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October  2019, 24(10): 5621-5632. doi: 10.3934/dcdsb.2019075

Exponential convergence for the 3D stochastic cubic Ginzburg-Landau equation with degenerate noise

Yan Zheng and  Jianhua Huang ,

College of Science, National University of Defense Technology, Changsha 410073, China

* Corresponding author: Jianhua Huang

Received  October 2018 Published  April 2019

Fund Project: The authors are supported by the NSF of China(No.11771449), NSF of Hunan(No.2018JJ2468), Fundamental Program of NUDT(No.ZK17-03-19).

The current paper is devoted to 3D stochastic Ginzburg-Landau equation with degenerate random forcing. We prove that the corresponding Markov semigroup possesses an exponentially attracting invariant measure. To accomplish this, firstly we establish a type of gradient inequality, which is also essential to proving asymptotic strong Feller property. Then we prove that the corresponding dynamical system possesses a strong type of Lyapunov structure and is of a relatively weak form of irreducibility.

Keywords: Stochastic Ginzburg-Landau equations, exponential mixing, ergodicity, degenerate noise.
Mathematics Subject Classification: Primary: 60H15; Secondary: 37A25.
Citation: Yan Zheng, Jianhua Huang. Exponential convergence for the 3D stochastic cubic Ginzburg-Landau equation with degenerate noise. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5621-5632. doi: 10.3934/dcdsb.2019075
References:
[1]

M. Barton-Smith, Invariant measure for the stochastic Ginzburg-Landau equation, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 29-52.  doi: 10.1007/s00030-003-1040-y.  Google Scholar

[2]

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P. Coullet and L. Lega, Defect-mediated turbulence in wave patterns, Europhys. Lett., 7 (1988), 511-516.  doi: 10.1209/0295-5075/7/6/006.  Google Scholar

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J. Eckmann and M. Haier, Invariant measures for stochastic partial differential equations in unbounded domains, Nonlinearity, 14 (2001), 133-151.  doi: 10.1088/0951-7715/14/1/308.  Google Scholar

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M. Hairer, Exponential mixing properties of stochastic pdes through asymptotic coupling, Probability Theory and Related Fields, 124 (2001), 345-380.  doi: 10.1007/s004400200216.  Google Scholar

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J. Mattingly, Exponential convergence for the stochastically forced navier-stokes equations and other partially dissipative dynamics, Communications in Mathematical Physics, 230 (2002), 421-462.  doi: 10.1007/s00220-002-0688-1.  Google Scholar

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M. Harier and C. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing, Ann. Math., 164 (2006), 993-1032.  doi: 10.4007/annals.2006.164.993.  Google Scholar

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M. Harier and C. Mattingly, Spectral gaps in Wasserstein distances and the 2D stochastic Navier-Stokes equations, The Annals of Probabiltiy, 36 (2008), 2050-2091.  doi: 10.1214/08-AOP392.  Google Scholar

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M. Hairer, J. Mattingly and M. Scheutzow, Asymptotic coupling and a general form of harris' theorem with applications to stochastic delay equations, Probability Theory and Related Fields, 149 (2011), 223-259.  doi: 10.1007/s00440-009-0250-6.  Google Scholar

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A. Joets and R. Ribotta, Defects in Non-linear Waves in Convection, Nonlinear Coherent Structures, 353 (2005), 157-169.  doi: 10.1007/BFb0033633.  Google Scholar

[13]

S. Kuksin, Randomly forced CGL equation: Stationary measures and the inviscid limit, J.Phys. A, 37 (2004), 3805-3822.  doi: 10.1088/0305-4470/37/12/006.  Google Scholar

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C. Odasso, Ergodicity for the stochastic complex Ginzburg-Landau equations, Ann. Inst. H. Poincaré Probab. Statist., 42 (2006), 417-454.  doi: 10.1016/j.anihpb.2005.06.002.  Google Scholar

[15]

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

[16]

