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References

  1. S. Albeverio, A.B. Cruzeiro, Global flow and invariant (Gibbs) measure for Euler and Navier–Stokes two dimensional fluids, Comm. Math. Phys. 129 (1990), 431–444.

    Article  MATH  MathSciNet  Google Scholar 

  2. S. Albeverio, B. Ferrario, Uniqueness results for the generators of the two-dimensional Euler and Navier–Stokes flows. The case of Gaussian invariant measures. J. Funct. Anal. 193 (2002), no. 1, 77–93.

    Article  MATH  MathSciNet  Google Scholar 

  3. S. Albeverio, B. Ferrario, Uniqueness of solutions of the stochastic Navier–Stokes equation with invariant measure given by the enstrophy. Ann. Probab. 32 (2004), no. 2, 1632–1649.

    Article  MATH  MathSciNet  Google Scholar 

  4. D. Barbato, M. Barsanti, H. Bessaih, F. Flandoli, Some rigorous results on a stochastic GOY model. J. Stat. Phys. 125 (2006), no. 3, 677–716.

    Article  MathSciNet  Google Scholar 

  5. V. Barbu, G. Da Prato, A. Debussche, Essential m-dissipativity of Kolmogorov operators corresponding to periodic 2D-Navier Stokes equations. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 15 (2004), no. 1, 29–38.

    Google Scholar 

  6. A. Bensoussan, Stochastic Navier–Stokes equations, Acta Appl. Math. 38 (1995), 267-304.

    Article  MATH  MathSciNet  Google Scholar 

  7. A. Bensoussan, R. Temam, Equations stochastique du type Navier–Stokes, J. Funct. Anal. 13 (1973), 195–222.

    Article  MATH  MathSciNet  Google Scholar 

  8. H. Bessaih, Martingale solutions for stochastic Euler equations. Stochastic Anal. Appl. 17 (1999), no. 5, 713–725.

    Article  MATH  MathSciNet  Google Scholar 

  9. H. Bessaih, F. Flandoli, Limit behaviour of a dense collection of vortex filaments, Math. Models Methods Appl. Sci. 14 (2004), no. 2, 189–215.

    Article  MATH  MathSciNet  Google Scholar 

  10. V.I. Bogachev, M. Röckner, Elliptic equations for measures on infinite-dimensional spaces and applications. Probab. Theory Related Fields 120 (2001), no. 4, 445–496.

    Article  MATH  MathSciNet  Google Scholar 

  11. H.I. Lisei-Breckner, Approximation and optimal control of the stochastic Navier–Stokes equations, Thesis, Halle 1999.

    Google Scholar 

  12. Z. Brzezniak, M. Capinski, F. Flandoli, Stochastic Navier–Stokes equations with multiplicative noise, Stochastic Anal. Appl. 10 (1992), no. 5, 523–532

    Article  MATH  MathSciNet  Google Scholar 

  13. Z. Brzezniak, S. Peszat, Stochastic two dimensional Euler equations. Ann. Probab. 29 (2001), no. 4, 1796–1832.

    Article  MATH  MathSciNet  Google Scholar 

  14. M. Capinski, N. Cutland, Stochastic Navier–Stokes equations, Acta Appl. Math. 25 (1991), 59–85.

    Article  MATH  MathSciNet  Google Scholar 

  15. M. Capinski, N. Cutland, Nonstandard Methods in Stochastic Fluid Mechanics, World Scientific, Singapore 1995.

    Google Scholar 

  16. M. Capinski, N. Cutland, Existence and global stochastic flow and attractors for Navier–Stokes equations, Probab. Theory Related Fields 115, (1999), 121–151.

    Article  MATH  MathSciNet  Google Scholar 

  17. M. Capinski, D. Gaterek, Stochastic equations in Hilbert space with application to Navier–Stokes equations in any dimension, J. Funct. Anal. 126 (1994), no. 1, 26–35.

    Article  MATH  MathSciNet  Google Scholar 

  18. M. Capinski, S. Peszat, On the existence of a solution to stochastic Navier–Stokes equations. Nonlinear Anal. 44 (2001), no. 2, Ser. A: Theory Methods, 141–177.

    Article  MATH  MathSciNet  Google Scholar 

  19. A. Chorin, Vorticity and Turbulence, Springer-Verlag, Berlin, 1993.

    Google Scholar 

  20. P.L. Chow, R.Z. Khasminskii, Stationary solutions of nonlinear stochastic evolution equations, Stochastic Anal. Appl. 15 (1997), no. 5, 671–699.

    Article  MATH  MathSciNet  Google Scholar 

  21. G. Da Prato, A. Debussche, Two-dimensional Navier–Stokes equations driven by a space-time white noise. J. Funct. Anal. 196 (2002), no. 1, 180–210.

