link to this document
https://projecteuclid.org/euclid.bj/1438777597

Digital Object Identifier
doi:10.3150/14-BEJ647

Mathematical Reviews number (MathSciNet)
MR3378470

Zentralblatt MATH identifier
1344.60030

Keywords
Brownian motions large deviations Poisson random measures Skorohod representation stochastic Navier–Stokes equations tightness

Citation

Zhai, Jianliang; Zhang, Tusheng. Large deviations for 2-D stochastic Navier–Stokes equations driven by multiplicative Lévy noises. Bernoulli 21 (2015), no. 4, 2351--2392. doi:10.3150/14-BEJ647. https://projecteuclid.org/euclid.bj/1438777597


Export citation

References

  • [1] Aldous, D. (1978). Stopping times and tightness. Ann. Probab. 6 335–340.
  • [2] Bensoussan, A. and Temam, R. (1973). Équations stochastiques du type Navier–Stokes. J. Funct. Anal. 13 195–222.
  • [3] Bessaih, H. and Millet, A. (2009). Large deviation principle and inviscid shell models. Electron. J. Probab. 14 2551–2579.
  • [4] Boué, M. and Dupuis, P. (1998). A variational representation for certain functionals of Brownian motion. Ann. Probab. 26 1641–1659.
  • [5] Brzeźniak, Z., Liu, W. and Zhu, J. (2014). Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise. Nonlinear Anal. Real World Appl. 17 283–310.
  • [6] Budhiraja, A., Chen, J. and Dupuis, P. (2013). Large deviations for stochastic partial differential equations driven by a Poisson random measure. Stochastic Process. Appl. 123 523–560.
  • [7] Budhiraja, A. and Dupuis, P. (2000). A variational representation for positive functionals of infinite dimensional Brownian motion. Probab. Math. Statist. 20 39–61.
  • [8] Budhiraja, A., Dupuis, P. and Maroulas, V. (2008). Large deviations for infinite dimensional stochastic dynamical systems. Ann. Probab. 36 1390–1420.
  • [9] Budhiraja, A., Dupuis, P. and Maroulas, V. (2011). Variational representations for continuous time processes. Ann. Inst. Henri Poincaré Probab. Stat. 47 725–747.
  • [10] Cardon-Weber, C. (1999). Large deviations for a Burgers’-type SPDE. Stochastic Process. Appl. 84 53–70.
  • [11] Cerrai, S. and Röckner, M. (2004). Large deviations for stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term. Ann. Probab. 32 1100–1139.
  • [12] Chenal, F. and Millet, A. (1997). Uniform large deviations for parabolic SPDEs and applications. Stochastic Process. Appl. 72 161–186.
  • [13] Chow, P.L. (1992). Large deviation problem for some parabolic Itô equations. Comm. Pure Appl. Math. 45 97–120.
  • [14] Chueshov, I. and Millet, A. (2010). Stochastic 2D hydrodynamical type systems: Well posedness and large deviations. Appl. Math. Optim. 61 379–420.
  • [15] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44. Cambridge: Cambridge Univ. Press.
  • [16] Duan, J. and Millet, A. (2009). Large deviations for the Boussinesq equations under random influences. Stochastic Process. Appl. 119 2052–2081.
  • [17] Flandoli, F. (1994). Dissipativity and invariant measures for stochastic Navier–Stokes equations. NoDEA Nonlinear Differential Equations Appl. 1 403–423.
  • [18] Flandoli, F. and Gatarek, D. (1995). Martingale and stationary solutions for stochastic Navier–Stokes equations. Probab. Theory Related Fields 102 367–391.
  • [19] Hairer, M. and Mattingly, J.C. (2006). Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing. Ann. of Math. (2) 164 993–1032.
  • [20] Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland Mathematical Library 24. Amsterdam: North-Holland.
  • [21] Jakubowski, A. (1986). On the Skorokhod topology. Ann. Inst. Henri Poincaré Probab. Stat. 22 263–285.
  • [22] Lions, J.-L. (1969). Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Paris: Dunod.
  • [23] Liu, W. (2010). Large deviations for stochastic evolution equations with small multiplicative noise. Appl. Math. Optim. 61 27–56.
  • [24] Manna, U., Sritharan, S.S. and Sundar, P. (2009). Large deviations for the stochastic shell model of turbulence. NoDEA Nonlinear Differential Equations Appl. 16 493–521.
  • [25] Mikulevicius, R. and Rozovskii, B.L. (2005). Global $L_{2}$-solutions of stochastic Navier–Stokes equations. Ann. Probab. 33 137–176.
  • [26] Ren, J. and Zhang, X. (2005). Freidlin–Wentzell’s large deviations for homeomorphism flows of non-Lipschitz SDEs. Bull. Sci. Math. 129 643–655.
  • [27] Ren, J. and Zhang, X. (2005). Schilder theorem for the Brownian motion on the diffeomorphism group of the circle. J. Funct. Anal. 224 107–133.
  • [28] Röckner, M. and Zhang, T. (2007). Stochastic evolution equations of jump type: Existence, uniqueness and large deviation principles. Potential Anal. 26 255–279.
  • [29] Röckner, M., Zhang, T. and Zhang, X. (2010). Large deviations for stochastic tamed 3D Navier–Stokes equations. Appl. Math. Optim. 61 267–285.
  • [30] Sowers, R.B. (1992). Large deviations for a reaction–diffusion equation with non-Gaussian perturbations. Ann. Probab. 20 504–537.
  • [31] Sritharan, S.S. and Sundar, P. (2006). Large deviations for the two-dimensional Navier–Stokes equations with multiplicative noise. Stochastic Process. Appl. 116 1636–1659.
  • [32] Świȩch, A. and Zabczyk, J. (2011). Large deviations for stochastic PDE with Lévy noise. J. Funct. Anal. 260 674–723.
  • [33] Temam, R. (1979). Navier–Stokes Equations: Theory and Numerical Analysis. Revised ed. Studies in Mathematics and Its Applications 2. Amsterdam: North-Holland. With an appendix by F. Thomasset.
  • [34] Temam, R. (1983). Navier–Stokes Equations and Nonlinear Functional Analysis. CBMS-NSF Regional Conference Series in Applied Mathematics 41. Philadelphia, PA: SIAM.
  • [35] Višik, M.I. and Fursikov, A.V. (1988). Mathematical Problems of Statistical Hydromechanics. Mathematics and Its Applications 9. Dordrecht: Kluver. Translated from 1980 Russian original Matematicheskie Zadachi Statisticheskoi Gidromekhaniki. Moscow: Nauka.
  • [36] Wang, W. and Duan, J. (2009). Reductions and deviations for stochastic partial differential equations under fast dynamical boundary conditions. Stoch. Anal. Appl. 27 431–459.
  • [37] Xu, T. and Zhang, T. (2009). Large deviation principles for 2-D stochastic Navier–Stokes equations driven by Lévy processes. J. Funct. Anal. 257 1519–1545.
  • [38] Yang, X., Zhai, J. and Zhang, T. (2014). Large deviations for SPDEs of jump type. Available at arXiv:1211.0466.
  • [39] Zhang, T.S. (2000). On the small time asymptotics of diffusion processes on Hilbert spaces. Ann. Probab. 28 537–557.
  • [40] Zhang, X. (2010). Stochastic Volterra equations in Banach spaces and stochastic partial differential equation. J. Funct. Anal. 258 1361–1425.