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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 & 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

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

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

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

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

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

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

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[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.  Google Scholar

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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. Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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. Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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