    Article  MATH  MathSciNet  Google Scholar 

  22. G. Da Prato, A. Debussche, Dynamic programming for the stochastic Navier–Stokes equations, M2AN Math. Model Numer. Anal. 34 (2000), n.2, 459–475.

    Google Scholar 

  23. G. Da Prato, A. Debussche, Ergodicity for the 3D stochastic Navier–Stokes equations, J. Math. Pures Appl. 82 (2003), 877–947.

    MATH  MathSciNet  Google Scholar 

  24. G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge 1992.

    MATH  Google Scholar 

  25. G. Da Prato, J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge 1996.

    MATH  Google Scholar 

  26. Weinan E, J.C. Mattingly, Ya. Sinai, Gibbsian dynamics and ergodicity for the stochastically forced Navier–Stokes equation, Comm. Math. Phys. 224 (2001), no. 1, 83–106.

    Google Scholar 

  27. Weinan E, J.C. Mattingly, Ergodicity for the Navier–Stokes equation with degenerate random forcing: finite-dimensional approximation, Comm. Pure Appl. Math., vol. 54 no. 11 (2001), pp. 1386–1402.

    Article  MATH  MathSciNet  Google Scholar 

  28. B. Ferrario, Stochastic Navier–Stokes equations: analysis of the noise to have a unique invariant measure. Ann. Mat. Pura Appl. (4) 177 (1999), 331–347.

    Article  MathSciNet  Google Scholar 

  29. B. Ferrario, Uniqueness result for the 2D Navier–Stokes equation with additive noise, Stoch. Stoch. Rep. 75 (2003), no. 6, 435–442.

    Article  MATH  MathSciNet  Google Scholar 

  30. F. Flandoli, Dissipativity and invariant measures for stochastic Navier–Stokes equations. Nonlinear Differential Equations Appl. 1 (1994), no. 4, 403–423.

    Article  MATH  MathSciNet  Google Scholar 

  31. F. Flandoli, Irreducibility of the 3D stochastic Navier–Stokes equation. J. Funct. Anal. 149 (1997), no. 1, 160–177.

    Article  MATH  MathSciNet  Google Scholar 

  32. F. Flandoli, Stochastic differential equations in fluid dynamics. Rend. Sem. Mat. Fis. Milano 66 (1996), 121–148 (1998).

    MATH  MathSciNet  Google Scholar 

  33. F. Flandoli, On the method of Da Prato and Debussche for the 3D stochastic Navier Stokes equations. J. Evol. Equ. 6 (2006), no. 2, 269–286.

    Article  MATH  MathSciNet  Google Scholar 

  34. F. Flandoli, D. Gaterek, Martingale and stationary solutions for stochastic Navier–Stokes equations, Probab. Theory Related Fields 102 (3), (1995), 367–391.

    Article  MATH  MathSciNet  Google Scholar 

  35. F. Flandoli, M. Gubinelli, The Gibbs ensemble of a vortex filament, Probab. Theory Related Fields 122 (2002) 317–340.

    Article  MATH  MathSciNet  Google Scholar 

  36. F. Flandoli, M. Gubinelli, Statistics of a vortex filament model. Electronic. J. Probab. 10 (2005), no. 25, 865–900.

    MathSciNet  Google Scholar 

  37. F. Flandoli, M. Gubinelli, M. Hairer, M. Romito, Remarks on scaling laws in turbulent fluids, to appear on Comm. Math. Phys..

    Google Scholar 

  38. F. Flandoli, M. Romito, Partial regularity for the stochastic Navier–Stokes equations, Trans. Amer. Math. Soc. 354, Nr. 6 (2002), 2207–2241.

    Article  MATH  MathSciNet  Google Scholar 

  39. F. Flandoli, M. Romito, Markov selections for the 3D stochastic Navier–Stokes equations, to appear on Probab. Theory Related Fields.

    Google Scholar 

  40. F. Flandoli, B. Schmalfuß, Weak solutions and attractors for three-dimensional Navier–Stokes equations with nonregular force. J. Dynam. Differential Equations 11 (1999), no. 2, 355–398.

    Article  MATH  MathSciNet  Google Scholar 

  41. C. Foias, C. Guillopé, R. Temam, New a priori estimates for Navier–Stokes equations in dimension 3, Comm. Partial Diff. Eq. 6 (1981), 329–259.

    Article  MATH  Google Scholar 

  42. U. Frisch, Turbulence, Cambridge University Press, Cambridge 1995.

    MATH  Google Scholar 

  43. H. Fujita Yashima, Equations de Navier–Stokes stochastiques non homogenes et applications, Scuola Normale Superiore, Pisa, 1992.

