September  2020, 25(9): 3553-3576. doi: 10.3934/dcdsb.2020072

Dynamics of non-autonomous fractional Ginzburg-Landau equations driven by colored noise

1. 

School of Mathematics and Statistics, Shandong University, Weihai, Shandong 264209, China

2. 

College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, China

3. 

Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA

* Corresponding author: Mingji Zhang

Received  August 2019 Published  September 2020 Early access  April 2020

Fund Project: This work was supported by the NSF of China (No. 11601278 and No.11601274), the NSF of Shandong Province (No. ZR2019MA050) and MPS Simons Foundation of USA (No. 628308)

In this work, the existence and uniqueness of random attractors of a class of non-autonomous non-local fractional stochastic Ginzburg-Landau equation driven by colored noise with a nonlinear diffusion term is established. We comment that compared to white noise, the colored noise is much easier to handle in examining the pathwise dynamics of stochastic systems. Additionally, we prove the attractors of the random fractional Ginzburg-Landau system driven by a linear multiplicative colored noise converge to those of the corresponding stochastic system driven by a linear multiplicative white noise.

Citation: Hong Lu, Mingji Zhang. Dynamics of non-autonomous fractional Ginzburg-Landau equations driven by colored noise. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3553-3576. doi: 10.3934/dcdsb.2020072
References:
[1]

S. Abe and S. Thurner, Anomalous diffusion in view of Einsteins 1905 theory of Brownian motion, Physica A, 356 (2005), 403-407. 

[2]

L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley-Interscience, New York, 1974.

[3]

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

[4]

L. CaffarelliJ. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151-1179.  doi: 10.4171/JEMS/226.

[5]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[6]

J. Doob, The Brownian movement and stochastic equations, Annals of Math., 43 (1942), 351-369.  doi: 10.2307/1968873.

[7]

C. Gal and M. Warma, Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions, Discrete Contin. Dyn. Syst. Ser. A, 36 (2016), 1279-1319.  doi: 10.3934/dcds.2016.36.1279.

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A. Garroni and S. Muller, A variational model for dislocations in the line tension limit, Arch. Ration. Mech. Anal., 181 (2006), 535-578.  doi: 10.1007/s00205-006-0432-7.

[9] W. GersterW. KistlerR. Naud and L. Paninski, Neuronal Dynamics: From Single Neurons to Networks and Models of Cognition, Cambridge University Press, Cambridge, 2014. 
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A. Gu and B. Wang, Asyptotic behavior of random Fitzhugh-Nagumo systems deriven by colored noise, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1689-1720.  doi: 10.3934/dcdsb.2018072.

[11]

Q. Guan, Integration by parts formula for regional fractional Laplacian, Comm. Math. Phys., 266 (2006), 289-329.  doi: 10.1007/s00220-006-0054-9.

[12]

Q. Guan and Z. Ma, Reflected symmetric $\alpha$-stable processes and regional fractional Laplacian, Probab. Theory Related Fields, 134 (2006), 649-694.  doi: 10.1007/s00440-005-0438-3.

[13]

Q. Guan and Z. Ma, Boundary problems for fractional Laplacians, Stoch. Dyn., 5 (2005), 385-424.  doi: 10.1142/S021949370500150X.

[14]

M. Jara, Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps, Comm. Pure Appl. Math., 62 (2009), 198-214.  doi: 10.1002/cpa.20253.

[15]

T. JiangX. Liu and J. Duan, Approximation for random stable manifolds under multiplicative correlated noise, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3163-3174.  doi: 10.3934/dcdsb.2016091.

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N. van Kampen, Stochastic Processes in Physics and Chemistry, Amsterdam-New York, 1981.

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M. Klosek-DygasB. Matkowsky and Z. Schuss, Colored noise in dynamical systems, SIAM J. Appl. Math., 48 (1988), 425-441.  doi: 10.1137/0148023.

