Skip to main content
SpringerLink
Log in

Noise Effect on the 2D Stochastic Burgers Equation

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

By comprehensive utilizing of the geometry structure of 2D Burgers equation and the stochastic noise, we find the decay properties of the solution to the stochastic 2D Burgers equation with Dirichlet boundary conditions. Consequently, the expected ergodicity for this turbulence model is established.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bakhtin, Y., Li, L.: Thermodynamic limit for directed polymers and stationary solutions of the Burgers equation, Comm. Pure Appl. Math., 72, 536–619 (2019)

    Article  MathSciNet  Google Scholar 

  2. Bakhtin, Y., Cator, E., Khanin, K.: Space-time stationary solutions for the Burgers equation, Journal of the American Mathematical Society, 27, 193–238 (2014)

    Article  MathSciNet  Google Scholar 

  3. Bec, J., Khanin, K.: Burgers turbulence. Phy. Rep., 447, 1–66 (2007)

    Article  MathSciNet  Google Scholar 

  4. Ben-Naim, E., Chen, S. Y., Doolen, G. D. et al.: Shocklike dynamics of inelastic gases, Phy. Rev. Lett., 83, 4069–4072 (1999)

    Article  Google Scholar 

  5. Bertini, L., Cancrini, N., Jona-Lasinio, G.: The stochastic Burgers equation, Commun. Math. Phys., 165, 211–232 (1994)

    Article  MathSciNet  Google Scholar 

  6. Beteman, H.: Some recent researches of the motion of fluid, Monthly Weather Rev., 43, 163–170 (1915)

    Article  Google Scholar 

  7. Brzezniak, Z., Goldys, B., Neklyudov, M.: Multidimensional stochastic Burgers equation, SIAM J. Math. Anal., 46, 871–889 (2014)

    Article  MathSciNet  Google Scholar 

  8. Bui, T.: Non-stationary Burgers flows with vanishing viscosity in bounded domains of R3, Math. Z., 145, 69–79 (1975)

    Article  MathSciNet  Google Scholar 

  9. Burgers, J. M.: Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion. Verh. Nederl. Akad. Wetensch. Afd. Natuurk. Sect. 1, 17, 1–53 (1939)

    MathSciNet  Google Scholar 

  10. Chan, T.: Scaling limits of Wick ordered KPZ equation, Commun. Math. Phys., 209, 671–690 (2000)

    Article  MathSciNet  Google Scholar 

  11. Cole, J. D.: On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math., 9, 225–236 (1951)

    Article  MathSciNet  Google Scholar 

  12. Da Prato, G., Debussche, A.: Stochastic Cahn–Hilliard equation, Nonlinear Anal., 26, 241–263 (1996)

    Article  MathSciNet  Google Scholar 

  13. Da Prato, G., Debussche, A., Temam, R.: Stochastic Burgers’ equation, NoDEA, 1, 389–402 (1994)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  15. Dirr, N., Souganidis, P. E.: Large-time behavior for viscous and nonviscous Hamilton–Jacobi equations forced by additive noise, SIAM J. Math. Anal., 37, 777–796 (2005)

    Article  MathSciNet  Google Scholar 

  16. Dong, Z.: On the uniqueness of invariant measure of the Burgers equation driven by Levy processes, J. Theoret. Probab., 21, 322–335 (2008)

    Article  MathSciNet  Google Scholar 

  17. Dong, Z., Guo, B. L., Wu, J.-L., et al.: Global well-posedness and regularity of 3D Burgers equation with multiplicative noise, SIAM Journal on Mathematical Analysis, 55, 1847–1882 (2023)

    Article  MathSciNet  Google Scholar 

  18. Dong, Z., Xu, L. H., Zhang, X. C.: Exponential ergodicity of stochastic Burgers equations driven by α-stable processes, J. Stat. Phys., 154, 929–949 (2014)

    Article  MathSciNet  Google Scholar 

  19. Dong, Z., Xu, T. G.: One-dimensional stochastic Burgers equation driven by Levy processes, J. Funct. Anal., 243, 631–678 (2007)

    Article  MathSciNet  Google Scholar 

  20. Dunlap, A., Graham, C., Ryzhik, L.: Stationary solutions to the stochastic Burgers equatin on the line, Commun. Math. Phys., 382, 875–949 (2021)

    Article  Google Scholar 

  21. Forsyth, A. R.: Theory of Differential Equations, Cambridge University Press, Cambridge, 1906

    Google Scholar 

  22. Glatt-Holtz, N., Ziane, M.: Strong pathwise solutions of the stochastic Navier–Stokes system, Advances in Differential Equations, 14, 567–600 (2009)

    Article  MathSciNet  Google Scholar 

  23. Gomes, D., Iturriaga, R., Khanin, K. et al.: Viscosity limit of stationary distributions for the random forced Burgers equation, Moscow Mathematical Journal, 5, 613–631 (2005)