    MATH  Google Scholar 

  44. G. Gallavotti, Foundations of Fluid Dynamics, Texts and Monographs in Physics, Springer-Verlag, Berlin, 2002. Translated from the Italian.

    Google Scholar 

  45. F. Gozzi, S.S. Sritharan, A. Swiech, Bellman equation associated to the optimal control of stochastic Navier–Stokes equations, Comm. Pure Appl. Math. LVIII (2005), 671–700.

    Article  MathSciNet  Google Scholar 

  46. A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, reprinted in Proc. Roy. Soc. London, Ser. A, 434 (1991), 9–13.

    Google Scholar 

  47. S.B. Kuksin, The Eulerian limit for 2D statistical hydrodynamics. J. Statist. Phys. 115 (2004), no. 1–2, 469–492.

    Article  MATH  MathSciNet  Google Scholar 

  48. S.B. Kuksin, O. Penrose, A family of balance relations for the two-dimensional Navier–Stokes equations with random forcing. J. Stat. Phys. 118 (2005), no. 3–4, 437–449.

    Article  MATH  MathSciNet  Google Scholar 

  49. S.B. Kuksin, A. Shirikyan, Ergodicity for the randomly forced 2D Navier–Stokes equations. Math. Phys. Anal. Geom. 4 (2001), no. 2, 147–195.

    Article  MATH  MathSciNet  Google Scholar 

  50. A. Kupiainen, Statistical theories of turbulence, In Advances in Mathematical Sciences and Applications, (Gakkotosho, Tokyo, 2003).

    Google Scholar 

  51. J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969.

    MATH  Google Scholar 

  52. P.L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1, Clarendon Press, Oxford 1996.

    MATH  Google Scholar 

  53. J.C. Mattingly and M. Hairer, Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing. Ann. of Math. (2) 164 (2006), no. 3, 993–1032.

    Google Scholar 

  54. J.L. Menaldi, S.S. Sritharan, Stochastic 2-D Navier–Stokes equation, Appl. Math. Optimiz. 46 (2002), 31–53.

    Article  MATH  MathSciNet  Google Scholar 

  55. M. Metivier, Stochastic Partial Differential Equations in Infinite Dimensional Spaces, Quaderni Scuola Normale Superiore di Pisa, Pisa 1988.

    MATH  Google Scholar 

  56. R. Mikulevicius, B.L. Rozovsky, Stochastic Navier–Stokes equations for turbulent flows. SIAM J. Math. Anal. 35 (2004), no. 5, 1250–1310.

    Article  MATH  MathSciNet  Google Scholar 

  57. R. Mikulevicius, B.L. Rozovsky, Global L2-solutions of stochastic Navier–Stokes equations, The Annals of Prob. 33 (2005), no. 1, 137–176.

    Article  MATH  Google Scholar 

  58. M. Röckner, Z. Sobol, Kolmogorov equations in infinite dimensions: well-posedness and regularity of solutions, with applications to stochastic generalized Burgers equations, Ann. Probab. 34 (2006), no. 2, 663–727

    Article  MATH  MathSciNet  Google Scholar 

  59. M. Romito, Ergodicity of the finite dimensional approximations of the 3D Navier–Stokes equations forced by a degenerate noise, J. Statist. Phys. 114 (1–2), (2004), 155–177.

    Article  MATH  MathSciNet  Google Scholar 

  60. B. Schmalfuß, Qualitative properties of the stochastic Navier Stokes equation, Nonlinear Analysis, TMA 28 (1997), 1545–1563.

    Article  MATH  Google Scholar 

  61. S.S. Sritharan, Deterministic and stochastic control of Navier–Stokes equations with linear, monotone and hyperviscosity, Appl. Math. Optimiz. 41 (2000), 255–308.

    Article  MATH  MathSciNet  Google Scholar 

  62. D.W. Stroock, S.R.S. Varadhan, Multidimensional Diffusion Processes, Springer-Verlag, Berlin 1979.

    MATH  Google Scholar 

  63. R. Temam, Navier–Stokes equations and Nonlinear Functional Analysis, SIAM, Philadelphia, Pennsylvania, 1983.

    Google Scholar 

  64. M. Viot, Solution faibles d’equations aux derivees partielles stochastique nonlineaires, These, Universite Pierre et Marie Curie, Paris, 1976.

    Google Scholar 

  65. M.J. Vishik, A.V. Fursikov, Mathematical Problems in Statistical Hydromechanics, Kluwer, Boston, 1988.

    Google Scholar 

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Flandoli, F. (2008). An Introduction to 3D Stochastic Fluid Dynamics. In: Da Prato, G., Rückner, M. (eds) SPDE in Hydrodynamic: Recent Progress and Prospects. Lecture Notes in Mathematics, vol 1942. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78493-7_2

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