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Limit Invariant Measures for the Modified Stochastic Swift–Hohenberg Equation in a 3D Thin Domain

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Abstract

This work is concerned with the modified stochastic Swift–Hohenberg equation in a 3D thin domain. Although the diffusion motion of molecules is irregular with the interference of the film-fluid fluctuation, the invariant measure on the trajectory space reveals delicate transition of the dynamical behavior when the interior forces change. We therefore prove that the invariant measure of the system converges weakly to the unique counterpart of the stochastic Swift–Hohenberg equation in a 2D bounded domain with a concrete convergence rate, as the modified parameter and the thickness of the thin domain tend to zero. Furthermore, we address that the smooth density of the limit invariant measure fulfills a Fokker–Planck equation.

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References

  1. Arrieta, J., Santamaría, E.: Distance of attractors of reaction-diffusion equations in thin domains. J. Differ. Equ. 263(9), 5459–5506 (2017)

    Article  MathSciNet  Google Scholar 

  2. Blömker, D., Hairer, M., Pavliotis, G.A.: Modulation equations: stochastic bifurcation in large domains. Commun. Math. Phys. 258(2), 479–512 (2005)

    Article  MathSciNet  Google Scholar 

  3. Caraballo, T., Chueshov, I., Kloeden, P.E.: Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain. SIAM J. Math. Anal. 38(5), 1489–1507 (2007)

    Article  MathSciNet  Google Scholar 

  4. Chueshov, I., Kuksin, S.: Stochastic 3D Navier-Stokes equations in a thin domain and its \(\alpha \)-approximation. Physica D 237(10), 1352–1367 (2008)

    Article  MathSciNet  Google Scholar 

  5. Da Prato, G., Zabczyk, J.: Ergodicity for Infinite Dimensional Systems. Cambridge University Press, Cambridge (1996)

    Book  Google Scholar 

  6. Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (2014)

    Book  Google Scholar 

  7. Faedo, S.: Il Principio di Minimo e Sue Applicazioni alle Equazioni Funzionali. Springer, New York (2011)

    Book  Google Scholar 

  8. Flandoli, F., Grotto, F., Luo, D.: Fokker-Planck equation for dissipative 2D Euler equations with cylindrical noise. Theory Probab. Math. Stat. 102, 117–143 (2020)

    Article  MathSciNet  Google Scholar 

  9. Gao, P.: The stochastic swift-hohenberg equation. Nonlinearity 30(9), 3516–3559 (2017)

    Article  MathSciNet  Google Scholar 

  10. Guo, Y., Duan, J., Li, D.: Approximation of random invariant manifolds for a stochastic Swift-Hohenberg equation. Discret. Cont. Dyn. Syst. 9(6), 1701–1715 (2016)

    Article  MathSciNet  Google Scholar 

  11. He, Y., Li, C., Wang, J.: Invariant measures and statistical solutions for the nonautonomous discrete modified Swift-Hohenberg equation. Bull. Malays. Math. Sci. Soc. 44(6), 3819–3837 (2021)

    Article  MathSciNet  Google Scholar 

  12. Hernández, M., Ong, K.W.: Stochastic Swift-Hohenberg equation with degenerate linear multiplicative noise. J. Math. Fluid Mech. 20(3), 1353–1372 (2018)

    Article  MathSciNet  Google Scholar 

  13. Kukavica, I., Ziane, M.: Regularity of the Navier-Stokes equation in a thin periodic domain with large data. Discret. Cont. Dyn. A. 16(1), 67–86 (2006)

    Article  MathSciNet  Google Scholar 

  14. Kukavica, I., Ziane, M.: On the regularity of the Navier-Stokes equation in a thin periodic domain. J. Differ. Equ. 234(2), 485–506 (2007)

    Article  MathSciNet  Google Scholar 

  15. Kuksin, S., Shirikyan, A.: A coupling approach to randomly forced nonlinear PDE’s I. Commun. Math. Phys. 221(2), 351–366 (2001)

    Article  MathSciNet  Google Scholar 

  16. Kuksin, S., Shirikyan, A.: Mathematics of Two-Dimensional Turbulence. Cambridge University Press, Cmbridge (2012)

    Book  Google Scholar 

  17. Li, Y., Wei, J., Lu, Z.: Random pullback attractor for a non-autonnmous modified Swift-Hohenberg equation with multiplicative noise. J. Appl. Anal. Comput. 11(1), 464–476 (2021)

    MathSciNet  Google Scholar 

  18. Li, Y., Wu, H., Zhao, T.: Random pullback attractor of a non-autonomous local modified stochastic Swift-Hohenberg equation with multiplicative noise. J. Math. Phys. 61(9), 092703 (2020)

    Article  MathSciNet  Google Scholar 

  19. Mohammed, W.W., Blömker, D., Klepel, K.: Modulation equation for stochastic Swift-Hohenberg equation. SIAM J. Math. Anal. 45(1), 14–30 (2013)

    Article  MathSciNet  Google Scholar 

  20. Nakasato, J.C., Pažanin, I., Pereira, M.: Reaction-diffusion problem in a thin domain with oscillating boundary and varying order of thickness. Z. Angew. Math. Phys. 72(1), 1–17 (2021)

    Article  MathSciNet  Google Scholar 

  21. Polat, M.: Global attractor for a modified Swift-Hohenberg equation. Comput. Math. Appl. 57(1), 62–66 (2009)

    Article  MathSciNet  Google Scholar 

  22. Qiao, H.: Limit theorems of SDEs driven by Lévy processes and application to nonlinear filtering problems. Nonlinear Differ. Equ. Appl. NoDEA 29(1), 8 (2022)

    Article  Google Scholar 

  23. Sell, G.R., You, Y.: Dynamics of Evolutionary Equations. Springer, New York (2002)

    Book  Google Scholar 

  24. Swift, J., Hohenberg, P.C.: Hydrodynamic fluctuations at the convective instability. Phys. Rev. A 15(1), 319 (1977)

    Article  Google Scholar 

  25. Taboada, M.: Finite-dimensional asymptotic behavior for the Swift-Hohenberg model of convection. Nonlinear Anal. 14(1), 43–54 (1990)

    Article  MathSciNet  Google Scholar 

  26. Vishik, M.I., Fursikov, A.V.: Mathematical Problems of Statistical Hydromechanics. Springer, New York (1988)

    Book  Google Scholar 

  27. Wang, Z., Du, X.: Pullback attractors for modified Swift-Hohenberg equation on unbounded domains with non-autonomous deterministic and stochastic forcing terms. J. Appl. Anal. Comput. 7(1), 207–223 (2017)

    MathSciNet  Google Scholar 

  28. Wang, J., Li, C., Yang, L., Jia, M.: Upper semi-continuity of random attractors and existence of invariant measures for nonlocal stochastic Swift-Hohenberg equation with multiplicative noise. J. Math. Phys. 62(11), 111507 (2021)

    Article  MathSciNet  Google Scholar 

  29. Wang, W., Sun, J., Duan, J.: Ergodic dynamics of the stochastic Swift-Hohenberg system. Nonlinear Anal. Real. 6(2), 273–295 (2005)

    Article  MathSciNet  Google Scholar 

  30. Wang, J., Zhang, X., Li, C.: Global martingale and pathwise solutions and infinite regularity of invariant measures for a stochastic modified Swift-Hohenberg equation. Nonlinearity 36(5), 2655–2707 (2023)

    Article  MathSciNet  Google Scholar 

  31. Xiao, Q., Gao, H.: Stochastic attractor bifurcation of the one-dimensional Swift-Hohenberg equation with multiplicative noise. J. Differ. Equ. 336, 565–588 (2022)

    Article  MathSciNet  Google Scholar 

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Funding

This work was supported by the National Natural Science Foundation of China (Grant No. 12171343), and the Sichuan Science and Technology Program (Grant No. 2022JDTD0019).

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Author notes

  1. Guanggan Chen, Wenhu Zhong and Yunyun Wei have contributed equally to this work.

Authors and Affiliations

  1. School of Mathematical Sciences and V.C. & V.R. Key Lab of Sichuan Province, Sichuan Normal University, Chengdu, 610066, China

    Guanggan Chen & Wenhu Zhong

  2. College of Mathematics and Physics, Chengdu University of Technology, Chengdu, 610059, China

    Yunyun Wei

Authors
  1. Guanggan Chen
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  2. Wenhu Zhong
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  3. Yunyun Wei
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Contributions

Guanggan Chen: Writing—original draft (lead). Wenhu Zhong: Writing—original draft (equal). Yunyun Wei: Writing—review & editing (equal).

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Correspondence to Wenhu Zhong.

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Chen, G., Zhong, W. & Wei, Y. Limit Invariant Measures for the Modified Stochastic Swift–Hohenberg Equation in a 3D Thin Domain. Appl Math Optim 89, 69 (2024). https://doi.org/10.1007/s00245-024-10140-7

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