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December  2016, 9(6): 1701-1715. doi: 10.3934/dcdss.2016071

Approximation of random invariant manifolds for a stochastic Swift-Hohenberg equation

Yanfeng Guo 1, , Jinqiao Duan 2, and  Donglong Li 1,

1. 

School of Science, Guangxi University of Science and Technology, Liuzhou, Guangxi 545006, China, China
2. 

Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616

Received  June 2015 Revised  September 2016 Published  November 2016

Random invariant manifolds are considered for a stochastic Swift-Hohenberg equation with multiplicative noise in the Stratonovich sense. Using a stochastic transformation and a technique of cut-off function, existence of random invariant manifolds and attracting property of the corresponding random dynamical system are obtained by Lyaponov-Perron method. Then in the sense of large probability, an approximation of invariant manifolds has been investigated and this is further used to describe the geometric shape of the invariant manifolds.
Keywords: stochastic transformation, Stochastic partial differential equations, approximation of invariant manifolds, Lyaponov-Perron method., random invariant manifolds.
Mathematics Subject Classification: Primary: 60H15, 37H05; Secondary: 37L55, 37L2.
Citation: Yanfeng Guo, Jinqiao Duan, Donglong Li. Approximation of random invariant manifolds for a stochastic Swift-Hohenberg equation. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1701-1715. doi: 10.3934/dcdss.2016071
References:
[1]

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

[2]

D. Blömker, M. Hairer and G. A. Pavliotis, Stochastic Swift-Hohenberg equaion near a change of stability, Proceedings of Equadiff, 11 (2005), 27-37.

[3]

D. Blömker and W. Wang, Qualitative properties of local random invariant manifolds for SPDE with quadratic nonlinearity, J. Dyn. Differ. Equ., 22 (2010), 677-695. doi: 10.1007/s10884-009-9145-6.

[4]

D. Blömker, Amplitude Equations for Stochastic Partial Differential Equations, vol.3 of Interdisciplinary Mathematical Sciences. World Scientific Publishing, Singapore, 2007. doi: 10.1142/9789812770608.

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G. Chen, J. Duan and J. Zhang, Geometric shape of invariant manifolds for a class of stochastic partial differential equations, J. Math. Phys., 52 (2011), 072702, 14pp. doi: 10.1063/1.3614777.

[6]

M. D. Chekroun, H. H. Liu and S. H. Wang, Stochastic Parameterizing Manifolds and non-Markovian Reduced Equations-Stochastic Manifolds for Nonlinear SPDEs II, Springer Briefs in Mathematics, Springer-Verlag, New York, 2015. doi: 10.1007/978-3-319-12520-6.

[7]

I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, New York, 2002. doi: 10.1007/b83277.

[8]

H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Differ. Equ., 9 (1997), 307-341. doi: 10.1007/BF02219225.

[9]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Rel. Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.

[10]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, New York, 1992. doi: 10.1017/CBO9780511666223.

[11]

J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic differential equations, Ann. Probab., 31 (2003), 2109-2135. doi: 10.1214/aop/1068646380.

[12]

J. Duan and K. Lu, Smooth stable and unstable manifolds for stoahcstic partial differential equations, J. Dyn. Diff. Equ., 16 (2004), 949-972. doi: 10.1007/s10884-004-7830-z.

[13]

J. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations, Elsevier, London, 2014.

[14]

J. P. Eckmann and C. E. Wayne, Propagating fronts and the center manifold theorem, Commun. Math. Phys, 136 (1991), 285-307. doi: 10.1007/BF02100026.

[15]

H. Fu, X. Liu and J. Duan, Slow manifolds for multi-time-scale stochastic evolutionary systems, Commun. Math. Sci., 11 (2013), 141-162. doi: 10.4310/CMS.2013.v11.n1.a5.

[16]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Varlag, Berlin, 1981.

[17]

D. Y. Hsieh, A. Q. Tang and X. P. Wang, On hydrodynamics instabilities, chaos, and phase transition, Acta Mech. Sin., 12 (1996), 1-14. doi: 10.1007/BF02486757.