[18]

M. KoslowskiA. Cuitino and M. Ortiz, A phasefield theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystal, J. Mech. Phys. Solids, 50 (2002), 2597-2635.  doi: 10.1016/S0022-5096(02)00037-6.

[19]

H. LuP. W. BatesS. Lu and M. Zhang, Dynamics of 3D fractional complex Ginzburg-Landau equation, J. Differential Equations, 259 (2015), 5276-5301.  doi: 10.1016/j.jde.2015.06.028.

[20]

H. LuP. W. BatesJ. Xin and M. Zhang, Asymptotic behavior of stochastic fractional power dissipative equations on $ \mathbb{R}^n$, Nonlinear Anal., 128 (2015), 176-198.  doi: 10.1016/j.na.2015.06.033.

[21]

H. LuP. W. BatesS. Lu and M. Zhang, Dynamics of the 3D fractional Ginzburg-Landau equation with multiplicative noise on an unbounded domain, Comm. Math. Sci., 14 (2016), 273-295.  doi: 10.4310/CMS.2016.v14.n1.a11.

[22]

H. LuS. Lv and M. Zhang, Fourier spectral approximation to the dynamical behavior of 3D fractional Ginzburg-Landau equation, Discrete Contin. Dyn. Syst. Ser. A, 37 (2017), 2539-2564.  doi: 10.3934/dcds.2017109.

[23]

H. Lu and M. Zhang, The spectral method for long-time behavior of a fractional power dissipative system., Taiwanese J. Math., 22 (2018), 453-483.  doi: 10.11650/tjm/170902.

[24]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162. 

[25]

X. Pu and B. Guo, Well-posedness and dynamics for the fractional Ginzburg-Landau equation, Applicable Analysis, 92 (2013), 318-334.  doi: 10.1080/00036811.2011.614601.

[26] L. RidolfiP. D'Odorico and F. Laio, Noise-Induced Phenomena in the Environmental Sciences, Cambridge University Press, New York, 2011.  doi: 10.1017/CBO9780511984730.
[27]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, Journal de Mathematiques Pures et Appliquees, 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.

[28]

R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 831-855.  doi: 10.1017/S0308210512001783.

[29]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst. Ser. A, 33 (2013), 2105-2137.  doi: 10.3934/dcds.2013.33.2105.

[30]

G. Uhlenbeck and L. Ornstein, On the theory of Brownian motion, Phys. Rev., 36 (1930), 823-841.  doi: 10.1103/PhysRev.36.823.

[31]

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

[32]

M. Wang and G. Uhlenbeck, On the theory of Brownian motion. Ⅱ, Rev. Modern Phys., 17 (1945), 323-342.  doi: 10.1103/RevModPhys.17.323.

show all references

References:
[1]

S. Abe and S. Thurner, Anomalous diffusion in view of Einsteins 1905 theory of Brownian motion, Physica A, 356 (2005), 403-407. 

[2]

L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley-Interscience, New York, 1974.

[3]

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

[4]

L. CaffarelliJ. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151-1179.  doi: 10.4171/JEMS/226.

[5]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[6]

J. Doob, The Brownian movement and stochastic equations, Annals of Math., 43 (1942), 351-369.  doi: 10.2307/1968873.

[7]

C. Gal and M. Warma, Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions, Discrete Contin. Dyn. Syst. Ser. A, 36 (2016), 1279-1319.  doi: 10.3934/dcds.2016.36.1279.

[8]

A. Garroni and S. Muller, A variational model for dislocations in the line tension limit, Arch. Ration. Mech. Anal., 181 (2006), 535-578.  doi: 10.1007/s00205-006-0432-7.

[9] W. GersterW. KistlerR. Naud and L. Paninski, Neuronal Dynamics: From Single Neurons to Networks and Models of Cognition, Cambridge University Press, Cambridge, 2014. 
[10]

A. Gu and B. Wang, Asyptotic behavior of random Fitzhugh-Nagumo systems deriven by colored noise, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1689-1720.  doi: 10.3934/dcdsb.2018072.

