Skip to main content
SpringerLink
Log in

Fractal dimension of random invariant sets and regular random attractors for stochastic hydrodynamical equations

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

The main aim of this work is to provide some general and unified results on the existence, regularity and finite fractal dimension estimates of pullback random attractors for a wide class of non-autonomous stochastic hydrodynamical systems arising from fluid dynamics. Using weaker conditions than that in the abstract results of Debussche (J Math Pures Appl 77:967–988, 1998) and Zhou et al. (Appl Math Comput 276:80–95, 2016), we obtain a better upper bound estimate of the fractal dimension of random invariant sets in both Banach space and its subspace. The existence, uniqueness and regularity of pullback random attractors for the systems are established by employing the famous idea of energy equation due to Ball (J Nonlinear Sci 7:475–502, 1997) as well as an approach of spectral decomposition. Based on our abstract results and some new uniform estimates on the difference of the solutions, we prove the finite fractal dimension of those random attractors in both initial space and regular subspace. To the best of our knowledge, this is the first general result on the fractal dimension estimate of random attractors in regular subspaces, which seems also new even in the autonomous and deterministic case.

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

Access this article

Log in via an institution

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

Data availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Notes

  1. In Table 1, we use \(\dim _H\mathcal A(\tau ,\omega )\) and \(\dim _V\mathcal A(\tau ,\omega )\) to denote the fractal dimension of \(\mathcal A\) in H and V, respectively, \(m_0,m_1\in \mathbb {N}\) with \(m_0\le m_1\) are independent of \(\tau \) and \(\omega \).

References

  1. Anh, C.T., Trang, P.H.: Pullback attractors for three-dimensional Navier–Stokes–Voigt equations in some unbounded domains. Proc. R. Soc. Edinb. Sect. A 143, 223–251 (2013)

    Google Scholar 

  2. Ball, J.M.: Continuity properties and global attractors of generalized semiflows and the Navier–Stokes equations. J. Nonlinear Sci. 7, 475–502 (1997)

    MathSciNet  Google Scholar 

  3. Bates, P.W., Lu, K., Wang, B.: Random attractors for stochastic reaction–diffusion equations on unbounded domains. J. Differ. Equ. 246, 845–869 (2009)

    MathSciNet  Google Scholar 

  4. Brzeźniak, Z., Capiński, M., Flandoli, F.: Pathwise global attractors for stationary random dynamical systems. Probabil. Theory Relat. Fields 95, 87–102 (1993)

    MathSciNet  Google Scholar 

  5. Caffarelli, L., Kohn, R., Nirenberg, L.: Partial regularity of suitable weak solutions of the Navier–Stokes equations. Commun. Pure Appl. Math. 35, 771–831 (1982)

    MathSciNet  Google Scholar 

  6. Caraballo, T., Carvalho, A.N., Langa, J.A., Rivero, F.: Existence of pullback attractors for pullback asymptotically compact processes. Nonlinear Anal. 72, 1967–1976 (2010)

    MathSciNet  Google Scholar 

  7. Caraballo, T., Marín-Rubio, P., Valero, J.: Autonomous and non-autonomous attractors for differential equations with delays. J. Differ. Equ. 208, 9–41 (2005)

    MathSciNet  Google Scholar 

  8. Caraballo, T., Herrera-Cobos, M., Marín-Rubio, P.: Time-dependent attractors for non-autonomous non-local reaction–diffusion equations. Proc. R. Soc. Edinb. Sect. A Math. 148(5), 957–981 (2018)

    MathSciNet  Google Scholar 

  9. Caraballo, T., Lukaszewicz, G., Real, J.: Pullback attractors for non-autonomous 2D Navier–Stokes equations in unbounded domains. C. R. Math. Acad. Sci. Paris 342, 263–268 (2006)

    MathSciNet  Google Scholar 

  10. Caraballo, T., Langa, J.A., Melnik, V.S., Valero, J.: Pullback attractors for nonautonomous and stochastic multivalued dynamical systems. Set-Valued Anal. 11, 153–201 (2003)

    MathSciNet  Google Scholar 

  11. Caraballo, T., Sonner, S.: Random pullback exponential attractors: general existence results for random dynamical systems in Banach spaces. Discret. Contin. Dyn. Syst. Ser. A 37(12), 6383–6403 (2017)

    MathSciNet  Google Scholar 

  12. Caraballo, T., Garrido-Atienza, M.J., Taniguchi, T.: The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion. Nonlinear Anal. 74, 3671–3684 (2011)

    MathSciNet  Google Scholar 

  13. Caraballo, T., Kloeden, P.E., Schmalfuss, B.: Exponentially stable stationary solutions for stochastic evolution equations and their perturbation. Appl. Math. Optim. 50, 183–207 (2004)

    MathSciNet  Google Scholar 

  14. Chueshov, I., Millet, A.: Stochastic 2D hydrodynamical type systems: well-posedness and large deviations. Appl. Math. Optim. 61, 379–420 (2010)

    MathSciNet  Google Scholar 

  15. Chueshov, I.: Dynamics of Quasi-Stable Dissipative Systems. Springer, Berlin (2015)

    Google Scholar 

  16. Crauel, H., Flandoli, F.: Attractors for random dynamical systems. Probabil. Theory Relat. Fields 100, 365–393 (1994)

    MathSciNet  Google Scholar 

  17. Crauel, H., Debussche, A., Flandoli, F.: Random attractors. J. Dyn. Differ. Equ. 9, 307–341 (1997)

    MathSciNet  Google Scholar 

  18. Crauel, H., Flandoli, F.: Hausdorff dimension of invariant sets for random dynamical systems. J. Dyn. Differ. Equ. 10, 449–474 (1998)

    MathSciNet  Google Scholar 

  19. Debussche, A.: On the finite dimensionality of random attractors. Stoch. Anal. Appl. 15, 473–491 (1997)

    MathSciNet  Google Scholar 

  20. Debussche, A.: Hausdorff dimension of a random invariant set. J. Math. Pures Appl. 77, 967–988 (1998)

    MathSciNet  Google Scholar 

  21. Flandoli, F., Schmalfuß, B.: Random attractors for the 3D stochastic Navier–Stokes equation with multiplicative noise. Stoch. Stoch. Rep. 59, 21–45 (1996)

    Google Scholar 

  22. Giga, Y.: Weak and strong solutions of the Navier–Stokes initial value problem. Publ. Res. Inst. Math. Sci. 19(3), 887–910 (1983)

    MathSciNet  Google Scholar 

  23. Giga, Y.: Solutions for semilinear parabolic equations in \(L^p\) and regularity of weak solutions of the Navier–Stokes system. J. Differ. Equ. 62(2), 186–212 (1986)

    Google Scholar 

  24. Gess, B., Liu, W., Röckner, M.: Random attractors for a class of stochastic partial differential equations driven by general additive noise. J. Differ. Equ. 251, 1225–1253 (2011)

    MathSciNet  Google Scholar 

  25. Gess, B., Liu, W., Schenke, A.: Random attractors for locally monotone stochastic partial differential equations. J. Differ. Equ. 269, 3414–3455 (2020)

    MathSciNet  Google Scholar 

  26. Gess, B.: Random attractors for singular stochastic evolution equations. J. Differ. Equ. 255, 524–559 (2013)

    MathSciNet  Google Scholar 

  27. Gess, B.: Random attractors for degenerate stochastic partial differential equations. J. Dyn. Differ. Equ. 25, 121–157 (2013)

    MathSciNet  Google Scholar 

  28. Gu, A., Li, D., Wang, B., Yang, H.: Regularity of random attractors for fractional stochastic reaction-diffusion equations on \(\mathbb{R} ^n\). J. Differ. Equ. 264, 7094–7137 (2018)

    Google Scholar 

  29. Guo, B., Chen, F.: Finite-dimensional behavior of global attractors for weakly damped nonlinear Schrödinger–Boussinesq equations. Phys. D 93, 101–118 (1996)

    MathSciNet  Google Scholar 

  30. Guo, B., Li, Y.: Attractor for dissipative Klein–Gordon-Schrödinger equations in \(\mathbb{R} ^3\). J. Differ. Equ. 136, 356–377 (1997)

    Google Scholar 

  31. García-Luengo, J., Marín-Rubio, P., Real, J.: Pullback attractors for 2D Navier–Stokes equations with delays and their regularity. Adv. Nonlinear Stud. 13, 331–357 (2013)

    MathSciNet  Google Scholar 

  32. Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (2013)

    Google Scholar 

  33. Lions, J.L.: Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires. Dunod, Paris (1969)

    Google Scholar 

  34. Ladyzhenskaya, O.A.: Attractors for Semigroups and Evolution Equations. Cambridge University Press, Cambridge (1991)

    Google Scholar 

  35. Ladyzhenskaya, O.A.: On the determination of minimal global attractors for the Navier–Stokes and other partial differential equations. Uspekhi Mat. Nauk 42(6), 25–60 (1987)

    MathSciNet  Google Scholar 

  36. Liu, W., Röckner, M.: SPDE in Hilbert space with locally monotone coefficients. J. Funct. Anal. 259, 2902–2922 (2010)

    MathSciNet  Google Scholar 

  37. Liu, W., Röckner, M.: Local and global well-posedness of SPDE with generalized coercivity conditions. J. Differ. Equ. 254, 725–755 (2013)

    MathSciNet  Google Scholar 

  38. Liu, W., Röckner, M.: Stochastic Partial Differential Equations: An Introduction. Springer, Berlin (2015)

    Google Scholar 

  39. Liu, W., Röckner, M., da Silva, J.L.: Quasi-linear (stochastic) partial differential equations with time-fractional derivatives. SIAM J. Math. Anal. 50, 2588–2607 (2018)

    MathSciNet  Google Scholar 

  40. Liu, W., Röckner, M., da Silva, J.L.: Strong dissipativity of generalized time-fractional derivatives and quasi-linear (stochastic) partial differential equations. J. Funct. Anal. 281, 109135 (2021)

    MathSciNet  Google Scholar 

  41. Miranville, A.: Exponential attractors for a class of evolution equations by a decomposition method. Comptes Rend. l’Acad. Sci. Ser. I Math. 328(2), 145–150 (1999)

    MathSciNet  Google Scholar 

  42. Robinson, J.C.: Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  43. Schmalfuss, B.: Backward cocycle and attractors of stochastic differential equations. In: Reitmann, V., Riedrich, T., Koksch, N. (eds.) International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, pp. 185–192. Technische Universität Dresden, Dresden (1992)

    Google Scholar 

  44. Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1997)

    Google Scholar 

  45. Temam, R.: Navier–Stokes Equations. North-Holland, Amsterdam (1977)

    Google Scholar 

  46. Wang, B.: Attractors for reaction–diffusion equations in unbounded domains. Phys. D 128, 41–52 (1999)

    MathSciNet  Google Scholar 

  47. Wang, B.: Random attractors for the stochastic Benjamin–Bona–Mahony equation on unbounded domains. J. Differ. Equ. 246, 2506–2537 (2009)

    MathSciNet  Google Scholar 

  48. Wang, B.: Asymptotic behavior of stochastic wave equations with critical exponents on \(\mathbb{R} ^{3}\). Trans. Am. Math. Soc. 363, 3639–3663 (2011)

    Google Scholar 

  49. Wang, B.: Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems. J. Differ. Equ. 253, 1544–1583 (2012)

    MathSciNet  Google Scholar 

  50. Wang, B.: Periodic random attractors for stochastic Navier–Stokes equations on unbounded domains. Electron. J. Differ. Equ. 59, 1–18 (2012)

    MathSciNet  Google Scholar 

  51. Wang, B.: Random attractors for non-autonomous stochastic wave equations with multiplicative noise. Discret. Contin. Dyn. Syst. 34, 269–300 (2014)

    MathSciNet  Google Scholar 

  52. Wang, B.: Well-posedness and long term behavior of supercritical wave equations driven by nonlinear colored noise on \(R^n\). J. Funct. Anal. 283, Paper No. 109498 (2022)

    Google Scholar 

  53. Wang, Z., Zhou, S.: Random attractor for stochastic non-autonomous damped wave equation with critical exponent. Discret. Contin. Dyn. Syst. 37, 545–573 (2017)

    MathSciNet  Google Scholar 

  54. Xu, J., Caraballo, T.: Long time behavior of stochastic nonlocal partial differential equations and Wong–Zakai approximations. SIAM J. Math. Anal. 54, 2792–2844 (2022)

    MathSciNet  Google Scholar 

  55. Zhou, S., Tian, Y., Wang, Z.: Fractal dimension of random attractor for stochastic non-autonomous reaction–diffusion equation. Appl. Math. Comput. 276, 80–95 (2016)

    MathSciNet  Google Scholar 

  56. Zhou, S., Zhao, M.: Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise. Discret. Cont. Dyn. Syst. 36, 2887–2914 (2016)

    MathSciNet  Google Scholar 

  57. Zhou, S., Wang, Z.: Finite fractal dimensions of random attractors for stochastic FitzHugh–Nagumo system with multiplicative white noise. J. Math. Anal. Appl. 441, 648–667 (2016)

    MathSciNet  Google Scholar 

  58. Zhou, S.: Random exponential attractor for cocycle and application to non-autonomous stochastic lattice systems with multiplicative white noise. J. Differ. Equ. 263, 2247–2279 (2017)

    MathSciNet  Google Scholar 

  59. Zhou, S.: Random exponential attractor for stochastic reaction–diffusion equation with multiplicative noise in \(\mathbb{R} ^3\). J. Differ. Equ. 263, 6347–6383 (2017)

    Google Scholar 

  60. Zhou, S., Yin, F., Ouyang, Z.: Random attractor for damped nonlinear wave equations with white noise. SIAM J. Appl. Dyn. Syst. 4, 883–903 (2005)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referees for some constructive suggestions and valuable comments.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Guizhou Normal University, Guiyang, 550001, China

    Renhai Wang

  2. Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China

    Boling Guo

  3. School of Mathematics and Statistics/RIMS, Jiangsu Normal University, Xuzhou, 221116, China

    Wei Liu

  4. Department of Natural Sciences, Hong Duc University, Thanh Hoa, 41000, Vietnam

    Da Tien Nguyen

Authors
  1. Renhai Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Boling Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wei Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Da Tien Nguyen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wei Liu.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supported by National Natural Science Foundation of China (Nos. 12171208, 12090011, 12090010, 11831014), Guizhou Normal University (11904/0522100) and Qianjiaoji [2022]124.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, R., Guo, B., Liu, W. et al. Fractal dimension of random invariant sets and regular random attractors for stochastic hydrodynamical equations. Math. Ann. 389, 671–718 (2024). https://doi.org/10.1007/s00208-023-02661-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-023-02661-3

Mathematics Subject Classification

Search

Navigation