Stochastic Navier-Stokes equations for turbulent flows

SIAM Journal on Mathematical Analysis

Volumn 35, Issue 5, 2004, Pages 1250-1310

  Mikulevicius, R ^a   Rozovskii, B L ^a  

^a UNIVERSITY OF SOUTHERN CALIFORNIA   (United States)

Author keywords

Kraichnan's turbulence; Moments; Stochastic Navier Stokes; Turbulence; Wiener chaos

Indexed keywords

EID: 1842643758     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036141002409167     Document Type: Article

References (52)

1. V. ARNOLD, Sur la geometrie differentielle des groupes de Lie de dimension infinie et ses applications a l'hydrodynamic des fluides parfaits, Ann. Inst. Grenoble, 16 (1966), pp. 319-361.
2. P. BAXENDALE AND T. E. HARRIS, Isotropic stochastic flows, Ann. Probab., 14 (1986), pp. 1155-1179.
3. A. BENSOUSSAN AND R. TEMAM, Equatios stochastique du type Navier-Stokes, J. Funct. Anal., 13 (1973), pp. 195-222.
4. Z. BRZEŹNIAK, M. CAPIŃSKI, AND F. FLANDOLI, Stochastic partial differential equations and turbulence, Math. Models Methods Appl. Sci., 1 (1991), pp. 41-59.
5. Z. BRZEZNIAK AND S. PESZAT, Strong local and global solutions to stochastic Navier-Stokes equations, in Infinite Dimensional Stochastic Analysis, Proceedings of the Coloquium of the Royal Netherlands Academy of Sciences, Amsterdam, 1999, North-Holland, Amsterdam, 2000, pp. 85-98.
6. R. H. CAMERON AND W. T. MARTIN, The orthogonal development of non-linear functional in a series of Fourier-Hermite functionals, Ann. of Math. (2), 48 (1947), pp. 385-392.
7. M. CAPIŃSKI AND N. J. CUTLAND, Stochastic Navier-Stokes equations, Acta Appl. Math., 25 (1991), pp. 59-85.
8. A. J. CHORIN AND J. MARSDEN, A mathematical introduction to fluid mechanics, Springer-Verlag, New York, 1990.
9. A. J. CHORIN, Lectures on Turbulence Theory, Berkeley Math. Lect. Notes 2, Berkeley, CA, 1993.
10. S. C. CROW AND G. H. CANAVAN, Relationship between a Wiener-Hermite expansion and an energy cascade, J. Fluid Mech., 41 (1970), pp. 387-403.
11. G. DAPRATO AND J. ZABCZYK, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, UK, 1992.
12. H. B. DA VEIGA, Existence and asymptotic behavior for the strong solutions of the Navier-Stokes equations in the whole space, Indiana Univ. Math. J., 36 (1987), pp. 149-166.
13. W. E. AND Y. SINAI, Recent results on mathematical & statistical hydrodynamics, Russian Math. Surveys, 55 (2000), pp. 25-57.
14. D. G. EBIN AND J. MARSDEN, Groups of diffeomorphisms and the notion of an incompressible fluid, Ann. of Math. (2), 92 (1970), pp. 102-163.
15. F. FLANDOLI AND D. GATAREK, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), pp. 367-391.
16. K. GAWEDZKI AND A. KUPIAINEN, Universality in Turbulence: An Exactly Solvable Model, in Low-Dimensional Models in Statistical Physics and Quantum Field Theory, H. Grosse and L. Pittner, eds., Springer-Verlag, Berlin, 1996, pp. 71-105.
17. K. GAWEDZKI AND M. VERGASSOLA, Phase transition in the passive scalar advection, Phys. D, 138 (2000), pp. 63-90.
18. r of the Navier-Stokes initial value problem, Arch. Ration. Mech. Anal., 89 (1985), pp. 251-265.
19. T. HIDA, H. H. KUO, J. POTTHOFF, AND L. STREIT, White Noise, Kluwer Academic, Dordrecht, The Netherlands, 1993.
20. H. HOLDEN, B. OKSENDAL, J. UBOE, AND T. ZHANG, Stochastic Partial Differential Equations. A Modeling, White Noise Functional Approach, Birkhäuser Boston, Boston, 1996.
21. A. INOUE AND T. FUNAKI, On a new derivation of the Navier-Stokes equation, Comm. Math. Phys., 6 (1979), pp. 83-90.
22. J. JACOD, Calcul stochastic et problemes de martingales, Lecture Notes in Math. 714, Springer-Verlag, New York, 1979.
23. M. JARDAK, C.-H. SU, AND G. E. KARNIADAKIS, Spectral polynomial solutions of the stochastic advection equation, J. Sci. Comput., 17 (2002), pp. 319-338.