[11]

Q. Guan, Integration by parts formula for regional fractional Laplacian, Comm. Math. Phys., 266 (2006), 289-329.  doi: 10.1007/s00220-006-0054-9.

[12]

Q. Guan and Z. Ma, Reflected symmetric $\alpha$-stable processes and regional fractional Laplacian, Probab. Theory Related Fields, 134 (2006), 649-694.  doi: 10.1007/s00440-005-0438-3.

[13]

Q. Guan and Z. Ma, Boundary problems for fractional Laplacians, Stoch. Dyn., 5 (2005), 385-424.  doi: 10.1142/S021949370500150X.

[14]

M. Jara, Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps, Comm. Pure Appl. Math., 62 (2009), 198-214.  doi: 10.1002/cpa.20253.

[15]

T. JiangX. Liu and J. Duan, Approximation for random stable manifolds under multiplicative correlated noise, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3163-3174.  doi: 10.3934/dcdsb.2016091.

[16]

N. van Kampen, Stochastic Processes in Physics and Chemistry, Amsterdam-New York, 1981.

[17]

M. Klosek-DygasB. Matkowsky and Z. Schuss, Colored noise in dynamical systems, SIAM J. Appl. Math., 48 (1988), 425-441.  doi: 10.1137/0148023.

[18]

M. KoslowskiA. Cuitino and M. Ortiz, A phasefield theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystal, J. Mech. Phys. Solids, 50 (2002), 2597-2635.  doi: 10.1016/S0022-5096(02)00037-6.

[19]

H. LuP. W. BatesS. Lu and M. Zhang, Dynamics of 3D fractional complex Ginzburg-Landau equation, J. Differential Equations, 259 (2015), 5276-5301.  doi: 10.1016/j.jde.2015.06.028.

[20]

H. LuP. W. BatesJ. Xin and M. Zhang, Asymptotic behavior of stochastic fractional power dissipative equations on $ \mathbb{R}^n$, Nonlinear Anal., 128 (2015), 176-198.  doi: 10.1016/j.na.2015.06.033.

[21]

H. LuP. W. BatesS. Lu and M. Zhang, Dynamics of the 3D fractional Ginzburg-Landau equation with multiplicative noise on an unbounded domain, Comm. Math. Sci., 14 (2016), 273-295.  doi: 10.4310/CMS.2016.v14.n1.a11.

[22]

H. LuS. Lv and M. Zhang, Fourier spectral approximation to the dynamical behavior of 3D fractional Ginzburg-Landau equation, Discrete Contin. Dyn. Syst. Ser. A, 37 (2017), 2539-2564.  doi: 10.3934/dcds.2017109.

[23]

H. Lu and M. Zhang, The spectral method for long-time behavior of a fractional power dissipative system., Taiwanese J. Math., 22 (2018), 453-483.  doi: 10.11650/tjm/170902.

[24]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162. 

[25]

X. Pu and B. Guo, Well-posedness and dynamics for the fractional Ginzburg-Landau equation, Applicable Analysis, 92 (2013), 318-334.  doi: 10.1080/00036811.2011.614601.

[26] L. RidolfiP. D'Odorico and F. Laio, Noise-Induced Phenomena in the Environmental Sciences, Cambridge University Press, New York, 2011.  doi: 10.1017/CBO9780511984730.
[27]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, Journal de Mathematiques Pures et Appliquees, 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.

[28]

R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 831-855.  doi: 10.1017/S0308210512001783.

[29]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst. Ser. A, 33 (2013), 2105-2137.  doi: 10.3934/dcds.2013.33.2105.

[30]

G. Uhlenbeck and L. Ornstein, On the theory of Brownian motion, Phys. Rev., 36 (1930), 823-841.  doi: 10.1103/PhysRev.36.823.

[31]

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

[32]

M. Wang and G. Uhlenbeck, On the theory of Brownian motion. Ⅱ, Rev. Modern Phys., 17 (1945), 323-342.  doi: 10.1103/RevModPhys.17.323.

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