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doi: 10.3934/dcdsb.2021063

Weak pullback attractors for stochastic Ginzburg-Landau equations in Bochner spaces

School of Mathematical Sciences, Laurent Mathematics Center and V.C. & V.R. Key Lab, Sichuan Normal University, Chengdu 610066, China

* Corresponding author: Ji Shu, shuji@sicnu.edu.cn

Received  October 2020 Revised  December 2020 Published  February 2021

Fund Project: The last author is supported by NSFC (11871138)

In this paper we discuss the weak pullback mean random attractors for stochastic Ginzburg-Landau equations defined in Bochner spaces. We prove the existence and uniqueness of weak pullback mean random attractors for the stochastic Ginzburg-Landau equations with nonlinear diffusion terms. We also establish the existence and uniqueness of such attractors for the deterministic Ginzburg-Landau equations with random initial data. In this case, the periodicity of the weak pullback mean random attractors is also proved whenever the external forcing terms are periodic in time.

Citation: Lu Zhang, Aihong Zou, Tao Yan, Ji Shu. Weak pullback attractors for stochastic Ginzburg-Landau equations in Bochner spaces. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021063
References:
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L. Arnold, Random Dynamical Systems, Springer-Verlag, New York, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

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M. BartuccelliP. ConstantinC. R. DoeringJ. Gibbon and M. Gisselfält, On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation, Physica D, 44 (1990), 421-444.  doi: 10.1016/0167-2789(90)90156-J.  Google Scholar

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C. R. DoeringJ. D. Gibbon and C. D. Levermore, Weak and strong solutions of the complex Ginzburg-Landau equation, Phys. D, 71 (1994), 285-318.  doi: 10.1016/0167-2789(94)90150-3.  Google Scholar

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J. DuanP. Holmes and E. S. Titi, Global existence theory for a generalized Ginzburg-Landau equation, Nonlinearity, 5 (1992), 1303-1314.   Google Scholar

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F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stochastics Stochastics Rep., 59 (1996), 21-45.  doi: 10.1080/17442509608834083.  Google Scholar

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M. Garrido-AtienzaK. Lu and B. Schmalfuss, Random dynamical systems for stochastic evolution equations driven by multiplicative fractional Brownian noise with Hurst parameters $H \in (1/3, 1/2]$, SIAM J. Appl. Dyn. Syst., 15 (2016), 625-654.  doi: 10.1137/15M1030303.  Google Scholar

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[14]

Y. Lan and J. Shu, Dynamics of non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise, Commun. Pure Appl. Anal., 18 (2019), 2409-2431.  doi: 10.3934/cpaa.2019109.  Google Scholar

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Y. Lan and J. Shu, Fractal dimension of random attractors for non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise, Dyn. Syst., 34 (2019), 274-300.  doi: 10.1080/14689367.2018.1523368.  Google Scholar

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[18]

D. LiK. LuB. Wang and X. Wang, Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains, Discrete Contin. Dyn. Syst., 38 (2018), 187-208.  doi: 10.3934/dcds.2018009.  Google Scholar

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D. LiK. LuB. Wang and X. Wang, Limiting dynamics for non-autonomous stochastic retarded reaction-diffusion equations on thin domains, Discrete Contin. Dyn. Syst., 39 (2019), 3717-3747.  doi: 10.3934/dcds.2019151.  Google Scholar

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K. Li, The uniqueness of the weak solutions for the complex Ginzburg-Landau Equation, J. Henan Norm. Univ. Nat. Sci., 41 (2013), 34-37.   Google Scholar

[21]

J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[22]

K. Lu and B. Wang, Wong-Zakai approximations and long term behavior of stochastic partial differential equations, J. Dynam. Differential Equations, 31 (2019), 1341-1371.  doi: 10.1007/s10884-017-9626-y.  Google Scholar

[23]

D. MaJ. Shu and L. Qin, Wong-Zakai approximations and asymptotic behavior of stochastic Ginzburg-Landau equations, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 4335-4359.  doi: 10.3934/dcdsb.2020100.  Google Scholar

[24]

C. Prevot and M. Rockner, A Concise Course on Stochastic Partial Differential Equations, Lecture notes in Mathematics, vol.1905, Springer, Berlin, 2007.  Google Scholar

[25]

X. Pu and B. Guo, Momentum estimates and ergodicity for the 3D stochastic cubic Ginzburg-Landau equation with degenerate noise, J. Differential Equations, 251 (2011), 1747-1777.  doi: 10.1016/j.jde.2011.06.011.  Google Scholar

[26]

D. Ruelle, Characteristic exponents for viscous fluid subjected to time dependent forces, Comm. Math. Phys., 93 (1984), 285-300.  doi: 10.1007/BF01258529.  Google Scholar

[27]

