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Ergodicity of 3D Stochastic Burgers Equation

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Abstract

3D Burgers equation is an important model for turbulence. It is natural to expect the long-time behaviour for this hydrodynamics equation. However, there is no result about the long-time behaviour for this deterministic model. Surprisingly, if the system is perturbed by stochastic noise, we establish the existence and uniqueness of invariant measure for 3D stochastic Burgers equation.

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References

  1. Bakhtin, Y., Cator, E., Khanin, K.: Space-time stationary solutions for the Burgers equation. J. Amer. Math. Soc., 27, 193–238 (2014)

    Article  MathSciNet  Google Scholar 

  2. Bakhtin, Y., and 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 

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

    Article  ADS  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  ADS  CAS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  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  ADS  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 Lévy 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 stochastic Burgers equations with multiplicative noise, arXiv:2108.08040 (2021)

  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  ADS  MathSciNet  Google Scholar 

  19. Dong, Z., Xu, T. G.: One-dimensional stochastic Burgers equation driven by Lévy 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  ADS  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  22. Forsyth, A. R.: Theory of Differential Eequations, Cambridge University Press, Cambridge, 1906

    Google Scholar 

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

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

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  34. 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 

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

    Book  Google Scholar 

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

    Google Scholar 

  37. Zhang, R., Zhou, G., Guo, B., et al.: Global well-posedness and large deviations for 3D stochastic Burgers equations. Z. Angew. Math. Phys., 71, 1–31 (2020)

    Article  ADS  MathSciNet  CAS  Google Scholar 

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Acknowledgements

We are grateful to Professor Z. Brzezniak for constant support and stimulating discussions. We also thank the referees for their time and comments.

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Authors and Affiliations

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

    Zhao Dong

  2. Department of Mathematics, Computational Foundry, Swansea University, Swansea, SA, 1 8EN, UK

    Jiang Lun Wu

  3. College of Mathematics and Statistics, Chongqing University, Chongqing, 400044, P. R. China

    Guo Li Zhou

Authors
  1. Zhao Dong
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  2. Jiang Lun Wu
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  3. Guo Li Zhou
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Correspondence to Guo Li Zhou.

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Conflict of Interest The authors declare no conflict of interest.

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Partially supported by National Key R&D Program of China (Grant Nos. 2020YFA0712700); NNSF of China (Grant Nos. 12090014, 11931004, 11971077); 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. cstc2020jcyj-msxmX0441); Fundamental Research Funds for the Central Universities (Grant No. 2020CDJ-LHZZ-027)

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Dong, Z., Wu, J.L. & Zhou, G.L. Ergodicity of 3D Stochastic Burgers Equation. Acta. Math. Sin.-English Ser. 40, 498–510 (2024). https://doi.org/10.1007/s10114-023-2055-4

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  • DOI: https://doi.org/10.1007/s10114-023-2055-4

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