An introduction to 3D stochastic fluid dynamics

Lecture Notes in Mathematics

Volumn 1942, Issue , 2008, Pages 51-150

  Flandoli, Franco ^a  

^a UNIVERSITY OF PISA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 41849117289     PISSN: 00758434     EISSN: None     Source Type: Book Series    
DOI: 10.1007/978-3-540-78493-7_2     Document Type: Article

References (65)

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.
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.
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.
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.
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.
6. A. Bensoussan, Stochastic Navier-Stokes equations, Acta Appl. Math. 38 (1995), 267-304.
7. A. Bensoussan, R. Temam, Equations stochastique du type Navier-Stokes, J. Funct. Anal. 13 (1973), 195-222.
8. H. Bessaih, Martingale solutions for stochastic Euler equations. Stochastic Anal. Appl. 17 (1999), no. 5, 713-725.
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.
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.
11. H.I. Lisei-Breckner, Approximation and optimal control of the stochastic Navier-Stokes equations, Thesis, Halle 1999.
12. Z. Brzezniak, M. Capinski, F. Flandoli, Stochastic Navier-Stokes equations with multiplicative noise, Stochastic Anal. Appl. 10 (1992), no. 5, 523-532
13. Z. Brzezniak, S. Peszat, Stochastic two dimensional Euler equations. Ann. Probab. 29 (2001), no. 4, 1796-1832.
14. M. Capinski, N. Cutland, Stochastic Navier-Stokes equations, Acta Appl. Math. 25 (1991), 59-85.
15. M. Capinski, N. Cutland, Nonstandard Methods in Stochastic Fluid Mechanics, World Scientific, Singapore 1995.
16. M. Capinski, N. Cutland, Existence and global stochastic flow and attractors for Navier-Stokes equations, Probab. Theory Related Fields 115, (1999), 121-151.
17. M. Capinski, D. Gaterek, Stochastic equations in Hubert space with application to Navier-Stokes equations in any dimension, J. Funct. Anal. 126 (1994), no. 1, 26-35.
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.
19. A. Chorin, Vorticity and Turbulence, Springer-Verlag, Berlin, 1993.
20. P.L. Chow, R.Z. Khasminskii, Stationary solutions of nonlinear stochastic evolution equations, Stochastic Anal. Appl. 15 (1997), no. 5, 671-699.
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.
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.
23. G. Da Prato, A. Debussche, Ergodicity for the 3D stochastic Navier-Stokes equations, J. Math. Pures Appl. 82 (2003), 877-947.
24. G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge 1992.
25. G. Da Prato, J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge 1996.
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.
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.
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.
29. B. Ferrario, Uniqueness result for the 2D Navier-Stokes equation with additive noise, Stoch. Stoch. Rep. 75 (2003), no. 6, 435-442.
30. F. Flandoli, Dissipativity and invariant measures for stochastic Navier-Stokes equations. Nonlinear Differential Equations Appl. 1 (1994), no. 4, 403-423.
31. F. Flandoli, Irreducibility of the 3D stochastic Navier-Stokes equation. J. Funct. Anal. 149 (1997), no. 1, 160-177.
32. F. Flandoli, Stochastic differential equations in fluid dynamics. Rend. Sem. Mat. Fis. Milano 66 (1996), 121-148 (1998).
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.
34. F. Flandoli, D. Gaterek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields 102 (3), (1995), 367-391.
35. F. Flandoli, M. Gubinelli, The Gibbs ensemble of a vortex filament, Probab. Theory Related Fields 122 (2002) 317-340.
36. F. Flandoli, M. Gubinelli, Statistics of a vortex filament model. Electronic. J. Probab. 10 (2005), no. 25, 865-900.
37. F. Flandoli, M. Gubinelli, M. Hairer, M. Romito, Remarks on scaling laws in turbulent fluids, to appear on Comm. Math. Phys..
38. F. Flandoli, M. Romito, Partial regularity for the stochastic Navier-Stokes equations, Trans. Amer. Math. Soc. 354, Nr. 6 (2002), 2207-2241.
39. F. Flandoli, M. Romito, Markov selections for the 3D stochastic Navier-Stokes equations, to appear on Probab. Theory Related Fields.
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.
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.
42. U. Frisch, Turbulence, Cambridge University Press, Cambridge 1995.
43. H. Fujita Yashima, Equations de Navier-Stokes stochastiques non homogenes et applications, Scuola Normale Superiore, Pisa, 1992.
44. G. Gallavotti, Foundations of Fluid Dynamics, Texts and Monographs in Physics, Springer-Verlag, Berlin, 2002. Translated from the Italian.
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.
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.
47. S.B. Kuksin, The Eulerian limit for 2D statistical hydrodynamics. J. Statist. Phys. 115 (2004), no. 1-2, 469-492.
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.
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.
50. A. Kupiainen, Statistical theories of turbulence, In Advances in Mathematical Sciences and Applications, (Gakkotosho, Tokyo, 2003).
51. J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969.
52. P.L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1, Clarendon Press, Oxford 1996.
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.
54. J.L. Menaldi, S.S. Sritharan, Stochastic 2-D Navier-Stokes equation, Appl. Math. Optimiz. 46 (2002), 31-53.
55. M. Metivier, Stochastic Partial Differential Equations in Infinite Dimensional Spaces, Quaderni Scuola Normale Superiore di Pisa, Pisa 1988.
56. R. Mikulevicius, B.L. Rozovsky, Stochastic Navier-Stokes equations for turbulent flows. SIAM J. Math. Anal. 35 (2004), no. 5, 1250-1310.
57. R. Mikulevicius, B.L. Rozovsky, Global L2-solutions of stochastic Navier-Stokes equations, The Annals of Prob. 33 (2005), no. 1, 137-176.
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
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.
60. B. Schmalfuß, Qualitative properties of the stochastic Navier Stokes equation, Nonlinear Analysis, TMA 28 (1997), 1545-1563.
61. S.S. Sritharan, Deterministic and stochastic control of Navier-Stokes equations with linear, monotone and hyperviscosity, Appl. Math. Optimiz. 41 (2000), 255-308.
62. D.W. Stroock, S.R.S. Varadhan, Multidimensional Diffusion Processes, Springer-Verlag, Berlin 1979.
63. R. Temam, Navier-Stokes equations and Nonlinear Functional Analysis, SIAM, Philadelphia, Pennsylvania, 1983.
64. M. Viot, Solution faibles d'equations aux derivees partielles stochastique non-lineaires, These, Universite Pierre et Marie Curie, Paris, 1976.
65. M.J. Vishik, A.V. Fursikov, Mathematical Problems in Statistical Hydromechanics, Kluwer, Boston, 1988.