B. Schmalfuss, V. Reitmann, T. Riedrich and N. Koksch (eds.), Backward cocycles and attractors of stochastic differential equations, in International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, Technische Universitat, Dresden, 1992,185–192. Google Scholar

[28]

T. Shen and J. Huang, Well-posedness and dynamics of stochastic fractional model for nonlinear optical fiber materials, Nonlinear Anal., 110 (2014), 33-46.  doi: 10.1016/j.na.2014.06.018.  Google Scholar

[29]

J. Shu, P. Li, J. Zhang and O. Liao, Random attractors for the stochastic coupled fractional Ginzburg-Landau equation with additive noise, J. Math. Phys., 56 (2015), 102702, 11 pp. doi: 10.1063/1.4934724.  Google Scholar

[30]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[31]

B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009, 31 pp. doi: 10.1142/S0219493714500099.  Google Scholar

[32]

B. Wang, Existence, stability and bifurcation of random complete and periodic solutions of stochastic parabolic equations, Nonlinear Anal., 103 (2014), 9-25.  doi: 10.1016/j.na.2014.02.013.  Google Scholar

[33]

B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269.  Google Scholar

[34]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.  Google Scholar

[35]

B. Wang, Weak pullback attractors for mean random dynamical systems in Bochner spaces, J. Dynam. Differential Equations, 31 (2019), 2177-2204.  doi: 10.1007/s10884-018-9696-5.  Google Scholar

[36]

B. Wang, Dynamics of stochastic reaction-diffusion lattice systems driven by nonlinear noise, J. Math. Anal. Appl., 477 (2019), 104-132.  doi: 10.1016/j.jmaa.2019.04.015.  Google Scholar

[37]

B. Wang, Weak pullback attractors for stochastic Navier-Stokes equations with nonlinear diffusion terms, Proc. Amer. Math. Soc., 147 (2019), 1627-1638.  doi: 10.1090/proc/14356.  Google Scholar

[38]

X. WangK. Lu and B. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 264 (2018), 378-424.  doi: 10.1016/j.jde.2017.09.006.  Google Scholar

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer-Verlag, New York, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[2]

M. BartuccelliP. ConstantinC. R. DoeringJ. Gibbon and M. Gisselfält, On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation, Physica D, 44 (1990), 421-444.  doi: 10.1016/0167-2789(90)90156-J.  Google Scholar

[3]

P. W. BatesK. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.  doi: 10.1016/j.jde.2008.05.017.  Google Scholar

[4]

T. CaraballoJ. A. LangaV. S. Melnik and J. Valero, Pullback attractors for nonautonomous and stochastic multivalued dynamical systems, Set-Valued Anal., 11 (2003), 153-201.  doi: 10.1023/A:1022902802385.  Google Scholar

[5]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab.Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar

[6]

C. R. DoeringJ. D. Gibbon and C. D. Levermore, Weak and strong solutions of the complex Ginzburg-Landau equation, Phys. D, 71 (1994), 285-318.  doi: 10.1016/0167-2789(94)90150-3.  Google Scholar

[7]

J. DuanP. Holmes and E. S. Titi, Global existence theory for a generalized Ginzburg-Landau equation, Nonlinearity, 5 (1992), 1303-1314.   Google Scholar

[8]

F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stochastics Stochastics Rep., 59 (1996), 21-45.  doi: 10.1080/17442509608834083.  Google Scholar

[9]

H. GaoM. Garrido-Atienza and B. Schmalfuss, Random attractors for stochastic evolution equations driven by fractional Brownian motion, SIAM J. Math. Anal., 46 (2014), 2281-2309.  doi: 10.1137/130930662.  Google Scholar

[10]

M. Garrido-AtienzaK. Lu and B. Schmalfuss, Random dynamical systems for stochastic evolution equations driven by multiplicative fractional Brownian noise with Hurst parameters $H \in (1/3, 1/2]$, SIAM J. Appl. Dyn. Syst., 15 (2016), 625-654.  doi: 10.1137/15M1030303.  Google Scholar

[11]

B. Guo and B. Wang, Finite dimensional behavior for the derivative Ginzburg-Landau equation in two spatial dimensions, Phys. D, 89 (1995), 83-99.  doi: 10.1016/0167-2789(95)00216-2.  Google Scholar

[12]

P. E. Kloeden and T. Lorenz, Mean-square random dynamical systems, J. Differential Equations, 253 (2012), 1422-1438.  doi: 10.1016/j.jde.2012.05.016.  Google Scholar

[13]

C. GuoJ. Shu and X. Wang, Fractal dimension of random attractors for non-autonomous fractional stochastic Ginzburg-Landau equations, Acta Math. Sin.(Engl. Ser.), 36 (2020), 318-336.  doi: 10.1007/s10114-020-8407-4.  Google Scholar

[14]

Y. Lan and J. Shu, Dynamics of non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise, Commun. Pure Appl. Anal., 18 (2019), 2409-2431.  doi: 10.3934/cpaa.2019109.  Google Scholar