    Article  MathSciNet  Google Scholar 

  24. Goldys, B., Maslowski, B.: Exponential ergodicity for stochastic Burgers and 2D Navier–Stokes equations, J. Funct. Anal., 226, 230–255 (2005)

    Article  MathSciNet  Google Scholar 

  25. Gourcy, M.: Large deviation principle of occupation measure for stochastic Burgers equation, Ann. I. H. Poincaré-PR., 43, 441–459 (2007)

    Article  MathSciNet  Google Scholar 

  26. Gyöngy, I., Nualart, D.: On the stochastic Burgers equation in the real line, Ann. Probab., 27, 782–802 (1999)

    Article  MathSciNet  Google Scholar 

  27. Hopf, E.: The partial differential equation ut + uux = uxx. Comm. Pure Appl. Math., 3, 201–230 (1950)

    Article  MathSciNet  Google Scholar 

  28. Hosokawa, I., Yamamoto, K.: Turbulence in the randomly forced one dimensional Burgers flow, J. Stat. Phys., 245, 245–272 (1975)

    Article  Google Scholar 

  29. Hsieh, P. F., Sibuya, Y.: Basic Theory of Ordinary Differential Equations, Higher Education Press, Beijing, 2007

    Google Scholar 

  30. Iturriaga, R., Khanin, K.: Burgers turbulence and random Lagrangian systems, Commun. Math. Phys., 232, 377–428 (2003)

    Article  MathSciNet  Google Scholar 

  31. Jeng, D. T.: Forced model equation for turbulence, The Physics of Fluids, 12, 2006–2010 (1969)

    Article  Google Scholar 

  32. Khanin, E. W., Mazel, K., Sinai, Ya. A.: Invariant measures for Burgers equation with stochastic forcing, Ann. Math., 151, 877–900 (2000)

    Article  MathSciNet  Google Scholar 

  33. Khanin, K., Zhang, K.: Hyperbolicity of minimizers and regularity of viscosity solutions for a random Hamilton–Jacobi equation, Commun. Math. Phys., 355, 803–837 (2017)

    Article  MathSciNet  Google Scholar 

  34. Kardar, M., Parisi, M., Zhang, J. C.: Dynamical scaling of growing interfaces, Phys. Rev. Lett., 56, 889–892 (1986)

    Article  Google Scholar 

  35. Kiselev, A., Ladyzhenskaya, O. A.: On the existence and uniqueness of the solution of the nonstationary problem for a viscous, incompressible fluid (Russian), Izv. Akad. Nauk SSSR. Ser. Mat., 21, 655–680 (1957)

    MathSciNet  Google Scholar 

  36. Lions, J., Magenes, B.: Nonhomogeneous Boundary Value Problems and Applications, Springer-Verlag, New York, 1972

    Book  Google Scholar 

  37. Temam, R.: Navier–Stokes Equations. Theory and Numerical Analysis, 3rd ed., Amer. Math. Soc., Providence, 2001

    Google Scholar 

  38. Zhang, R., Zhou, G. L., Guo, B. L., et al.: Global well-posedness and large deviations for 3D stochastic Burgers equations. Z. Angew. Math. Phys., 71:30 (2020)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We would like to express our deep appreciation to Professor Z. Brzezniak for constant help and encouragement.

Author information

Authors and Affiliations

  1. RCSDS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, P. R. China

    Zhao Dong

  2. Beijing Normal University-Hong Kong Baptist University United International College, Zhuhai, 519087, P. R. China

    Jiang Lun Wu

  3. Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science (IRADS), UIC, Zhuhai, 519087, P. R. China

    Jiang Lun Wu

  4. School of Statistics and Mathematics, Chongqing University and Key Laboratory of Nonlinear Analysis and Its Applications (Chongqing University), Ministry of Education, Chongqing, 400044, P. R. China

    Guo Li Zhou

Authors
  1. Zhao Dong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jiang Lun Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Guo Li Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guo Li Zhou.

Ethics declarations

Conflict of Interest The authors declare no conflict of interest.

Additional information

Supported by National Key R and D Program of China (Grant No. 2020YFA0712700), NSFC (Grant Nos. 11931004, 11971077, 12090014), Key Laboratory of Random Complex Structures and Data Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences (Grant No. 2008DP173182), Chongqing Key Laboratory of Analytic Mathematics and Applications, Chongqing University, Natural Science Foundation Project of CQ (Grant No. cstc2020jcyjmsxmX0441), Fundamental Research Funds for the Central Universities (Grant No. 2020CDJ-LHZZ-027), UIC Start-up Research Fund (Grant No. UICR0700072-24)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dong, Z., Wu, J.L. & Zhou, G.L. Noise Effect on the 2D Stochastic Burgers Equation. Acta. Math. Sin.-English Ser. 40, 2065–2090 (2024). https://doi.org/10.1007/s10114-024-3079-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-024-3079-0

Keywords

MR(2010) Subject Classification

Search

Navigation