On recent progress for the stochastic Navier Stokes equations
Journées équations aux dérivées partielles (2003), article no. 11, 52 p.

We give an overview of the ideas central to some recent developments in the ergodic theory of the stochastically forced Navier Stokes equations and other dissipative stochastic partial differential equations. Since our desire is to make the core ideas clear, we will mostly work with a specific example : the stochastically forced Navier Stokes equations. To further clarify ideas, we will also examine in detail a toy problem. A few general theorems are given. Spatial regularity, ergodicity, exponential mixing, coupling for a SPDE, and hypoellipticity are all discussed.

     author = {Mattingly, Jonathan},
     title = {On recent progress for the stochastic Navier Stokes equations},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {11},
     publisher = {Universit\'e de Nantes},
     year = {2003},
     doi = {10.5802/jedp.625},
     zbl = {02079446},
     mrnumber = {2050597},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jedp.625/}
Mattingly, Jonathan. On recent progress for the stochastic Navier Stokes equations. Journées équations aux dérivées partielles (2003), article  no. 11, 52 p. doi : 10.5802/jedp.625. http://www.numdam.org/articles/10.5802/jedp.625/

[Arn98] Arnold. Random dynamical systems. Springer-Verlag, Berlin,1998 | MR 1723992 | Zbl 0834.58026

[AS03] Andrei Acrachev And Andrey Sarychev. Navier-stokes equation controlled by degenerate forcing: Controllabillity in finite-dimentional projections. Preprint, 2003. | MR 2082964

[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. | MR 2001790 | Zbl 1054.60062

[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. | MR 1144097 | Zbl 0744.60063

[Bel95] Denis R. Bell. Degenerate stochastic differential equations and hypoellipticity. Longman, Harlow, 1995. | MR 1471702 | Zbl 0859.60051

[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. | MR 1788483 | Zbl 0972.60044

[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. | MR 1868991 | Zbl 0994.60066

[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. | MR 1930573 | Zbl 1033.76011

[BM03] Yuri Bakhtin And Jonathan C. Mattingly. Stationary solutions of stochastic differential equation with memory and stochastic partial differential equations. Preprint, 2003. | Zbl 1098.34063

[CDF97] Hans Crauel, Arnaud Debussche, And Franco Flandoli. Random attractors. J. Dynam. Differential Equations, 9(2):307-341, 1997. | MR 1451294 | Zbl 0884.58064

[Cer99] Sandra Cerrai. Ergodicity for stochastic reaction-diffusion systems with polynomial coefficients. Stochastics Stochastics Rep., 67(1-2):17-51, 1999. | MR 1717811 | Zbl 0935.60049

[CF88] Peter Constantin And Ciprian Foiaș. Navier-Stokes Equations. University of Chicago Press, Chicago, 1988. | MR 972259 | Zbl 0687.35071

[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. | MR 966192 | Zbl 0683.58002

[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. | MR 832433 | Zbl 0493.28007

[CK97] Pao-Liu Chow And Rafail Z. Khasminskii. Stationary solutions of nonlinear stochastic evolution equations. Stochastic Anal. Appl., 15(5):671-699, 1997. | MR 1478880 | Zbl 0899.60056

[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. | MR 1325465 | Zbl 0838.76016

[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. | Numdam | MR 960525 | Zbl 0647.60076

[DPZ92] Giuseppe Da Prato And Jerzy Zabczyk. Stochastic Equations in Infinite Dimensions. Cambridge, 1992. | MR 1207136 | Zbl 0761.60052

[DPZ96] Giuseppe Da Prato And Jerzy Zabczyk. Ergodicity for Infinite Dimensional Systems. Cambridge, 1996.

[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. | MR 1985790 | Zbl 1012.35001

[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. | MR 1331063 | Zbl 1023.76513

[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. | MR 488202 | Zbl 0355.60004

[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. | MR 1335230 | Zbl 0842.58056

[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. | MR 1838749 | Zbl 0983.60058

[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. | MR 1779561 | Zbl 0972.35196

[EL02] Weinan E And Di Liu. Gibbsian dynamics and invariant measures for stochastic dissipative PDEs. J. Statist. Phys., 108(5/6):1125-1156, 2002. | MR 1933448 | Zbl 1023.60057

[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. | MR 1846802 | Zbl 1024.76012

[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. | MR 1868992 | Zbl 0994.60065

[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. | MR 1771642 | Zbl 0967.76038

[Fer97] Benedetta Ferrario. Ergodic results for stochastic Navier-Stokes equation. Stochastics and Stochastics Reports, 60(3-4):271-288, 1997. | MR 1467721 | Zbl 0882.60059

[FG98] F. Flandoli And F. Gozzi. Kolmogorov equation associated to a stochastic Navier-Stokes equation. J. Funct. Anal., 160(1):312-336, 1998. | MR 1658680 | Zbl 0928.60044

[Fla94] Franco Flandoli. Dissipativity and invariant measures for stochastic Navier-Stokes equations. NoDEA, 1:403-426, 1994. | MR 1300150 | Zbl 0820.35108

