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doi: 10.3934/dcdsb.2021020
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Limiting behavior of unstable manifolds for spdes in varying phase spaces

Lin Shi 1,, , Dingshi Li 1, and  Kening Lu 2,

1. 

School of Mathematics, Southwest Jiaotong University, Chengdu 610031, China
2. 

Department of Mathematics, Brigham Young University, Provo, Utah 84602, USA

* Corresponding author: Lin Shi

Received  June 2020 Early access January 2021

Fund Project: The first author is supported by National Natural Science Foundation of China (NO. 11701475, NO. 12071384, NO. 11971394 and NO. 11971330)

In this paper, we study a class of singularly perturbed stochastic partial differential equations in terms of the phase spaces. We establish the smooth convergence of unstable manifolds of these equations. As an example, we study the stochastic reaction-diffusion equations on thin domains.

Keywords: Stochastic partial differential equation, random dynamical systems, unstable manifold, and thin domain.
Mathematics Subject Classification: Primary: 37D45, 37C40.
Citation: Lin Shi, Dingshi Li, Kening Lu. Limiting behavior of unstable manifolds for spdes in varying phase spaces. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021020
References:
[1]

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

[2]

J. M. Arrieta and E. Santamaría, Estimates on the distance of inertial manifolds, Discrete Contin. Dyn. Syst., 34 (2014), 3921-3944.  doi: 10.3934/dcds.2014.34.3921.  Google Scholar

[3]

P. W. Bates, K. Lu and C. Zeng, Persistence of overflowing manifolds for semiflow, Comm. Pure Appl. Math., 52 (1999), 983-1046.  doi: 10.1002/(SICI)1097-0312(199908)52:8<983::AID-CPA4>3.0.CO;2-O.  Google Scholar

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P. W. Bates, K. Lu and C. Zeng, Existence and persistence of invariant manifolds for semiflows in Banach space, Mem. Amer. Math. Soc., 135 (1998), 645. doi: 10.1090/memo/0645.  Google Scholar

[5]

P. W. Bates, K. Lu and C. Zeng, Approximately invariant manifolds and global dynamics of spike states, Invent. Math., 174 (2008), 355-433.  doi: 10.1007/s00222-008-0141-y.  Google Scholar

[6]

A. Bensoussan and F. Flandoli, Stochastic inertial manifold, Stochastics Rep., 53 (1995), 13-39.  doi: 10.1080/17442509508833981.  Google Scholar

[7]

T. Caraballo, I. Chueshov and J. A. Langa, Existence of invariant manifolds for coupled parabolic and hyperbolic stochastic partial differential equations, Nonlinearity, 18 (2005), 747-767.  doi: 10.1088/0951-7715/18/2/015.  Google Scholar

[8]

T. Caraballo, I. D. Chueshov and P. E. Kloeden, Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain, SIAM J. Math. Anal. 38 (2007) 1489–1507. doi: 10.1137/050647281.  Google Scholar

[9]

S.-N. Chow, X.-B. Lin and K. Lu, Smooth invariant foliations in infinite-dimensional spaces, J. Diff. Eqs., 94 (1991) 266–291. doi: 10.1016/0022-0396(91)90093-O.  Google Scholar

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T. V. Girya and I. D. Chueshov, Inertial manifolds and stationary measures for stochastically perturbed dissipative dynamical systems, Sb. Math., 186 (1995) 29–45. doi: 10.1070/SM1995v186n01ABEH000002.  Google Scholar

[11]

I. Chueshov and S. Kuksin, Random kick-forced 3D Navier-Stokes equations in a thin domain, Arch. Ration. Mech. Anal., 188 (2008) 117–153. doi: 10.1007/s00205-007-0068-2.  Google Scholar

[12]

I. Chueshov and S. Kuksin, Stochastic 3D Navier-Stokes equations in a thin domain and its $\alpha$-approximation, Phys. D, 237 (2008) 1352–1367. doi: 10.1016/j.physd.2008.03.012.  Google Scholar

[13] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimension, University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.  Google Scholar
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G. Da Prato and A. Debussche, Construction of stochastic inertial manifolds using backward integration, Stochastics Rep., 59 (1996) 305–324. doi: 10.1080/17442509608834094.  Google Scholar

[15]

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

[16]

