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Exponential convergence for the 3D stochastic cubic Ginzburg-Landau equation with degenerate noise
College of Science, National University of Defense Technology, Changsha 410073, China |
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.
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. |
[2] |
Coullet, Elphick, Gil and Lega,
Topological defects of wave patterns, Phys. Rev. Lett., 59 (1987), 884-887.
doi: 10.1103/PhysRevLett.59.884. |
[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. |
[4] |
G. Da Parto and J. Zabcyzk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, 1996.
doi: 10.1017/CBO9780511662829.![]() ![]() |
[5] |
G. Da Parto and J. Zabcyzk, Stochastic Equations in Infinite Dimensionals, Cambridge University Press, Cambridge, 1992.
doi: 10.1017/CBO9780511666223.![]() ![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[12] |
A. Joets and R. Ribotta,
Defects in Non-linear Waves in Convection, Nonlinear Coherent Structures, 353 (2005), 157-169.
doi: 10.1007/BFb0033633. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[2] |
Coullet, Elphick, Gil and Lega,
Topological defects of wave patterns, Phys. Rev. Lett., 59 (1987), 884-887.
doi: 10.1103/PhysRevLett.59.884. |
[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. |
[4] |
G. Da Parto and J. Zabcyzk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, 1996.
doi: 10.1017/CBO9780511662829.![]() ![]() |
[5] |
G. Da Parto and J. Zabcyzk, Stochastic Equations in Infinite Dimensionals, Cambridge University Press, Cambridge, 1992.
doi: 10.1017/CBO9780511666223.![]() ![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[12] |
A. Joets and R. Ribotta,
Defects in Non-linear Waves in Convection, Nonlinear Coherent Structures, 353 (2005), 157-169.
doi: 10.1007/BFb0033633. |
[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. |
[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. |
[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. |
[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. |
[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. |
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