[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. | MR 1346374 | Zbl 0845.35080

[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. | Numdam | MR 223716 | Zbl 0176.54103

[FST88] Ciprian Foias, George R. Sell, And Roger Temam. Inertial manifolds for nonlinear evolutionary equations. J. Differential Equations, 73(2):309-353s, 1988. | MR 943945 | Zbl 0643.58004

[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. | MR 1026858 | Zbl 0702.35203

[Hai02] M. Hairer. Exponential mixing properties of stochastic PDEs through asymptotic coupling. Probab. Theory Related Fields, 124(3):345-380, 2002. | MR 1939651 | Zbl 1032.60056

[Jur97] Velimir Jurdjevic. Geometric control theory, volume 52 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1997. | MR 1425878 | Zbl 0940.93005

[Kif86] Yuri Kifer. Ergodic theory of random transformations. Birkhäuser Boston Inc., Boston, MAs, 1986. | MR 884892 | Zbl 0604.28014

[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. | MR 1927233 | Zbl 1010.60066

[KS00] Sergei Kuksin And Armen Shirikyan. Stochastic dissipative PDEs and Gibbs measures. Comm. Math. Phys., 213(2):291-330, 2000. | MR 1785459 | Zbl 0974.60046

[KS02] Sergei Kuksin And Armen Shirikyan. Coupling approach to white-forced nonlinear PDEs. J. Math. Pures Appl. (9), 81(6):567-602, 2002. | MR 1912412 | Zbl 1021.37044

[Kuk03] Sergei Kuksin. Eulerian limit for 2d statistical hydrodynamics. Preprint, 2003. | MR 2070104

[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. | MR 780762 | Zbl 0546.60056

[LJ87] Y. Le Jan. Équilibre statistique pour les produits de difféomorphismes aléatoires indépendants. Ann. Inst. H. Poincaré Probab. Statist., 23(1):111-120, 1987. | Numdam | MR 877387 | Zbl 0614.60047

[LO97] C. David Levermore And Marcel Oliver. Analyticity of solutions for a generalized Euler equation. J. Differential Equations, 133(2):321-339, 1997. | MR 1427856 | Zbl 0876.35090

[Mat98] Jonathan C. Mattingly. The Stochastically forced Navier-Stokes equations: energy estimates and phase space contraction}. PhD thesis, Princeton University, 1998.

[Mat99] Jonathan C. Mattingly. Ergodicity of 2D Navier-Stokes equations with random forcing and large viscosity. Comm. Math. Phys., 206(2):273-288, 1999. | MR 1722141 | Zbl 0953.37023

[Mat02a] Jonathan C. Mattingly. Contractivity and ergodicity of the random map x|x-θ|. Theory of Probability and its Applications, 47(2):388-397, 2002. | MR 2003207 | Zbl 1032.60064

[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. | MR 1933449 | Zbl 1030.60049

[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. | MR 1937652 | Zbl 1054.76020

[MP03] Jonathan C. Mattingly And Étienne Pardoux. Malliavin calculus and the randomly forced Navier Stokes equation. Preprint, 2003.

[MR] R. Mikulevicius And B. L. Rozovskii. Stochastic navier-stokes equations for turbulent flows. Preprint. | MR 2050201

[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. | MR 1719695 | Zbl 0961.35112

[MS03] Jonathan C. Mattingly And Toufic M. Suidan. The small scales of the stochastic navier stokes equations under rough forcing. Preprint, 2003.

[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. | MR 1931266 | Zbl 1075.60072

[MT93] S. P. Meyn And R. L. Tweedie. Markov Chains and Stochastic Stability. Springer-Verlag, 1993. | MR 1287609 | Zbl 0925.60001

[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. | MR 1910827 | Zbl 1009.37049

[Nor86] James Norris. Simplified Malliavin calculus. In Séminaire de Probabilités, XX, 1984/85, pages 101-130. Springer, Berlin,s 1986. | Numdam | MR 942019 | Zbl 0609.60066

[Oks92] Bernt Oksendal. Stochastic Differential Equations. Springer-Verlag, 3nd edition, 1992. | MR 1217084 | Zbl 0747.60052

[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. | MR 1749867 | Zbl 0960.35081

[Rom02] Marco Romito. Ergodicity of the finite dimensional approximation of the 3d navier-stokes equations forced by a degenerate. Peprint, 2002.

[RY94] Daniel Revuz And Marc Yor. Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, second edition, 1994. | MR 1303781 | Zbl 0731.60002

[Sch97] Björn Schmalfuss. Qualitative properties for the stochastic Navier-Stokes equation. Nonlinear Anal., 28(9):1545-1563, 1997. | MR 1431206 | Zbl 0882.60058

[Shi02] Armin Shirikyan. A version of the law of large number and applications. In Probabilistic Methods in Fluids. World Scientific, 2002. | MR 2083377 | Zbl 1066.76020

[Sin94] Ya. G. Sinaĭ. Topics in ergodic theory, volume 44 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1994. | MR 1258087 | Zbl 0805.58005

[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. | MR 1318914 | Zbl 0833.35110

[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. | Zbl 0688.35077