[18]

M. F. Hilali, S. Metens, P. Borckmans and G. Dewel, Pattern selection in the generalized Swift-Hohenberg equation, Phys. Rev. E, 51 (1995), 2046-2052.

[19]

P. Imkeller and A. Monahan, Conceptual stochastic climate dynamics, Stoch. Dynam., 2 (2002), 311-326.

[20]

G. Lin, H. Gao, J. Duan and V. J. Ervin, Asymptotic dynamical difference between the nonlocal and local Swift-Hohenberg models, J. Math. Phys., 41 (2000), 2077-2089. doi: 10.1063/1.533228.

[21]

A. Mielke and G. Schneider, Attractors for modulation equations on unbounded domains: Existence and comparison, Nonlinearity, 8 (1995), 734-768. doi: 10.1088/0951-7715/8/5/006.

[22]

J. Oh, J. M. Ortiz de Zárate, J. V. Sengers and G. Ahlers, Dynamics of fluctuations in a fluid below the onset of Rayleigh-Bnárd convection, Phys. Rev. E, 69 (2004), 021106.

[23]

J. Oh and G. Ahlers, Thermal-noise effect on the transition to Rayleigh-Bnárd convection, Phys. Rev. Lett., 91 (2003), 094501.

[24]

A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer-Varlag, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1.

[25]

I. Rehberg, S. Rasenat, M. de la Torre Juárez, W. Schöpf, F. Hörner, G. Ahlers and H. R. Brand, Thermally induced hydrodynamic fluctuations below the onset of electroconvection, Phys. Rev. Lett., 67 (1991), 596-599. doi: 10.1103/PhysRevLett.67.596.

[26]

X. Sun, J. Duan and X. Li, An impact of noise on invariant manifolds in nonlinear dynamical systems, J. Math. Phys., 51 (2010), 042702, 12pp. doi: 10.1063/1.3371010.

[27]

J. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 15 (1977), 319-328. doi: 10.1103/PhysRevA.15.319.

[28]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York Berlin Heidelberg, 1988. doi: 10.1007/978-1-4684-0313-8.

[29]

W. Wang and J. Duan, A dynamical approximation for stochastic partial differential equations, J. Math. Phys., 48 (2007), 102701, 14 pp. doi: 10.1063/1.2800164.

[30]

W. Wang, J. Sun and J. Duan, Ergodic dynamics of the stochastic Swift-Hohenberg system, Nonlinear Analysis: Real World Appl., 6 (2005), 273-295. doi: 10.1016/j.nonrwa.2004.08.009.

[31]

E. Waymire and J. Duan, Probability and Partial Differential Equations in Modern Applied Mathematics, IMA, Vol. 140, Springer, New York, 2005. doi: 10.1007/978-0-387-29371-4.

show all references

References:
[1]

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

[2]

D. Blömker, M. Hairer and G. A. Pavliotis, Stochastic Swift-Hohenberg equaion near a change of stability, Proceedings of Equadiff, 11 (2005), 27-37.

[3]

D. Blömker and W. Wang, Qualitative properties of local random invariant manifolds for SPDE with quadratic nonlinearity, J. Dyn. Differ. Equ., 22 (2010), 677-695. doi: 10.1007/s10884-009-9145-6.

[4]

D. Blömker, Amplitude Equations for Stochastic Partial Differential Equations, vol.3 of Interdisciplinary Mathematical Sciences. World Scientific Publishing, Singapore, 2007. doi: 10.1142/9789812770608.

[5]

G. Chen, J. Duan and J. Zhang, Geometric shape of invariant manifolds for a class of stochastic partial differential equations, J. Math. Phys., 52 (2011), 072702, 14pp. doi: 10.1063/1.3614777.

[6]

M. D. Chekroun, H. H. Liu and S. H. Wang, Stochastic Parameterizing Manifolds and non-Markovian Reduced Equations-Stochastic Manifolds for Nonlinear SPDEs II, Springer Briefs in Mathematics, Springer-Verlag, New York, 2015. doi: 10.1007/978-3-319-12520-6.

[7]

I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, New York, 2002. doi: 10.1007/b83277.