M. Rochner and X. Zhang, Stochastic tamed 3D Navier-Stokes equations: Existence, uniqueness and ergodicity, Probab. Theory Related Fields, 145 (1009), 211-267.  doi: 10.1007/s00440-008-0167-5.  Google Scholar

[17]

D. Yang and Z. Hou, Large deviations for the stochastic derivative Ginzburg-Landau equation with multiplicative noise, Phys. D, 237 (2008), 82-91.  doi: 10.1016/j.physd.2007.08.015.  Google Scholar

show all references

References:
[1]

M. Barton-Smith, Invariant measure for the stochastic Ginzburg-Landau equation, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 29-52.  doi: 10.1007/s00030-003-1040-y.  Google Scholar

[2]

Coullet, Elphick, Gil and Lega, Topological defects of wave patterns, Phys. Rev. Lett., 59 (1987), 884-887.  doi: 10.1103/PhysRevLett.59.884.  Google Scholar

[3]

P. Coullet and L. Lega, Defect-mediated turbulence in wave patterns, Europhys. Lett., 7 (1988), 511-516.  doi: 10.1209/0295-5075/7/6/006.  Google Scholar

[4] G. Da Parto and J. Zabcyzk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, 1996.  doi: 10.1017/CBO9780511662829.  Google Scholar
[5] G. Da Parto and J. Zabcyzk, Stochastic Equations in Infinite Dimensionals, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.  Google Scholar
[6]

J. Eckmann and M. Haier, Invariant measures for stochastic partial differential equations in unbounded domains, Nonlinearity, 14 (2001), 133-151.  doi: 10.1088/0951-7715/14/1/308.  Google Scholar

[7]

M. Hairer, Exponential mixing properties of stochastic pdes through asymptotic coupling, Probability Theory and Related Fields, 124 (2001), 345-380.  doi: 10.1007/s004400200216.  Google Scholar

[8]

J. Mattingly, Exponential convergence for the stochastically forced navier-stokes equations and other partially dissipative dynamics, Communications in Mathematical Physics, 230 (2002), 421-462.  doi: 10.1007/s00220-002-0688-1.  Google Scholar

[9]

M. Harier and C. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing, Ann. Math., 164 (2006), 993-1032.  doi: 10.4007/annals.2006.164.993.  Google Scholar

[10]

M. Harier and C. Mattingly, Spectral gaps in Wasserstein distances and the 2D stochastic Navier-Stokes equations, The Annals of Probabiltiy, 36 (2008), 2050-2091.  doi: 10.1214/08-AOP392.  Google Scholar

[11]

M. Hairer, J. Mattingly and M. Scheutzow, Asymptotic coupling and a general form of harris' theorem with applications to stochastic delay equations, Probability Theory and Related Fields, 149 (2011), 223-259.  doi: 10.1007/s00440-009-0250-6.  Google Scholar

[12]

A. Joets and R. Ribotta, Defects in Non-linear Waves in Convection, Nonlinear Coherent Structures, 353 (2005), 157-169.  doi: 10.1007/BFb0033633.  Google Scholar

[13]

S. Kuksin, Randomly forced CGL equation: Stationary measures and the inviscid limit, J.Phys. A, 37 (2004), 3805-3822.  doi: 10.1088/0305-4470/37/12/006.  Google Scholar

[14]

C. Odasso, Ergodicity for the stochastic complex Ginzburg-Landau equations, Ann. Inst. H. Poincaré Probab. Statist., 42 (2006), 417-454.  doi: 10.1016/j.anihpb.2005.06.002.  Google Scholar

[15]

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

[16]

M. Rochner and X. Zhang, Stochastic tamed 3D Navier-Stokes equations: Existence, uniqueness and ergodicity, Probab. Theory Related Fields, 145 (1009), 211-267.  doi: 10.1007/s00440-008-0167-5.  Google Scholar

[17]

D. Yang and Z. Hou, Large deviations for the stochastic derivative Ginzburg-Landau equation with multiplicative noise, Phys. D, 237 (2008), 82-91.  doi: 10.1016/j.physd.2007.08.015.  Google Scholar

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