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March  2017, 22(2): 605-625. doi: 10.3934/dcdsb.2017029

Ergodicity of the stochastic coupled fractional Ginzburg-Landau equations driven by α-stable noise

Tianlong Shen and  Jianhua Huang ,

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

* Corresponding author: Jianhua Huang

Received  December 2015 Revised  November 2016 Published  December 2016

The current paper is devoted to the ergodicity of stochastic coupled fractional Ginzburg-Landau equations driven by $α$-stable noise on the Torus $\mathbb{T}$. By the maximal inequality for stochastic $α$-stable convolution and commutator estimates, the well-posedness of the mild solution for stochastic coupled fractional Ginzburg-Landau equations is established. Due to the discontinuous trajectories and non-Lipschitz nonlinear term, the existence and uniqueness of the invariant measures are obtained by the strong Feller property and the accessibility to zero.

Keywords: Stochastic, coupled fractional Ginzburg-Landau equations, ergodicity, α-stable noise, invariant measure.
Mathematics Subject Classification: Primary: 35R60, 60H15; Secondary: 37A2.
Citation: 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
References:
[1]

S. Albeverio, J. Wu and T. Zhang, Parabolic SPDEs driven by Poissson white noise, Stoch. Proc. Appl., 74 (1998), 21-36.  doi: 10.1016/S0304-4149(97)00112-9.  Google Scholar

[2]

D. Applebaum, Lévy Processes and Stochastic Calculus in Combridge Studies in Advance Mathematics, Cambridge University Press, 2004. Google Scholar

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

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D. Dai, Z. Li and Z. Liu, Exact homoclinic wave and soliton solutions for the 2D Ginzburg-Landau equation, Physics Letters A, 372 (2008), 3010-3014.  doi: 10.1016/j.physleta.2008.01.015.  Google Scholar

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Z. Dong, L. Xu and X. Zhang, Exponential ergodicity of stochastic Burgers equations, J. Stat. Phys., 154 (2014), 929-949.  doi: 10.1007/s10955-013-0881-y.  Google Scholar

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Z. Dong, L. Xu and X. Zhang, Invariance measures of stochastic 2D Navier-Stokes equations driven by α-stable processes, Electron. Comm. Probab., 16 (2011), 678-688.  doi: 10.1214/ECP.v16-1664.  Google Scholar

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Z. Dong and Y. Xie, Ergodicity of stochastic 2D Navier-stokes equations with Lévy noise, J. Diff. Eqns., 251 (2011), 196-222.  doi: 10.1016/j.jde.2011.03.015.  Google Scholar

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J. Ginibre and G. Velo, The Cauchy problem in local spaces for the complex Ginzburg-Landau equation, Ⅰ. Compactness method, Physica D, 95 (1996), 191-228.  doi: 10.1016/0167-2789(96)00055-3.  Google Scholar

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J. Ginibre and G. Velo, The Cauchy problem in local spaces for the complex Ginzburg-Landau equation, Ⅱ. Contraction method, Commun. Math. Phys., 187 (1997), 45-79.  doi: 10.1007/s002200050129.  Google Scholar

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B. Guo, H. Huang and M. Jiang, Ginzberg-Landau Equation, Science Press, Beijing, 2002. Google Scholar

[12]

M. Hairer, Ergodicity theory for stochastic PDEs, 2008. Available from: http://www.haier.org/notes/Imperial.pdf. Google Scholar

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T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm, Pure Appl. Math., 41 (1998), 891-907.  doi: 10.1002/cpa.3160410704.  Google Scholar

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C. Kening, G. Ponce and L. Vega, Well-poesdness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347.  doi: 10.2307/2939277.  Google Scholar

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K. Porsezian, R. Murali and A. Malomed, Modulational instability in linearly coupled complex cubie-quintic Ginzberg-Landau equations, Chaos, Soliton and Fractals, 40 (2009), 1907-1913.   Google Scholar

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S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy noise: An Evolution Equation Approach, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, 2007. Google Scholar

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E. Priola and J. Zabcyzk, Structrual properties of semilinear SPDEs driven by cylindrical stable processes, Probab. Theorey Relat. Fields, 149 (2011), 97-137.  doi: 10.1007/s00440-009-0243-5.  Google Scholar

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E. Priola, L. Xu and J. Zabczyk, Exponential mixing for some SPDEs with L$\acute{e}$vy noise, Stoch.& Dynam., 11 (2011), 521-534.  doi: 10.1142/S0219493711003425.  Google Scholar

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E. Priola, A. Shirkyan, L. Xu and J. Zabczyk, Exponential ergodicity and regularity for equations with Lévy noise, Stoch. Proc. Appl., 122 (2012), 106-133.  doi: 10.1016/j.spa.2011.10.003.  Google Scholar

[20]

H. Sakaguchi and B. Malomed, Stable solitons in coupled Ginzberg-Landau equations describing Bose-Einstein condensates and nonlinear optical waveguides and cavities, PhySica D, 183 (2003), 282-292.  doi: 10.1016/S0167-2789(03)00181-7.  Google Scholar

[21]

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

[22]

E. Stein, Singular Integrals and Differentiablity Properties of Functions, Princeton University Press, 1970. Google Scholar

[23]

X. Sun and Y. Xie, Ergodicity of stochastic dissipative equations driven by α-stable process, Stoch. Anal. Appl., 32 (2014), 61-76.  doi: 10.1080/07362994.2013.843141.  Google Scholar

[24]

F. Wang, Gradient estimate for Ornstein-Uhlenbeck jump processes, Stoch. Proc. Appl., 121 (2011), 466-478.  doi: 10.1016/j.spa.2010.12.002.  Google Scholar

[25]

B. Wang, The limit behaviour for the Cauchy problem of the complex Ginzburg-Landau equation, Comm. Pure Appl. Math., 55 (2002), 481-508.  doi: 10.1002/cpa.10024.  Google Scholar

[26]

L. Xu, Ergodicity of the stochastic real Ginzburg-Landau equations driven by α-stable noises, Stoch. Proc. Appl., 123 (2013), 3710-3736.  doi: 10.1016/j.spa.2013.05.002.  Google Scholar

[27]

L. Xu, Exponential mixing of 2D SDEs forced by degenerate Lévy noise, J. Evol. Equ., 14 (2014), 249-272.  doi: 10.1007/s00028-013-0212-4.  Google Scholar

show all references

References:
[1]

S. Albeverio, J. Wu and T. Zhang, Parabolic SPDEs driven by Poissson white noise, Stoch. Proc. Appl., 74 (1998), 21-36.  doi: 10.1016/S0304-4149(97)00112-9.  Google Scholar

[2]

D. Applebaum, Lévy Processes and Stochastic Calculus in Combridge Studies in Advance Mathematics, Cambridge University Press, 2004. Google Scholar

[3]

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

[4]

D. Dai, Z. Li and Z. Liu, Exact homoclinic wave and soliton solutions for the 2D Ginzburg-Landau equation, Physics Letters A, 372 (2008), 3010-3014.  doi: 10.1016/j.physleta.2008.01.015.  Google Scholar

[5]

Z. Dai and M. Jiang, Exponential attractors of the Ginzburg-Landau-BBM equations, J. Math. Res. Expo., 21 (2001), 317-322.   Google Scholar

[6]

Z. Dong, L. Xu and X. Zhang, Exponential ergodicity of stochastic Burgers equations, J. Stat. Phys., 154 (2014), 929-949.  doi: 10.1007/s10955-013-0881-y.  Google Scholar

[7]

Z. Dong, L. Xu and X. Zhang, Invariance measures of stochastic 2D Navier-Stokes equations driven by α-stable processes, Electron. Comm. Probab., 16 (2011), 678-688.  doi: 10.1214/ECP.v16-1664.  Google Scholar

[8]

Z. Dong and Y. Xie, Ergodicity of stochastic 2D Navier-stokes equations with Lévy noise, J. Diff. Eqns., 251 (2011), 196-222.  doi: 10.1016/j.jde.2011.03.015.  Google Scholar