[8]

H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Differ. Equ., 9 (1997), 307-341. doi: 10.1007/BF02219225.

[9]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Rel. Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.

[10]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, New York, 1992. doi: 10.1017/CBO9780511666223.

[11]

J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic differential equations, Ann. Probab., 31 (2003), 2109-2135. doi: 10.1214/aop/1068646380.

[12]

J. Duan and K. Lu, Smooth stable and unstable manifolds for stoahcstic partial differential equations, J. Dyn. Diff. Equ., 16 (2004), 949-972. doi: 10.1007/s10884-004-7830-z.

[13]

J. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations, Elsevier, London, 2014.

[14]

J. P. Eckmann and C. E. Wayne, Propagating fronts and the center manifold theorem, Commun. Math. Phys, 136 (1991), 285-307. doi: 10.1007/BF02100026.

[15]

H. Fu, X. Liu and J. Duan, Slow manifolds for multi-time-scale stochastic evolutionary systems, Commun. Math. Sci., 11 (2013), 141-162. doi: 10.4310/CMS.2013.v11.n1.a5.

[16]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Varlag, Berlin, 1981.

[17]

D. Y. Hsieh, A. Q. Tang and X. P. Wang, On hydrodynamics instabilities, chaos, and phase transition, Acta Mech. Sin., 12 (1996), 1-14. doi: 10.1007/BF02486757.

[18]

M. F. Hilali, S. Metens, P. Borckmans and G. Dewel, Pattern selection in the generalized Swift-Hohenberg equation, Phys. Rev. E, 51 (1995), 2046-2052.

[19]

P. Imkeller and A. Monahan, Conceptual stochastic climate dynamics, Stoch. Dynam., 2 (2002), 311-326.

[20]

G. Lin, H. Gao, J. Duan and V. J. Ervin, Asymptotic dynamical difference between the nonlocal and local Swift-Hohenberg models, J. Math. Phys., 41 (2000), 2077-2089. doi: 10.1063/1.533228.

[21]

A. Mielke and G. Schneider, Attractors for modulation equations on unbounded domains: Existence and comparison, Nonlinearity, 8 (1995), 734-768. doi: 10.1088/0951-7715/8/5/006.

[22]

J. Oh, J. M. Ortiz de Zárate, J. V. Sengers and G. Ahlers, Dynamics of fluctuations in a fluid below the onset of Rayleigh-Bnárd convection, Phys. Rev. E, 69 (2004), 021106.

[23]

J. Oh and G. Ahlers, Thermal-noise effect on the transition to Rayleigh-Bnárd convection, Phys. Rev. Lett., 91 (2003), 094501.

[24]

A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer-Varlag, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1.

[25]

I. Rehberg, S. Rasenat, M. de la Torre Juárez, W. Schöpf, F. Hörner, G. Ahlers and H. R. Brand, Thermally induced hydrodynamic fluctuations below the onset of electroconvection, Phys. Rev. Lett., 67 (1991), 596-599. doi: 10.1103/PhysRevLett.67.596.

[26]

X. Sun, J. Duan and X. Li, An impact of noise on invariant manifolds in nonlinear dynamical systems, J. Math. Phys., 51 (2010), 042702, 12pp. doi: 10.1063/1.3371010.

[27]

J. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 15 (1977), 319-328. doi: 10.1103/PhysRevA.15.319.

[28]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York Berlin Heidelberg, 1988. doi: 10.1007/978-1-4684-0313-8.

[29]

W. Wang and J. Duan, A dynamical approximation for stochastic partial differential equations, J. Math. Phys., 48 (2007), 102701, 14 pp. doi: 10.1063/1.2800164.

[30]

W. Wang, J. Sun and J. Duan, Ergodic dynamics of the stochastic Swift-Hohenberg system, Nonlinear Analysis: Real World Appl., 6 (2005), 273-295. doi: 10.1016/j.nonrwa.2004.08.009.

[31]

E. Waymire and J. Duan, Probability and Partial Differential Equations in Modern Applied Mathematics, IMA, Vol. 140, Springer, New York, 2005. doi: 10.1007/978-0-387-29371-4.

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