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Continuous Wong–Zakai Approximations of Random Attractors for Quasi-linear Equations with Nonlinear Noise

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Abstract

We consider a family of random quasi-linear equations driven by nonlinear Wong–Zakai noise and parameterized by the non-zero size \(\lambda \) of noise. After proving the existence of a random attractor \(A_\lambda (\omega )\) in the square Lebesgue space, we then show that there is a residual dense subset of the space of nonzero real numbers such that, under the Hausdorff metric, the map \(\lambda \rightarrow A_\lambda (\theta _s\omega )\) is continuous at all points of the residual dense set, where \(\theta _s\) is a group of self-transformations on the probability space. We also prove that as \(\lambda \rightarrow \pm \infty \) the random attractor converges upper-semicontinuously to the global attractor of the deterministic quasi-linear equation. The upper semi-continuity result is new for nonlinear noise, while, the lower semi-continuity result is new even for linear noise. The theory of Baire category is the main tool used to prove the residual continuity.

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References

  1. Aida, S., Sasaki, K.: Wong–Zakai approximation of solutions to reflecting stochastic differential equations on domains in Euclidean spaces. Stoch. Process Appl. 123, 3800–3827 (2013)

    MathSciNet  MATH  Google Scholar 

  2. Arrieta, J.M., Carvalho, A.N., Langa, J.A.: Continuity of dynamical structures for nonautonomous evolution equations under singular perturbations. J. Dyn. Differ. Equ. 24, 427–481 (2012)

    MathSciNet  MATH  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  MATH  Google Scholar 

  4. Brzezniak, Z., Manna, U., Mukherjee, D.: Wong–Zakai approximation for the stochastic Landau–Lifshitz–Gilbert equations. J. Differ. Equ. 267, 776–825 (2019)

    MathSciNet  MATH  Google Scholar 

  5. Caraballo, T., Carvalho, A.N., Da Costa, H.B.: Equi-attraction and continuity of attractors for skew-product semiflows. Discrete Contin. Dyn. Syst. Ser. B 21, 2949–2967 (2016)

    MathSciNet  MATH  Google Scholar 

  6. Cui, H., Langa, J.A., Li, Y.: Measurability of random attractors for quasi strong-to-weak continuous random dynamical systems. J. Dyn. Differ. Equ. 30, 1873–1898 (2018)

    MathSciNet  MATH  Google Scholar 

  7. Cui, H., Kloeden, P.E., Wu, F.: Pathwise upper semi-continuity of random pullback attractors along the time axis. Physica D 374, 21–34 (2018)

    MathSciNet  MATH  Google Scholar 

  8. Fernandez-Martinez, M., Guirao, J.L.G., Vera Lopez, J.A.: Fractal dimension for IFS-attractors revisited. Qual. Theory Dyn. Syst. 17(3), 709–722 (2018)

    MathSciNet  MATH  Google Scholar 

  9. Flandoli, F., Gubinelli, M., Priola, E.: Well-posedness of the transport equation by stochastic perturbation. Invent. Math. 180, 1–53 (2010)

    MathSciNet  MATH  Google Scholar 

  10. Freitas, M.M., Kalita, P., Langa, J.A.: Continuity of non-autonomous attractors for hyperbolic perturbation of parabolic equations. J. Differ. Equ. 264, 1886–1945 (2018)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  13. Gu, A.: Asymptotic behavior of random lattice dynamical systems and their Wong–Zakai approximations. Discrete Contin. Dyn. Syst. B 24, 5737–5767 (2019)

    MathSciNet  MATH  Google Scholar 

  14. Gu, A., Lu, K., Wang, B.: Asymptotic behavior of random Navier–Stokes equations driven by Wong–Zakai approximations. Discrete Cont. Dyn. Syst. 39, 185–218 (2019)

    MathSciNet  MATH  Google Scholar 

  15. Han, X., Kloeden, P.E., Usman, B.: Upper semi-continuous convergence of attractors for a Hopfield-type lattice model. Nonlinearity 33, 1881–1906 (2020)