24. 2), Rev. Mat. Iberoamericana, 2 (1986), pp. 73-88.
25. 2), Duke Math. J., 55 (1987), pp. 487-489.
26. R. H. KRAICHNAN, Small-scale structure of a scalar field convected by turbulence, Phys. Fluids, 11 (1968), pp. 945-963.
27. p-theory of stochastic partial differential equations in the whole space, SIAM J. Math. Anal., 27 (1996), pp. 313-340.
28. N. V. KRYLOV, An analytic approach to SPDEs, in Stochastic Partial Differential Equations: Six Perspectives, Math. Surveys Monogr. 64, R. Carmona and B. Rozovskii, eds., AMS, Providence, RI, 1999, pp. 185-242.
29. H. KUNITA, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, New York, 1990.
30. L. D. LANDAU AND E. M. LIFSHITZ, Fluid Mechanics, Addison-Wesley, Reading, MA, 1959.
31. Y. LE JAN AND O. RAIMOND, Integration of Brownian Vector Fields, Ann. Probab., 30 (2002) pp. 826-873.
32. O. P. LE MAÎTRE, O. M. KINO, H. N. NAJIM, AND R. G. GHANEM, A stochastic projection method for fluid flow, J. Comput. Phys., 173 (2001), pp. 481-511.
33. S. LOTOTSKY, R. MIKULEVICIUS, AND B. L. ROZOVSKII, Nonlinear filtering revisited: A spectral approach, SIAM J. Control Optim., 35 (1997), pp. 435-461.
34. P. MAJOR, Multiple Wiener-Ito Integrals, 2nd ed., Lecture Notes in Math. 849, Springer-Verlag, Berlin, 1981.
35. J. MATTINGLY, The Stochastic Navier-Stokes Equation. Energy Estimates and Phase Space Construction, Thesis, Princeton University, Princeton, NJ, 1998.
36. P.-A. MEYER, Quantum Probability for Probabilists, 2nd ed., Lecture Notes in Math. 1538, Springer-Verlag, Berlin, 1995.
37. R. MIKULEVICIUS, On the Cauchy problem for stochastic Stokes equations, SIAM J. Math. Anal., 34 (2002), pp. 121-141.
38. p-theory for systems of SPDEs, Electron. J. Probab., 6 (2001), pp. 1-35.
39. R. MIKULEVICIUS AND B. L. ROZOVSKII, Stochastic Navier-Stokes equations. Propagation of chaos and statistical moments, in Optimal Control and Partial Differential Equations, J. L. Menaldi, E. Rofmann, and A. Sulem, eds., IOS Press, Amsterdam, 2001, pp. 258-267.
40. R. MIKULEVICIUS AND B. L. ROZOVSKII, On equations of stochastic fluid mechanics, in Stochastics in Finite and Infinite Dimensions, T. Hida, R. Karandikar, H. Kunita, B. Rajput, S. Watanabe, and J. Xiong, eds., Birkhäuser Boston, Boston, MA, 2001, pp. 285-302.
41. R. MIKULEVICIUS AND B. L. ROZOVSKII, Martingale problems for stochastic PDE's, in Stochastic Partial Differential Equations: Six Perspectives, Math. Surveys Monogr. 64, R. Carmona and B. L. Rozovskii, eds., AMS, Providence, RI, 1998, pp. 243-325.
42. R. MIKULEVICIUS AND B. L. ROZOVSKII, Linear parabolic stochastic PDE's and Wiener chaos, SIAM J. Math. Anal., 29 (1998), pp. 452-480.
43. 2-solutions of stochastic Navier-Stokes equations, Ann. Probab., to appear.
44. d, Electron. J. Probab., 5 (2000), pp. 1-20.
45. D. NUALART AND B. L. ROZOVSKII, Weighted Wiener chaos and linear SPDE's driven by a space-time white noise, J. Funct. Anal., 149 (1997), pp. 200-225.
46. S. A. ORSZAG AND L. R. BISSONNETTE, Dynamical properties of truncated Wiener-Hermite expansions, Phys. Fluids, 10 (1967), pp. 2603-2613.
47. B. L. ROZOVSKII, Stochastic Evolution Systems, Kluwer Academic, Dordrecht, Boston, 1990.
48. A. SIEGEL, T. IMAMURA, AND W. C. MEECHAM, Wiener-Hermite expansion in model turbulence in the late decay stage, J. Math. Phys., 6 (1965), pp. 707-721.
49. E. STEIN, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970.
50. H. TRIEBEL, Theory of Function Spaces, Birkhäuser Verlag, Basel, 1983.
51. M. VIOT, Solutions faibles d'equation aux derives partielles stochastiques non lineaires, Thèse de doctorat, University of Paris VI, Paris, France, 1976.
52. M. I. VISHIK AND A. V. FURSIKOV, Mathematical Problems of Statistical Hydromechanics, Kluwer Academic Press, Dordrecht, Boston, London, 1979.