On recent progress for the stochastic Navier Stokes equations

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1. [Arn98] Arnold.Random dynamical systems. Springer-Verlag, Berlin,1998 Zbl0834.58026MR1723992
2. [AS03] Andrei Acrachev and Andrey Sarychev. Navier-stokes equation controlled by degenerate forcing: Controllabillity in finite-dimentional projections. Preprint, 2003. MR2082964
3. [Bak02] Yu. Yu. Bakhtin. Existence and uniqueness of stationary solution of %nonlinear stochastic differential equation with memory. Theory Probab. Appl, 47(4):764-769, 2002. Zbl1054.60062MR2001790
4. [Bax91] Peter H. Baxendale. Statistical equilibrium and two-point motion for a stochastic flow of diffeomorphisms. In Spatial stochastic processes, volume 19 of Progress in Probability, pages 189-218. Birkhäuser Boston, Boston, MA, 1991. Zbl0744.60063MR1144097
5. [Bel95] Denis R. Bell. Degenerate stochastic differential equations and hypoellipticity. Longman, Harlow, 1995. Zbl0859.60051MR1471702
6. [BKL00] J. Bricmont, A. Kupiainen, and R. Lefevere. Probabilistic estimates for the two-dimensional stochastic Navier-Stokes equations.J. Statist. Phys., 100(3-4):743-756, 2000. Zbl0972.60044MR1788483
7. [BKL01] J. Bricmont, A. Kupiainen, and R. Lefevere. Ergodicity of the 2D Navier-Stokes equations with random forcing. Comm. Math. Phys., 224(1):65-81, 2001. Dedicated to Joel L. Lebowitz. Zbl0994.60066MR1868991
8. [BKL02] J. Bricmont, A. Kupiainen, and R. Lefevere. Exponential mixing of the 2D stochastic Navier-Stokes dynamics. Comm. Math. Phys., 230(1):87-132, 2002. Zbl1033.76011MR1930573
9. [BM03] Yuri Bakhtin and Jonathan C. Mattingly. Stationary solutions of stochastic differential equation with memory and stochastic partial differential equations. Preprint, 2003. Zbl1098.34063
10. [CDF97] Hans Crauel, Arnaud Debussche, and Franco Flandoli. Random attractors. J. Dynam. Differential Equations, 9(2):307-341, 1997. Zbl0884.58064MR1451294
11. [Cer99] Sandra Cerrai. Ergodicity for stochastic reaction-diffusion systems with polynomial coefficients. Stochastics Stochastics Rep., 67(1-2):17-51, 1999. Zbl0935.60049MR1717811
12. [CF88] Peter Constantin and Ciprian Foiaș. Navier-Stokes Equations. University of Chicago Press, Chicago, 1988. Zbl0687.35071MR972259
13. [CFNT89] P. Constantin, C. Foiaș, B. Nicolaenko, and R. Temam. Integral manifolds and inertial manifolds for dissipative partial differential equations, volume 70 of Applied Mathematical Sciences. Springer-Verlag, New York-Berlin, 1989. Zbl0683.58002MR966192
14. [CFS82] I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinaĭ. Ergodic theory, volume 245 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, New York-Berlins, 1982. Zbl0493.28007MR832433
15. [CK97] Pao-Liu Chow and Rafail Z. Khasminskii. Stationary solutions of nonlinear stochastic evolution equations. Stochastic Anal. Appl., 15(5):671-699, 1997. Zbl0899.60056MR1478880
16. [DG95] Charles R. Doering and J. D. Gibbon. Applied analysis of the Navier-Stokes equations}. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 1995. Zbl0838.76016MR1325465
17. [DLJ88] R. W. R. Darling and Yves Le Jan. The statistical equilibrium of an isotropic stochastic flow with negative Lyapounov exponents is trivial. In Séminaire de Probabilités, XXII, volume 1321 of Lecture Notes in Math., pages 175-185. Springer, Berlins, 1988. Zbl0647.60076MR960525
18. [DPZ92] Giuseppe Da Prato and Jerzy Zabczyk. Stochastic Equations in Infinite Dimensions. Cambridge, 1992. Zbl0761.60052MR1207136
19. [DPZ96] Giuseppe Da Prato and Jerzy Zabczyk. Ergodicity for Infinite Dimensional Systems. Cambridge, 1996. Zbl0849.60052
20. [DPZ02] Giuseppe Da Prato and Jerzy Zabczyk. Second order partial differential equations in Hilbert spaces, volume 293 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2002. Zbl1012.35001MR1985790
21. [DT95] Charles R. Doering and Edriss S. Titi. Exponential decay rate of the power spectrum for solutions of the Navier-Stokes equations. Phys. Fluids, 7(6):1384-1390, 1995. Zbl1023.76513MR1331063
22. [Dud76] R. M. Dudley. Probabilities and metrics. Matematisk Institut, Aarhus Universitet, Aarhus, 1976. Convergence of laws on metric spaces, with a view to statistical testing, Lecture Notes Series, No. 45. Zbl0355.60004MR488202
23. [EFNT94] A. Eden, C. Foias, B Nicolaenko, and R. Temam. Exponential Attractors for dissipative Evolution equations. Research in Applied Mathematics. John Wiley and Sons and Masson, 1994. Zbl0842.58056MR1335230
24. [EH01] J.-P. Eckmann and M. Hairer. Uniqueness of the invariant measure for a stochastic PDE driven by degenerate noise. Comm. Math. Phys., 219(3):523-565, 2001. Zbl0983.60058MR1838749
25. [EKMS00] Weinan E, K. Khanin, A. Mazel, and Ya. Sinai. Invariant measures for Burgers equation with stochastic forcing. Ann. of Math. (2), 151(3):877-960, 2000. Zbl0972.35196MR1779561
26. [EL02] Weinan E and Di Liu. Gibbsian dynamics and invariant measures for stochastic dissipative PDEs. J. Statist. Phys., 108(5/6):1125-1156, 2002. Zbl1023.60057MR1933448
27. [EM01] Weinan E and Jonathan C. Mattingly. Ergodicity for the Navier-Stokes equation with degenerate random forcing: finite-dimensional approximation. Comm. Pure Appl. Math., 54(11):1386-1402, 2001. Zbl1024.76012MR1846802
28. [EMS01] Weinan E, J. C. Mattingly, and Ya G. Sinai. Gibbsian dynamics and ergodicity for the stochastic forced navier-stokes equation. Comm. Math. Phys., 224(1), 2001. Zbl0994.60065MR1868992
29. [EVE00] Weinan E and Eric Vanden Eijnden. Generalized flows, intrinsic stochasticity, and turbulent transport. Proc. Natl. Acad. Sci. USA, 97(15):8200-8205 (electronic), 2000. Zbl0967.76038MR1771642
30. [Fer97] Benedetta Ferrario. Ergodic results for stochastic Navier-Stokes equation. Stochastics and Stochastics Reports, 60(3-4):271-288, 1997. Zbl0882.60059MR1467721
31. [FG98] F. Flandoli and F. Gozzi. Kolmogorov equation associated to a stochastic Navier-Stokes equation. J. Funct. Anal., 160(1):312-336, 1998. Zbl0928.60044MR1658680
32. [Fla94] Franco Flandoli. Dissipativity and invariant measures for stochastic Navier-Stokes equations. NoDEA, 1:403-426, 1994. Zbl0820.35108MR1300150
33. [FM95] Franco Flandoli and B. Maslowski. Ergodicity of the 2-D Navier-Stokes equation under random perturbations. Comm. in Math. Phys., 171:119-141, 1995. Zbl0845.35080MR1346374
34. [FP67] C. Foiaș and G. Prodi. Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension 2. Rend. Sem. Mat. Univ. Padova, 39:1-34, 1967. Zbl0176.54103MR223716
35. [FST88] Ciprian Foias, George R. Sell, and Roger Temam. Inertial manifolds for nonlinear evolutionary equations. J. Differential Equations, 73(2):309-353s, 1988. Zbl0643.58004MR943945
36. [FT89] C. Foiaș and R. Temam. Gevrey class regularity for the solutions of the Navier-Stokes equations. J. Funct. Anal., 87(2):359-369, 1989. Zbl0702.35203MR1026858
37. [Hai02] M. Hairer. Exponential mixing properties of stochastic PDEs through asymptotic coupling. Probab. Theory Related Fields, 124(3):345-380, 2002. Zbl1032.60056MR1939651
38. [Jur97] Velimir Jurdjevic. Geometric control theory, volume 52 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1997. Zbl0940.93005MR1425878
39. [Kif86] Yuri Kifer. Ergodic theory of random transformations. Birkhäuser Boston Inc., Boston, MAs, 1986. Zbl0604.28014MR884892
40. [KPS02] Sergei Kuksin, Andrey Piatnitski, and Armen Shirikyan. A coupling approach to randomly forced nonlinear PDEs. II. Comm. Math. Phys., 230(1):81-85, 2002. Zbl1010.60066MR1927233
41. [KS00] Sergei Kuksin and Armen Shirikyan. Stochastic dissipative PDEs and Gibbs measures. Comm. Math. Phys., 213(2):291-330, 2000. Zbl0974.60046MR1785459
42. [KS02] Sergei Kuksin and Armen Shirikyan. Coupling approach to white-forced nonlinear PDEs. J. Math. Pures Appl. (9), 81(6):567-602, 2002. Zbl1021.37044MR1912412
43. [Kuk03] Sergei Kuksin. Eulerian limit for 2d statistical hydrodynamics. Preprint, 2003. MR2070104
44. [KS84] Shigeo Kusuoka and Daniel Stroock. Applications of the Malliavin calculus. I. In Stochastic analysis (Katata/Kyoto, 1982), spages 271-306. North-Holland, Amsterdam, 1984. Zbl0546.60056MR780762
45. [LJ87] Y. Le Jan. Équilibre statistique pour les produits de difféomorphismes aléatoires indépendants. Ann. Inst. H. Poincaré Probab. Statist., 23(1):111-120, 1987. Zbl0614.60047MR877387
46. [LO97] C. David Levermore and Marcel Oliver. Analyticity of solutions for a generalized Euler equation. J. Differential Equations, 133(2):321-339, 1997. Zbl0876.35090MR1427856
47. [Mat98] Jonathan C. Mattingly. The Stochastically forced Navier-Stokes equations: energy estimates and phase space contraction}. PhD thesis, Princeton University, 1998. 
48. [Mat99] Jonathan C. Mattingly. Ergodicity of $2$D Navier-Stokes equations with random forcing and large viscosity. Comm. Math. Phys., 206(2):273-288, 1999. Zbl0953.37023MR1722141
49. [Mat02a] Jonathan C. Mattingly. Contractivity and ergodicity of the random map $x\mapsto |x-\theta |$. Theory of Probability and its Applications, 47(2):388-397, 2002. Zbl1032.60064MR2003207
50. [Mat02b] Jonathan C. Mattingly. The dissipative scale of the stochastics Navier-Stokes equation: regularization and analyticity. J. Statist. Phys., 108(5-6):1157-1179, 2002. Zbl1030.60049MR1933449
51. [Mat02c] Jonathan C. Mattingly. Exponential convergence for the stochastically forced Navier-Stokes equations and other partially dissipative dynamics. Comm. Math. Phys., 230(3):421-462, 2002. Zbl1054.76020MR1937652
52. [MP03] Jonathan C. Mattingly and Étienne Pardoux. Malliavin calculus and the randomly forced Navier Stokes equation. Preprint, 2003. Zbl1113.60058
53. [MR] R. Mikulevicius and B. L. Rozovskii. Stochastic navier-stokes equations for turbulent flows. Preprint. Zbl1062.60061MR2050201
54. [MS99] J. C. Mattingly and Ya. G. Sinai. An elementary proof of the existence and uniqueness theorem for the Navier-Stokes equations. Commun. Contemp. Math., 1(4):497-516, 1999. Zbl0961.35112MR1719695
55. [MS03] Jonathan C. Mattingly and Toufic M. Suidan. The small scales of the stochastic navier stokes equations under rough forcing. Preprint, 2003. 
56. [MSH02] J. C. Mattingly, A.M. Stuart, and D. J. Higham. Ergodicity for SDEs and approximations: Locally lipschitz vector fields and degenerate noise. Stochastic Process. Appl. 101, no. 2, 185-232, 2002. Zbl1075.60072MR1931266
57. [MT93] S. P. Meyn and R. L. Tweedie. Markov Chains and Stochastic Stability. Springer-Verlag, 1993. Zbl0925.60001MR1287609
58. [MY02] Nader Masmoudi and Lai-Sang Young. Ergodic theory of infinite dimensional systems with applications to dissipative parabolic PDEs. Comm. Math. Phys., 227(3):461-481, 2002. Zbl1009.37049MR1910827
59. [Nor86] James Norris. Simplified Malliavin calculus. In Séminaire de Probabilités, XX, 1984/85, pages 101-130. Springer, Berlin,s 1986. Zbl0609.60066MR942019
60. [Oks92] Bernt Oksendal. Stochastic Differential Equations. Springer-Verlag, 3nd edition, 1992. Zbl0747.60052MR1217084
61. [OT00] Marcel Oliver and Edriss S. Titi. Remark on the rate of decay of higher order derivatives for solutions to the Navier-Stokes equations in ${𝐫}^n$. J. Funct. Anal., 172(1):1-18, 2000. Zbl0960.35081MR1749867
62. [Rom02] Marco Romito. Ergodicity of the finite dimensional approximation of the 3d navier-stokes equations forced by a degenerate. Peprint, 2002. 
63. [RY94] Daniel Revuz and Marc Yor. Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, second edition, 1994. Zbl0731.60002MR1303781
64. [Sch97] Björn Schmalfuss. Qualitative properties for the stochastic Navier-Stokes equation. Nonlinear Anal., 28(9):1545-1563, 1997. Zbl0882.60058MR1431206
65. [Shi02] Armin Shirikyan. A version of the law of large number and applications. In Probabilistic Methods in Fluids. World Scientific, 2002. Zbl1066.76020MR2083377
66. [Sin94] Ya. G. Sinaĭ. Topics in ergodic theory, volume 44 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1994. Zbl0805.58005MR1258087
67. [Tem95] Roger Temam. Navier-Stokes equations and nonlinear functional analysis, volume 66 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, second edition, 1995. Zbl0833.35110MR1318914
68. [VF88] M.J. Vishik and A.V. Fursikov. Mathematical Problems of Statistical Hydrodynamics. Kluwer Academic Publishers, 1988. Updated version of Russian original of same name. Zbl0688.35077

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