Exponential ergodicity for stochastic Burgers and 2D Navier-Stokes equations

Journal of Functional Analysis

Volumn 226, Issue 1, 2005, Pages 230-255

  Goldys, B ^a   Maslowski, B ^b  

^a UNIVERSITY OF NEW SOUTH WALES   (Australia)

^b MATHEMATICAL INSTITUTE   (Czech Republic)

Author keywords

Additive noise; Burgers equation; Invariant measure; Navier Stokes equation; Spectral gap; Uniform exponential ergodicity

Indexed keywords

EID: 23344452942     PISSN: 00221236     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jfa.2004.12.009     Document Type: Article

References (31)

1. J. Bricmont A. Kupiainen R. Lefevere Exponential mixing of the 2D stochastic Navier-Stokes dynamics Comm. Math. Phys. 230 2002 87-132
2. M. Capinski D. Gatarek Stochastic equations in Hilbert space with application to Navier-Stokes equations in any dimension J. Funct. Anal. 126 1994 26-35
3. G. Da Prato A. Debussche Differentiability of the transition semigroup of the stochastic Burgers equation, and application to the corresponding Hamilton-Jacobi equation Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9), Mat. Appl. 9 1998 267-277
4. G. Da Prato, A. Debussche, Maximal dissipativity of the Dirichlet operator corresponding to the Burgers equation, in: Stochastic Processes, Physics and Geometry: New Interplays, vol. I (Leipzig, 1999), CMS Conference Proceedings, vol. 28, American Mathematical Society, Providence, RI, 2000, pp. 85-98.
5. G. Da Prato A. Debussche R. Temam Stochastic Burgers' equation NoDEA Nonlinear Differential Equations Appl. 1 1994 389-402
6. G. Da Prato D. Gatarek Stochastic Burgers equation with correlated noise Stochastics Stochastics Rep. 52 1995 29-41
7. G. Da Prato S. Kwapien J. Zabczyk Regularity of solutions of linear stochastic equations in Hilbert spaces Stochastics 23 1987 1-23
8. G. Da Prato J. Zabczyk Stochastic Equations in Infinite Dimensions 1992 Cambridge University Press Cambridge
9. G. Da Prato J. Zabczyk Ergodicity for Infinite Dimensional Systems 1996 Cambridge University Press Cambridge
10. E. W., J.C. Mattingly, Ya. Sinai, Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation, Comm. Math. Phys. 224 (2001) 83-106.
11. B. Ferrario Ergodic results for stochastic Navier-Stokes equation Stochastics Stochastics Rep. 60 1997 271-288
12. B. Ferrario Stochastic Navier-Stokes equations: Analysis of the noise to have a unique invariant measure Annali Mat. Pura Appl. 177 1999 331-347
13. F. Flandoli Dissipativity and invariant measures for stochastic Navier-Stokes equations NoDEA 1 1994 252-286
14. F. Flandoli D. Ga̧tarek Martingale and stationary solutions for stochastic Navier-Stokes equations Probab. Theory Related Fields 102 1995 367-391
15. F. Flandoli B. Maslowski Ergodicity of the 2-D Navier-Stokes equation under random perturbations Comm. Math. Phys. 172 1995 119-141
16. r of the Navier-Stokes initial value problem Arch. Rational Mech. Anal. 89 1985 267-281
17. B. Goldys, B. Maslowski, Lower estimates of transition densities and bounds on exponential ergodicity for stochastic PDE's, preprint, School of Mathematics, The University of New South Wales, 2004, http://www.maths.unsw.edu.au/statistics/pubs/statspreprints2004.html.
18. M. Hairer J.C. Mattingly Ergodic properties of highly degenerate 2D stochastic Navier-Stokes equations C. R. Acad. Sci. Paris, Ser. I 1339 2004 879-882
19. M. Hairer, J.C. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing, preprint, math.PR/0409057(2004), http://front.math.ucdavis.edu/author/Hairer-M*&c=PR.
20. M. Hairer, J.C. Mattingly, E. Pardoux, Malliavin calculus and ergodic properties of highly degenerate 2D stochastic Navier-Stokes equation, preprint, math.PR/0409057 (2004), http://front.math.ucdavis.edu/author/Hairer-M*&c=PR.
21. N. Jain B. Jamison Contributions to Doeblin's theory of Markov processes Z. Wahrscheinlichkeitstheorie Verw. Geb. 8 1967 19-40
22. R.Z. Khasminskii Ergodic properties of recurrent diffusions and stabilization of the Cauchy problem for parabolic equations Teor. Veroyat. Primen. 5 1960 7-28 (in Russian)
23. S. Kuksin A. Shirikyan A coupling approach to randomly forced nonlinear PDE's I Comm. Math. Phys. 221 2001 351-366
24. S. Kuksin A. Shirikyan Coupling approach to white-forced nonlinear PDE's J. Math. Pures Appl. 81 6 2002 567-602
25. B. Maslowski I. Simao Asymptotic properties of stochastic semilinear equations by the method of lower measures Colloquium Math. 72 1997 147-171
26. B. Maslowski I. Simao Long-time behaviour of non-autonomous SPDE's Stochastic Processes Appl. 95 2001 285-309
27. J.C. Mattingly Exponential convergence for the stochastically forced Navier-Stokes equations and other partially dissipative dynamics Comm. Math. Phys. 230 2002 421-462
28. S.P. Meyn R.L. Tweedie Markov Chains and Stochastic Stability 1993 Springer Berlin
29. T. Shardlow Geometric ergodicity for stochastic PDE's Stochastic Anal. Appl. 17 1999 857-869
30. R. Temam Navier-Stokes Equations 1984 North-Holland Amsterdam
31. M.J. Vishik A.V. Fursikov Mathematical Problems of Statistical Hydromechanics 1980 Kluwer Dordrecht