[9]

J. Ginibre and G. Velo, The Cauchy problem in local spaces for the complex Ginzburg-Landau equation, Ⅰ. Compactness method, Physica D, 95 (1996), 191-228.  doi: 10.1016/0167-2789(96)00055-3.  Google Scholar

[10]

J. Ginibre and G. Velo, The Cauchy problem in local spaces for the complex Ginzburg-Landau equation, Ⅱ. Contraction method, Commun. Math. Phys., 187 (1997), 45-79.  doi: 10.1007/s002200050129.  Google Scholar

[11]

B. Guo, H. Huang and M. Jiang, Ginzberg-Landau Equation, Science Press, Beijing, 2002. Google Scholar

[12]

M. Hairer, Ergodicity theory for stochastic PDEs, 2008. Available from: http://www.haier.org/notes/Imperial.pdf. Google Scholar

[13]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm, Pure Appl. Math., 41 (1998), 891-907.  doi: 10.1002/cpa.3160410704.  Google Scholar

[14]

C. Kening, G. Ponce and L. Vega, Well-poesdness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347.  doi: 10.2307/2939277.  Google Scholar

[15]

K. Porsezian, R. Murali and A. Malomed, Modulational instability in linearly coupled complex cubie-quintic Ginzberg-Landau equations, Chaos, Soliton and Fractals, 40 (2009), 1907-1913.   Google Scholar

[16]

S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy noise: An Evolution Equation Approach, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, 2007. Google Scholar

[17]

E. Priola and J. Zabcyzk, Structrual properties of semilinear SPDEs driven by cylindrical stable processes, Probab. Theorey Relat. Fields, 149 (2011), 97-137.  doi: 10.1007/s00440-009-0243-5.  Google Scholar

[18]

E. Priola, L. Xu and J. Zabczyk, Exponential mixing for some SPDEs with L$\acute{e}$vy noise, Stoch.& Dynam., 11 (2011), 521-534.  doi: 10.1142/S0219493711003425.  Google Scholar

[19]

E. Priola, A. Shirkyan, L. Xu and J. Zabczyk, Exponential ergodicity and regularity for equations with Lévy noise, Stoch. Proc. Appl., 122 (2012), 106-133.  doi: 10.1016/j.spa.2011.10.003.  Google Scholar

[20]

H. Sakaguchi and B. Malomed, Stable solitons in coupled Ginzberg-Landau equations describing Bose-Einstein condensates and nonlinear optical waveguides and cavities, PhySica D, 183 (2003), 282-292.  doi: 10.1016/S0167-2789(03)00181-7.  Google Scholar

[21]

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

[22]

E. Stein, Singular Integrals and Differentiablity Properties of Functions, Princeton University Press, 1970. Google Scholar

[23]

X. Sun and Y. Xie, Ergodicity of stochastic dissipative equations driven by α-stable process, Stoch. Anal. Appl., 32 (2014), 61-76.  doi: 10.1080/07362994.2013.843141.  Google Scholar

[24]

F. Wang, Gradient estimate for Ornstein-Uhlenbeck jump processes, Stoch. Proc. Appl., 121 (2011), 466-478.  doi: 10.1016/j.spa.2010.12.002.  Google Scholar

[25]

B. Wang, The limit behaviour for the Cauchy problem of the complex Ginzburg-Landau equation, Comm. Pure Appl. Math., 55 (2002), 481-508.  doi: 10.1002/cpa.10024.  Google Scholar

[26]

L. Xu, Ergodicity of the stochastic real Ginzburg-Landau equations driven by α-stable noises, Stoch. Proc. Appl., 123 (2013), 3710-3736.  doi: 10.1016/j.spa.2013.05.002.  Google Scholar

[27]

L. Xu, Exponential mixing of 2D SDEs forced by degenerate Lévy noise, J. Evol. Equ., 14 (2014), 249-272.  doi: 10.1007/s00028-013-0212-4.  Google Scholar

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