[15]

Y. Lan and J. Shu, Fractal dimension of random attractors for non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise, Dyn. Syst., 34 (2019), 274-300.  doi: 10.1080/14689367.2018.1523368.  Google Scholar

[16]

D. LiZ. Dai and X. Liu, Long time behaviour for generalized complex Ginzburg-Landau equation, J. Math. Anal. Appl., 330 (2007), 934-948.  doi: 10.1016/j.jmaa.2006.07.095.  Google Scholar

[17]

D. Li and B. Guo, Asymptotic behavior of the 2D generalized stochastic Ginzburg-Landau equation with additive noise, Appl. Math. Mech. (English Ed.), 30 (2009), 945-956.  doi: 10.1007/s10483-009-0801-x.  Google Scholar

[18]

D. LiK. LuB. Wang and X. Wang, Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains, Discrete Contin. Dyn. Syst., 38 (2018), 187-208.  doi: 10.3934/dcds.2018009.  Google Scholar

[19]

D. LiK. LuB. Wang and X. Wang, Limiting dynamics for non-autonomous stochastic retarded reaction-diffusion equations on thin domains, Discrete Contin. Dyn. Syst., 39 (2019), 3717-3747.  doi: 10.3934/dcds.2019151.  Google Scholar

[20]

K. Li, The uniqueness of the weak solutions for the complex Ginzburg-Landau Equation, J. Henan Norm. Univ. Nat. Sci., 41 (2013), 34-37.   Google Scholar

[21]

J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[22]

K. Lu and B. Wang, Wong-Zakai approximations and long term behavior of stochastic partial differential equations, J. Dynam. Differential Equations, 31 (2019), 1341-1371.  doi: 10.1007/s10884-017-9626-y.  Google Scholar

[23]

D. MaJ. Shu and L. Qin, Wong-Zakai approximations and asymptotic behavior of stochastic Ginzburg-Landau equations, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 4335-4359.  doi: 10.3934/dcdsb.2020100.  Google Scholar

[24]

C. Prevot and M. Rockner, A Concise Course on Stochastic Partial Differential Equations, Lecture notes in Mathematics, vol.1905, Springer, Berlin, 2007.  Google Scholar

[25]

X. Pu and B. Guo, Momentum estimates and ergodicity for the 3D stochastic cubic Ginzburg-Landau equation with degenerate noise, J. Differential Equations, 251 (2011), 1747-1777.  doi: 10.1016/j.jde.2011.06.011.  Google Scholar

[26]

D. Ruelle, Characteristic exponents for viscous fluid subjected to time dependent forces, Comm. Math. Phys., 93 (1984), 285-300.  doi: 10.1007/BF01258529.  Google Scholar

[27]

B. Schmalfuss, V. Reitmann, T. Riedrich and N. Koksch (eds.), Backward cocycles and attractors of stochastic differential equations, in International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, Technische Universitat, Dresden, 1992,185–192. Google Scholar

[28]

T. Shen and J. Huang, Well-posedness and dynamics of stochastic fractional model for nonlinear optical fiber materials, Nonlinear Anal., 110 (2014), 33-46.  doi: 10.1016/j.na.2014.06.018.  Google Scholar

[29]

J. Shu, P. Li, J. Zhang and O. Liao, Random attractors for the stochastic coupled fractional Ginzburg-Landau equation with additive noise, J. Math. Phys., 56 (2015), 102702, 11 pp. doi: 10.1063/1.4934724.  Google Scholar

[30]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[31]

B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009, 31 pp. doi: 10.1142/S0219493714500099.  Google Scholar

[32]

B. Wang, Existence, stability and bifurcation of random complete and periodic solutions of stochastic parabolic equations, Nonlinear Anal., 103 (2014), 9-25.  doi: 10.1016/j.na.2014.02.013.  Google Scholar

[33]

B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269.  Google Scholar

[34]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.  Google Scholar

[35]

B. Wang, Weak pullback attractors for mean random dynamical systems in Bochner spaces, J. Dynam. Differential Equations, 31 (2019), 2177-2204.  doi: 10.1007/s10884-018-9696-5.  Google Scholar

[36]

B. Wang, Dynamics of stochastic reaction-diffusion lattice systems driven by nonlinear noise, J. Math. Anal. Appl., 477 (2019), 104-132.  doi: 10.1016/j.jmaa.2019.04.015.  Google Scholar

[37]

B. Wang, Weak pullback attractors for stochastic Navier-Stokes equations with nonlinear diffusion terms, Proc. Amer. Math. Soc., 147 (2019), 1627-1638.  doi: 10.1090/proc/14356.  Google Scholar

[38]

X. WangK. Lu and B. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 264 (2018), 378-424.  doi: 10.1016/j.jde.2017.09.006.  Google Scholar

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