    MathSciNet  MATH  Google Scholar 

  16. Hoang, L.T., Olson, E.J., Robinson, J.C.: On the continuity of global attractors. Proc. Am. Math. Soc. 143, 4389–4395 (2015)

    MathSciNet  MATH  Google Scholar 

  17. Hoang, L.T., Olson, E.J., Robinson, J.C.: Continuity of pullback and uniform attractors. J. Differ. Equ. 264, 4067–4093 (2018)

    MathSciNet  MATH  Google Scholar 

  18. Kloeden, P.E., Simsen, J.: Attractors of asymptotically autonomous quasi-linear parabolic equation with spatially variable exponents. J. Math. Anal. Appl. 425, 911–918 (2015)

    MathSciNet  MATH  Google Scholar 

  19. Kloeden, P.E., Simsen, J., Simsen, M.S.: Asymptotically autonomous multivalued cauchy problems with spatially variable exponents. J. Math. Anal. Appl. 445, 513–531 (2017)

    MathSciNet  MATH  Google Scholar 

  20. Krause, A., Lewis, M., Wang, B.: Dynamics of the non-autonomous stochastic p-Laplace equation driven by multiplicative noise. Appl. Math. Comput. 246, 365–376 (2014)

    MathSciNet  MATH  Google Scholar 

  21. Langa, J.A., Robinson, J.C., Suarez, A.: The stability of attractors for non-autonomous perturbations of gradient-like systems. J. Differ. Equ. 234, 607–625 (2007)

    MathSciNet  MATH  Google Scholar 

  22. Li, F.Z., Li, Y.R., Wang, R.H.: Regular measurable dynamics for reaction–diffusion equations on narrow domains with rough noise. Discrete Contin. Dyn. Syst. 38, 3663–3685 (2018)

    MathSciNet  MATH  Google Scholar 

  23. Li, F.Z., Li, Y.R., Wang, R.H.: Limiting dynamics for stochastic reaction–diffusion equations on the Sobolev space with thin domains. Comput. Math. Appl. 79, 457–475 (2020)

    MathSciNet  MATH  Google Scholar 

  24. Li, Y.N., Yang, Z.J., Da, F.: Robust attractors for a perturbed non-autonomous extensible beam equation with nonlinear nonlocal damping. Discrete Contin. Dyn. Syst. 39, 5975–6000 (2019)

    MathSciNet  MATH  Google Scholar 

  25. Li, Y.R., Gu, A., Li, J.: Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations. J. Differ. Equ. 258, 504–534 (2015)

    MathSciNet  MATH  Google Scholar 

  26. Li, Y.R., Li, F.Z.: Limiting dynamics for stochastic FitzHugh-Nagumo equations on large domains. Stoch. Dyn. 19, 2019 (2019)

    MathSciNet  MATH  Google Scholar 

  27. Li, Y.R., She, L.B., Wang, R.H.: Asymptotically autonomous dynamics for parabolic equations. J. Math. Anal. Appl. 459, 1106–1123 (2018)

    MathSciNet  MATH  Google Scholar 

  28. Li, Y.R., Yin, J.Y.: Existence, regularity and approximation of global attractors for weakly dissipative p-Laplace equations. Discrete Contin. Dyn. Syst. Ser. S 9, 1939–1957 (2016)

    MathSciNet  MATH  Google Scholar 

  29. Liu, L.F., Fu, X.: Existence and upper semicontinuity of (L-2, L-q) pullback attractors for a stochastic p-Laplacian equation. Commun. Pure Appl. Anal. 16, 443–473 (2017)

    MathSciNet  MATH  Google Scholar 

  30. Lu, K., Wang, B.: Wong–Zakai approximations and long term behavior of stochastic partial differential equations. J. Dyn. Differ. Equ. 31, 1341–1371 (2019)

    MathSciNet  MATH  Google Scholar 

  31. Manna, U., Mukherjee, D., Panda, A.A.: Wong–Zakai approximation for the stochastic Landau–Lifshitz–Gilbert equations with anisotropy energy. J. Math. Anal. Appl. 480, 1–13 (2019)

