American Institute of Mathematical Sciences

Advanced
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]

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

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

[1]

Tianlong Shen, Jianhua Huang. Ergodicity of the stochastic coupled fractional Ginzburg-Landau equations driven by α-stable noise. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 605-625. doi: 10.3934/dcdsb.2017029
[2]

Yun Lan, Ji Shu. Dynamics of non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2409-2431. doi: 10.3934/cpaa.2019109
[3]

Shujuan Lü, Hong Lu, Zhaosheng Feng. Stochastic dynamics of 2D fractional Ginzburg-Landau equation with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 575-590. doi: 10.3934/dcdsb.2016.21.575
[4]

Hans G. Kaper, Bixiang Wang, Shouhong Wang. Determining nodes for the Ginzburg-Landau equations of superconductivity. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 205-224. doi: 10.3934/dcds.1998.4.205
[5]

Dingshi Li, Xiaohu Wang. Asymptotic behavior of stochastic complex Ginzburg-Landau equations with deterministic non-autonomous forcing on thin domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 449-465. doi: 10.3934/dcdsb.2018181
[6]

Dingshi Li, Lin Shi, Xiaohu Wang. Long term behavior of stochastic discrete complex Ginzburg-Landau equations with time delays in weighted spaces. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5121-5148. doi: 10.3934/dcdsb.2019046
[7]

Alessia Berti, Valeria Berti, Ivana Bochicchio. Global and exponential attractors for a Ginzburg-Landau model of superfluidity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 247-271. doi: 10.3934/dcdss.2011.4.247
[8]

Dmitry Glotov, P. J. McKenna. Numerical mountain pass solutions of Ginzburg-Landau type equations. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1345-1359. doi: 10.3934/cpaa.2008.7.1345
[9]

Dmitry Turaev, Sergey Zelik. Analytical proof of space-time chaos in Ginzburg-Landau equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1713-1751. doi: 10.3934/dcds.2010.28.1713
[10]

Noboru Okazawa, Tomomi Yokota. Smoothing effect for generalized complex Ginzburg-Landau equations in unbounded domains. Conference Publications, 2001, 2001 (Special) : 280-288. doi: 10.3934/proc.2001.2001.280
[11]

N. I. Karachalios, H. E. Nistazakis, A. N. Yannacopoulos. Remarks on the asymptotic behavior of solutions of complex discrete Ginzburg-Landau equations. Conference Publications, 2005, 2005 (Special) : 476-486. doi: 10.3934/proc.2005.2005.476
[12]

Yuta Kugo, Motohiro Sobajima, Toshiyuki Suzuki, Tomomi Yokota, Kentarou Yoshii. Solvability of a class of complex Ginzburg-Landau equations in periodic Sobolev spaces. Conference Publications, 2015, 2015 (special) : 754-763. doi: 10.3934/proc.2015.0754
[13]

Bixiang Wang, Shouhong Wang. Gevrey class regularity for the solutions of the Ginzburg-Landau equations of superconductivity. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 507-522. doi: 10.3934/dcds.1998.4.507
[14]

Gregory A. Chechkin, Vladimir V. Chepyzhov, Leonid S. Pankratov. Homogenization of trajectory attractors of Ginzburg-Landau equations with randomly oscillating terms. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1133-1154. doi: 10.3934/dcdsb.2018145
[15]

Kolade M. Owolabi, Edson Pindza. Numerical simulation of multidimensional nonlinear fractional Ginzburg-Landau equations. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 835-851. doi: 10.3934/dcdss.2020048
[16]

Mickaël Dos Santos, Oleksandr Misiats. Ginzburg-Landau model with small pinning domains. Networks & Heterogeneous Media, 2011, 6 (4) : 715-753. doi: 10.3934/nhm.2011.6.715
[17]

Fanghua Lin, Ping Zhang. On the hydrodynamic limit of Ginzburg-Landau vortices. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 121-142. doi: 10.3934/dcds.2000.6.121
[18]

N. I. Karachalios, Hector E. Nistazakis, Athanasios N. Yannacopoulos. Asymptotic behavior of solutions of complex discrete evolution equations: The discrete Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 711-736. doi: 10.3934/dcds.2007.19.711
[19]

Iuliana Oprea, Gerhard Dangelmayr. A period doubling route to spatiotemporal chaos in a system of Ginzburg-Landau equations for nematic electroconvection. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 273-296. doi: 10.3934/dcdsb.2018095
[20]

Leonid Berlyand, Volodymyr Rybalko, Nung Kwan Yip. Renormalized Ginzburg-Landau energy and location of near boundary vortices. Networks & Heterogeneous Media, 2012, 7 (1) : 179-196. doi: 10.3934/nhm.2012.7.179

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (61)
  • HTML views (336)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences