Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Markov selections for the 3D stochastic Navier–Stokes equations
Download PDF
Download PDF
  • Published: 31 March 2007

Markov selections for the 3D stochastic Navier–Stokes equations

  • Franco Flandoli1 &
  • Marco Romito2 

Probability Theory and Related Fields volume 140, pages 407–458 (2008)Cite this article

  • 472 Accesses

  • 84 Citations

  • Metrics details

Abstract

We investigate the Markov property and the continuity with respect to the initial conditions (strong Feller property) for the solutions to the Navier–Stokes equations forced by an additive noise. First, we prove, by means of an abstract selection principle, that there are Markov solutions to the Navier–Stokes equations. Due to the lack of continuity of solutions in the space of finite energy, the Markov property holds almost everywhere in time. Then, depending on the regularity of the noise, we prove that any Markov solution has the strong Feller property for regular initial conditions. We give also a few consequences of these facts, together with a new sufficient condition for well-posedness.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Ash R.B. (1972). Real Analysis and Probability. Probability and Mathematical Statistics, vol. 11. Academic, New York

    Google Scholar 

  2. Blömker, D., Flandoli, F., Romito, M.: Markovianity and ergodicity for a surface growth PDE. (preprint) (2006)

  3. Chow P.L. and Khasminskii R.Z. (1997). Stationary solutions of nonlinear stochastic evolution equations. Stochastic Anal. Appl. 15(5): 671–699

    Article  MATH  MathSciNet  Google Scholar 

  4. Chung K.L. (1974). A Course in Probability Theory Probability and Mathematical Statistics, vol. 21. Academic, New York

    Google Scholar 

  5. Constantin P., Weinan E. and Titi E.S. (1994). Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Commun. Math. Phys. 165(1): 207–209

    Article  MATH  Google Scholar 

  6. Da Prato G. and Debussche A. (2003). Ergodicity for the 3D stochastic Navier–Stokes equations. J. Math. Pure. Appl. 82: 877–947

    MATH  MathSciNet  Google Scholar 

  7. Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications, vol. 44. Cambridge University Press, Cambridge (1992)

  8. Da Prato G. and Zabczyk J. (1996). Ergodicity for Infinite Dimensional Systems. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  9. Debussche, A., Odasso, C.: Markov solutions for the 3D stochastic Navier–Stokes equations with state dependent noise. Available on the arXiv preprint archive at the web address http://www.arxiv.org/abs/math.AP/0512361

  10. Duchon J. and Robert R. (2000). Inertial energy dissipation for weak solutions of incompressible Euler and Navier–Stokes equations. Nonlinearity 13(1): 249–255

    Article  MATH  MathSciNet  Google Scholar 

  11. Fefferman, C.L.: Existence and smoothness of the Navier–Stokes equation. Paper available at http://www.claymath.org/millennium/Navier-Stokes_Equations/

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

    MATH  MathSciNet  Google Scholar 

  13. Flandoli, F.: On the method of Da Prato and Debussche for the 3D stochastic Navier–Stokes equations. J. Evol. Equ. (to appear)

  14. Flandoli, F.: An introduction to 3D stochastic fluid dynamics. In: Proceedings of the CIME course on SPDE in hydrodynamics: recent progress and prospects. Lecture Notes in Mathematics, Springer, Berlin (to appear). Available on the web page of CIME at the address http://www.cime.unifi.it

  15. Flandoli F. (1997). Irreducibility of the 3-D stochastic Navier–Stokes equation. J. Funct. Anal. 149(1): 160–177

    Article  MATH  MathSciNet  Google Scholar 

  16. Flandoli F. and Maslowski B. (1995). Ergodicity of the 2-D Navier–Stokes equations under random perturbations. Commun. Math. Phys. 171: 119–141

    Article  MathSciNet  Google Scholar 

  17. Flandoli F. and Gatarek D. (1995). Martingale and stationary solutions for stochastic Navier–Stokes equations. Probab. Theory Relat. Fields 102: 367–391

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  19. Flandoli F. and Romito M. (2006). Markov selections and their regularity for the three-dimensional stochastic Navier–Stokes equations. C. R. Math. Acad. Sci. Paris Ser. I 343: 47–50

    MATH  MathSciNet  Google Scholar 

  20. Flandoli, F., Romito, M.: Regularity of transition semigroups associated to a 3D stochastic Navier–Stokes equation. (preprint) (2006)

  21. Krylov N.V. (1973). The selection of a Markov process from a Markov system of processes and the construction of quasidiffusion processes (in Russian). Izv. Akad. Nauk SSSR Ser. Mat. 37: 691–708

    MathSciNet  Google Scholar 

  22. Revuz D. and Yor M. (1991). Continuous Martingales and Brownian Motion Grundlehren der Mathematischen Wissenschaften, vol. 293. Springer, Berlin

    Google Scholar 

  23. Serrin, J.: The initial value problem for the Navier–Stokes equations. In: Proceedings of Symposium on Nonlinear Problems, Madison, pp. 69–98. University of Wisconsin Press, Madison (1963)

  24. Stroock D.W. and Yor M. (1980). On extremal solutions of martingale problems. Ann. Sci. École Norm. Sup. (4) 13(1): 95–164

    MATH  MathSciNet  Google Scholar 

  25. Stroock D.W. and Varadhan S.R.S. (1979). Multidimensional Diffusion Processes. Springer, Berlin

    MATH  Google Scholar 

  26. Temam, R.: Navier–Stokes Equations and Nonlinear Functional Analysis, 2nd edn. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 66. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1995)

  27. Temam, R.: Some Developments on Navier–Stokes Equations in the Second Half of the 20th Century. Development of Mathematics 1950–2000, 1049–1106. Birkhäuser, Basel (2000)

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica Applicata, Università di Pisa, via Buonarroti 1, 56127, Pisa, Italy

    Franco Flandoli

  2. Dipartimento di Matematica, Università di Firenze, Viale Morgagni 67/a, 50134, Firenze, Italy

    Marco Romito

Authors
  1. Franco Flandoli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marco Romito
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marco Romito.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Flandoli, F., Romito, M. Markov selections for the 3D stochastic Navier–Stokes equations. Probab. Theory Relat. Fields 140, 407–458 (2008). https://doi.org/10.1007/s00440-007-0069-y

Download citation

  • Received: 27 February 2006

  • Revised: 05 March 2007

  • Published: 31 March 2007

  • Issue Date: March 2008

  • DOI: https://doi.org/10.1007/s00440-007-0069-y

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Stochastic Navier–Stokes equations
  • Martingale problem
  • Markov property
  • Markov selections
  • Strong Feller property
  • Well posedness

Mathematics Subject Classification (2000)

  • 76D05
  • 60H15
  • 35Q30
  • 60H30
  • 76M35
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature