Skip to main content
SpringerLink
Log in

Fractal Dimension of Random Attractors for Non-autonomous Fractional Stochastic Ginzburg—Landau Equations

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

Abstract

This paper considers the dynamical behavior of solutions for non-autonomous stochastic fractional Ginzburg-Landau equations driven by additive noise with α ε (0,1). First, we give some conditions for bounding the fractal dimension of a random invariant set of non-autonomous random dynamical system. Second, we derive uniform estimates of solutions and establish the existence and uniqueness of tempered pullback random attractors for the equation in H. At last, we prove the finiteness of fractal dimension of random attractors.

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

References

  1. Arnold, L.: Random Dynamical Systems, Springer-Verlag, New York, 1998

    Book  MATH  Google Scholar 

  2. Bartuccelli, M., Constantin, P., Doering, C., et al.: On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation. Phys. D, 44, 421–444 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bates, P. W., Lisei, H., Lu, K.: Attractors for stochastic lattice dynamical system. Stock. Dyn., 6, 1–21 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Brzezniak, Z., Li, Y.: Asymptotic compactness and absorbing sets for 2D stochastic Navier—Stokes equations on some unbounded domains. Trans. Amer. Math. Soc., 358, 5587–5629 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Crauel, H., Debussche, A., Flandoli, F.: Random attractors. J. Dynam. Differential Equations, 9, 307–341 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Crauel, H., Flandoli, F.: Attractors for random dynamical systems. Probab. Theory Related Fields, 100, 365–393 (1994)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math., 136, 521–573 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Doering, C., Gibbon, J., Levermore, C.: Weak and strong solutions of the complex Ginzburg-Landau equation. Phys. D, 71, 285–318 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dong, J., Xu, M.: Space-time fractional Schrödinger equation with time-independent potentials. J. Math. Anal. Appl, 344, 1005–1017 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fan, X., Wang, Y.: Attractors for a second order nonautonomous lattice dynamical systems with nonlinear damping. Phys. Lett. A, 365, 17–27 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Flandoli, F., Schmalfuss, B.: Random attractors for the 3D stochastic Navier—Stokes equation with multiplicative white noise. Stock. Stock. Rep., 59, 21–45 (1996)

    Article  MATH  Google Scholar 

  14. Garrido-Atienza, M., Lu, K., Schmalfuss, B.: Random dynamical systems for stochastic equations driven by a fractional Brownian motion. Discrete Contin. Dyn. Syst. Ser. B, 14, 473–493 (2010)

    MathSciNet  MATH  Google Scholar 

  15. Guo, B., Han, Y., Xin, J.: Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation. Appl. Math. Comput., 204, 458–477 (2008)

    Google Scholar 

  16. Guo, B., Huo, Z.: Global well-posedness for the fractional nonlinear Schrödinger equation. Commun. Partial Differential Equations, 36, 247–255 (2011)

    Article  MATH  Google Scholar 

  17. Guo, B., Pu, X., Huang, F.: Fractional Partial Differential Equations and their Numerical Solutions, Science Press, Beijing, 2011

    MATH  Google Scholar 

  18. Guo, B., Wang, X.: Finite dimensional behavior for the derivative Ginzburg-Landau equation in two soatial dimensions. Phys. D, 89, 83–99 (1995)

    Article  MathSciNet  Google Scholar 

  19. Han, X., Shen, W., Zhou, S.: Random attractors for stochastic lattice dynamical system in weighted space. J. Differential Equations, 250, 1235–1266 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Li, D., Dai, Z., Liu, X.: Long time behavior for generalized complex Ginzburg-Landau equation. J. Math. Anal. Appl, 330, 938–948 (2007)

    Google Scholar 

  21. Li, D., Guo, B.: Asymptotic behavior of the 2D generalized stochastic Ginzburg-Landau equation with additive noise. Appl. Math. Mech., 30, 883–894 (2009)

    MathSciNet  Google Scholar 

  22. Lu, H., Bates, P. W., Lu, S., et al.: Dynamics of 3-D fractional complex Ginzburg-Landau equation. J. Differential Equations, 259, 5276–5301 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lu, H., Bates, P. W., Lu, S., et al.: Dynamics of the 3D fractional Ginzburg-Landau equation with multiplicative noise on an unbounded domain. Commmu. Math. Sci., 14, 273–295 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lu, H., Lu, S.: Random attrator for fractional Ginzburg—Laudau equation with multiplicative noise. Taiwanese J. Math., 18, 435–450 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Pu, X., Guo, B.: Global weak soltuions of the fractional Landau—Lifshitz—Maxwell equation. J. Math. Anal. Appl., 372, 86–98 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Pu, X., Guo, B.: Well-posedness and dynamics for the fractional Ginzburg—Laudau equation. Appl. Anal., 92, 318–334 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Shen, T., Huang, J.: Well-posedness and dynamics of stochastic fractional model for nonlinear optical fiber materials. Nonlinear Anal., 110, 33–46 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. Shu, J.: Random attractors for stochastic discrete Klein—Gordon—Schrödinger equations driven by fractional Brownian motions. Discrete Contin. Dyn. Syst. Ser. B, 22, 1587–1599 (2017)

    MathSciNet  MATH  Google Scholar 

  29. Shu, J., Guo, C.: Well-posedness and dynamics for non-autonomous fractional stochastic Ginzburg—Landau equations, preprint

  30. Shu, J., Li, P., Zhang, J., et al.: Random attractors for the stochastic coupled fractional Ginzburg—Landau equation with additive noise. J. Math. Phys., 56, 102702 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Tarasov Vasily, E., Zaslavsky George, M.: Fractional Ginzburg—Laudau equation for fractal media. Phys. A, 354, 249–261 (2005)

    Article  Google Scholar 

  32. Walters, P.: Introduction to Ergodic Theory, Springer-Verlag, New York, 2000

    MATH  Google Scholar 

  33. Wang, B.: Upper semicontinuity of random attractors for non-compact random systems. Electron. J. Differential Equations, 139, 1–18 (2009)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  36. Wang, B.: Existence and upper-semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms. Stock. Dyn., 14, 1450009 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wang, B.: Asymptotic behavior of non-autonomous fractional stochastic reaction-diffusion equations. Nonlinear Anal., 158, 60–82 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  38. Yin, F., Liu, L.: D-pullback attractor for a non-autonomous wave equation with additive noise on unbounded domains. Comput. Math. Appl., 68, 424–438 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  39. Yin, J., Li, Y., Gu, A.: Backwards compact attractors and periodic attractors for non-autonomous damped wave equations on an unbounded domain. Comput. Math. Appl., 74, 744–758 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  40. Zhao, C., Zhou, S.: Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications. J. Math. Anal. Appl., 354, 78–95 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  42. Zhou, S., Zhao, M.: Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise. Discrete Contin. Dyn. Syst., 36, 2887–2914 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  43. Zhou, S., Zhao, M.: Random attractors for damped non-autonomous wave equations with memory and white noise. Nonlinear Anal, 120, 202–226 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  44. Zhou, S., Zhao, M.: Fractal dimension of random invariant sets for non-autonomous random dynamical systems and random attractor for stochastic damped wave equation. Nonlinear Anal., 133, 292–318 (2016)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the reviewers for their helpful comments.

Author information

Authors and Affiliations

  1. Department of Mathematics, China University of Mining and Technology Beijing, Beijing, 100083, P. R. China

    Chun Xiao Guo

  2. School of Mathematical Science and V. C & V. R. Key Lab, Sichuan Normal University, Chengdu, 610068, P. R. China

    Ji Shu

  3. Department of Mathematics, Sichuan University, Chengdu, 610064, P. R. China

    Xiao Hu Wang

Authors
  1. Chun Xiao Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ji Shu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiao Hu Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ji Shu.

Additional information

Supported by National Natural Science Foundation of China (Grant Nos. 11571245, 11771444, 11871138 and 11871049), funding of V. C. & V. R. Key Lab of Sichuan Province, the Yue Qi Young Scholar Project, China University of Mining and Technology (Beijing) and China Scholarship Council (CSC)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guo, C.X., Shu, J. & Wang, X.H. Fractal Dimension of Random Attractors for Non-autonomous Fractional Stochastic Ginzburg—Landau Equations. Acta. Math. Sin.-English Ser. 36, 318–336 (2020). https://doi.org/10.1007/s10114-020-8407-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-020-8407-4

Keywords

MR(2010) Subject Classification

Search

Navigation