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 Electronic Journal of Probability > Vol. 14 (2009) > Paper 89 open journal systems 


Large deviation principle and inviscid shell models

Hakima Bessaih, University of Wyoming
Annie Millet, University Paris 1


Abstract
LDP is proved for the inviscid shell model of turbulence. As the viscosity coefficient converges to 0 and the noise intensity is multiplied by its square root, we prove that some shell models of turbulence with a multiplicative stochastic perturbation driven by a H-valued Brownian motion satisfy a LDP in C([0,T], V) for the topology of uniform convergence on [0,T], but where V is endowed with a topology weaker than the natural one. The initial condition has to belong to V and the proof is based on the weak convergence of a family of stochastic control equations. The rate function is described in terms of the solution to the inviscid equation.


Full text: PDF

Pages: 2551--2579

Published on: November 26, 2009
Bibliography
  1. D. Barbato, M. Barsanti, H. Bessaih and F. Flandoli, Some rigorous results on a stochastic Goy model. Journal of Statistical Physics , 125 (2006) 677-716. Math. Review 2009c:76034
  2. V. Barbu, G. Da Prato, Existence and ergodicity for the two-dimensional stochastic magneto-hydrodynamics equations. Appl. Math. Optim. 56(2) (2007), 145-168. Math. Review 2008i:76157
  3. L. Biferale, Shell models of energy cascade in turbulence. Annu. Rev. Fluid Mech., Annual Reviews, Palo Alto, CA 35 (2003), 441-468. Math. Review 2004b:76074
  4. A. Budhiraja, P. Dupuis, A variational representation for positive functionals of infinite dimensional Brownian motion. Prob. and Math. Stat. 20 (2000), 39-61. Math. Review 2001j:60048
  5. A. Budhiraja, P. Dupuis and V. Maroulas, Large deviations for infinite dimensional stochastic dynamical systems. Ann. Prob. 36 (2008), 1390-1420. Math. Review 2009k:60145
  6. M. Capinskyi, D. Gatarek, Stochastic equations in Hilbert space with application to Navier-Stokes equations in any dimension. J. Funct. Anal. 126 (1994) 26-35. Math. Review 95j:60094
  7. S. Cerrai, M. Röckner, Large deviations for stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction terms. Ann. Probab. 32 (2004), 1100-1139. Math. Review 2005b:60063
  8. M. H. Chang, Large deviations for the Navier-Stokes equations with small stochastic perturbations. Appl. Math. Comput. 76 (1996), 65-93. Math. Review 96m:35246
  9. F. Chenal, A. Millet, Uniform large deviations for parabolic SPDEs and applications. Stochastic Process. Appl. 72 (1997), no. 2, 161--186. Math. Review 98m:60038
  10. I. Chueshov, A. Millet, Stochastic 2D hydrodynamical type systems: Well posedness and large deviationsi. Appl. Math. Optim. (to appear), DOI 10.1007/s00245-009-9091-z, Preprint arXiv:0807.1810.v2, July 2008. Math. Review number not available.
  11. P. Constantin, B. Levant and E. S. Titi, Analytic study of the shell model of turbulence. Physica D 219 (2006), 120-141. Math.Review 2007h:76055
  12. P. Constantin, B. Levant and E. S. Titi, Regularity of inviscid shell models of turbulence. Phys. Rev. E 3 (1) (2007), 016304, 10 pp. Math. Review 2008e:76085
  13. G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions , Cambridge University Press, 1992. Math. Review 95g:60073
  14. A. Dembo, O. Zeitouni, Large Deviations Techniques and Applications Springer-Verlag, New York, 2000. Math. Review 99d:60030
  15. A. Du, J. Duan and H. Gao, Small probability events for two-layer geophysical flows under uncertainty. preprint arXiv:0810.2818, October 2008. Math. Review number not available.
  16. J. Duan, A. Millet, Large deviations for the Boussinesq equations under random influences. Stochastic Process. Appl. 119-6 (2009), 2052-2081. Math. Review MR2519356
  17. P. Dupuis, R.S. Ellis, A Weak Convergence Approach to the Theory of Large Deviations , Wiley-Interscience, New York, 1997. Math. Review 99f:60057
  18. N. H. Katz, N. Pavlovi'c, Finite time blow-up for a dyadic model of the Euler equations. Trans. Amer. Math. Soc. 357 (2005), 695-708. Math. Review 2005h:35284
  19. H. Kunita, Stochastic flows and stochastic differential equations , Cambridge University Press, Cambridge-New York, 1990. Math. Review 91m:60107
  20. W. Liu, Large deviations for stochastic evolution equations with small multiplicative noise, Appl. Math. Optim. (to appear), doi:10.1007/s00245-009-9072-2, preprint arXiv:0801.1443-v4, September 2009 Math. Review number not available.
  21. V. S. L'vov, E. Podivilov, A. Pomyalov, I. Procaccia and D. Vandembroucq, Improved shell model of turbulence, Physical Review E , 58 (1998), 1811-1822. Math. Review MR1637121
  22. U. Manna, S.S. Sritharan and P. Sundar, Large deviations for the stochastic shell model of turbulence, preprint arXiv:0802.0585-v1, February 2008. Math. Review number not available.
  23. M. Mariani, Large deviations principles for stochastic scalar conservation laws. Probab. Theory Related Fields , to appear. preprint arXiv:0804.0997v3 Math. Review number not available.
  24. J.L. Menaldi, S.S. Sritharan, Stochastic 2-D Navier-Stokes equation. Appl. Math. Optim. 46 (2002), 31-53. Math. Review 2003k:35190
  25. K. Ohkitani, M. Yamada, Temporal intermittency in the energy cascade process and local Lyapunov analysis in fully developed model of turbulence. Prog. Theor. Phys. 89 (1989), 329-341. Math. Review 90j:76065
  26. R. Sowers, Large deviations for a reaction-diffusion system with non-Gaussian perturbations. Ann. Prob. 20 (1992), 504-537. Math. Review 93e:60125
  27. S. S. Sritharan, P. Sundar, Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise. Stoch. Proc. and Appl. 116 (2006), 1636-1659. Math. Review 2008d:60079
















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Electronic Journal of Probability. ISSN: 1083-6489