    MathSciNet  MATH  Google Scholar 

  32. Novruzov, E., Hagverdiyev, A.: On long-time dynamics of the solution of doubly nonlinear equation. Qual. Theory Dyn. Syst. 15, 127–155 (2016)

    MathSciNet  MATH  Google Scholar 

  33. Robinson, J.C.: Linear embeddings of finite-dimensional subsets of Banach spaces into Euclidean spaces. Nonlinearity 22, 753–746 (2009)

    MathSciNet  Google Scholar 

  34. Robinson, J.C.: Stability of random attractors under perturbation and approximation. J. Differ. Equ. 186, 652–669 (2002)

    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)

    MathSciNet  MATH  Google Scholar 

  36. Wang, F., Li, J., Li, Y.: Random attractors for Ginzburg–Landau equations driven by difference noise of a Wiener-like process. Adv. Differ. Equ. 2019, 1–17 (2019)

    MathSciNet  MATH  Google Scholar 

  37. Wang, R., Shi, L., Wang, B.: Asymptotic behavior of fractional nonclassical diffusion equations driven by nonlinear colored noise on \({\mathbb{R}}^N\). Nonlinearity 32, 4524–4556 (2019)

    MathSciNet  MATH  Google Scholar 

  38. Wang, S., Li, Y.: Longtime robustness of pullback random attractors for stochastic magneto-hydrodynamics equations. Physica D 382, 46–57 (2018)

    MathSciNet  MATH  Google Scholar 

  39. Wang, S., Li, Y.: Probabilistic continuity of a pullback random attractor in time-sample. Discrete Contin. Dyn. Syst. B 25, 2699–2722 (2020)

    MathSciNet  MATH  Google Scholar 

  40. Wang, X., Lu, K., Wang, B.: Wong–Zakai approximations and attractors for stochastic reaction–diffusion equations on unbounded domains. J. Differ. Equ. 264, 378–424 (2018)

    MathSciNet  MATH  Google Scholar 

  41. Wang, X., Lu, K., Wang, B.: Random attractors for delay parabolic equations with additive noise and deterministic nonautonomous forcing. SIAM J. Appl. Dyn. Syst. 14, 1018–1047 (2015)

    MathSciNet  MATH  Google Scholar 

  42. Wong, E., Zakai, M.: On the convergence of ordinary integrals to stochastic integrals. Ann. Math. Stat. 36, 1560–1564 (1965)

    MathSciNet  MATH  Google Scholar 

  43. Yin, J., Li, Y.: Two types of upper semi-continuity of bi-spatial attractors for non-autonomous stochastic p-Laplacian equations on R-n. Math. Methods Appl. Sci. 40, 4863–4879 (2017)

    MathSciNet  MATH  Google Scholar 

  44. Yosida, K.: Functional Analysis, 6th edn. Springer, Berlin (1980)

    MATH  Google Scholar 

  45. Zaj, M., Ghane, F.H.: Non-hyperbolic solenoidal thick Bony attractors. Qual. Theory Dyn. Syst. 18, 35–55 (2019)

    MathSciNet  MATH  Google Scholar 

  46. Zhao, W.: Random dynamics of stochastic p-Laplacian equations on R-N with an unbounded additive noise. J. Math. Anal. Appl. 455, 1178–1203 (2017)

    MathSciNet  MATH  Google Scholar 

  47. Zhao, W., Zhang, Y., Chen, S.: Higher-order Wong–Zakai approximations of stochastic reaction–diffusion equations on R-N. Physica D 401, 1–15 (2020)

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, People’s Republic of China

    Yangrong Li, Shuang Yang & Qiangheng Zhang

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  1. Yangrong Li
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  2. Shuang Yang
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  3. Qiangheng Zhang
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Correspondence to Yangrong Li.

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This work is supported by National Natural Science Foundation of China Grant 11571283.

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Li, Y., Yang, S. & Zhang, Q. Continuous Wong–Zakai Approximations of Random Attractors for Quasi-linear Equations with Nonlinear Noise. Qual. Theory Dyn. Syst. 19, 87 (2020). https://doi.org/10.1007/s12346-020-00423-z

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