J. Duan, K. Lu and B. Schmalfuß, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynam. Diff. Eqns., 16 (2004) 949–972. doi: 10.1007/s10884-004-7830-z.  Google Scholar

[17]

J. K. Hale and G. Raugel, Reaction-diffusion equation on the thin domain, J. Math. pures et. appl., 71 (1992) 33–95.  Google Scholar

[18]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, volume 840 of Lecture Notes in Mathematics, Springer-Verlag, New York, 1981.  Google Scholar

[19]

D. A. Jones, A. M. Stuart and E. S. Titi, Persistence of invariant sets for dissipative evolution equations, J. Math. Anal. Appl., 219 (1998) 479–502. doi: 10.1006/jmaa.1997.5847.  Google Scholar

[20]

D. Li, B. Wang and X. Wang, Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains, J. Diff. Eqs., 262 (2017) 1575–1602. doi: 10.1016/j.jde.2016.10.024.  Google Scholar

[21]

D. Li, K. Lu, B. Wang, and X. Wang, Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains, Discrete Contin. Dyn. Syst., 38 (2018) 187–208. doi: 10.3934/dcds.2018009.  Google Scholar

[22]

K. Lu and B. Schmalfuß, Invariant foliations for stochastic partial differential equations, Stoch. Dyn., 8 (2008) 505–518. doi: 10.1142/S0219493708002421.  Google Scholar

[23]

K. Lu and B. Schmalfuß, Invariant manifolds for stochastic wave equations, J. Diff. Eqs., 236 (2007) 460–492. doi: 10.1016/j.jde.2006.09.024.  Google Scholar

[24]

S.-E. A. Mohammed and M. K. R. Scheutzow, The stable manifold theorem for stochastic differential equations., The Annals of Probability, 27 (1999) 615–652, . doi: 10.1214/aop/1022677380.  Google Scholar

[25]

S. Mohammed, T. Zhang and H. Zhao, The stable manifold theorem for semilinear SPDEs, Memoirs of AMS, , 196 (2008) 1–105. Google Scholar

[26]

P. S. Ngiamsunthorn, Invariant manifolds for parabolic equations under perturbation of the domain, Nonlinear Analysis TMA, 80 (2013) 23–48. doi: 10.1016/j.na.2012.12.001.  Google Scholar

[27]

M. Prizzi and K. P. Rybakowski, The effect of domain squeezing upon the dynamics of reaction-diffusion equations, J. Diff. Eqs. 173 (2001) 271–320. doi: 10.1006/jdeq.2000.3917.  Google Scholar

[28]

M. Prizzi and K. P. Rybakowski, Inertial manifolds on squeezed domains, J. Dynam. Diff. Eqs., 15 (2003) 1–48. doi: 10.1023/A:1026151910637.  Google Scholar

[29]

M. Prizzi and K. P. Rybakowski, On inertial manifolds for reaction-diffusion equations on genuinely high-dimensional thin domains, Studia Math., 154 (2003) 253–275. doi: 10.4064/sm154-3-6.  Google Scholar

[30]

E. Santamaría, Distance of Attractors of Evolutionary Equations, Universidad Complutense de Madrid, Ph.D thesis, 2014. Google Scholar

[31]

B. Schmalfuß, A random fixed point theorem and the random graph transformation, J. Math. Anal. Appl., 225 (1998) 91–113. doi: 10.1006/jmaa.1998.6008.  Google Scholar

[32]

N. Varchon, Domain perturbation and invariant manifolds, J. Evol. Equ., 12 (2012) 547–569. doi: 10.1007/s00028-012-0144-4.  Google Scholar

[33]

T. Wanner, Linearization of random dynamical systems, Dynamics Rep., 4 (1995) 203–269.  Google Scholar

show all references

References:
[1]

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

[2]

J. M. Arrieta and E. Santamaría, Estimates on the distance of inertial manifolds, Discrete Contin. Dyn. Syst., 34 (2014), 3921-3944.  doi: 10.3934/dcds.2014.34.3921.  Google Scholar

[3]

P. W. Bates, K. Lu and C. Zeng, Persistence of overflowing manifolds for semiflow, Comm. Pure Appl. Math., 52 (1999), 983-1046.  doi: 10.1002/(SICI)1097-0312(199908)52:8<983::AID-CPA4>3.0.CO;2-O.  Google Scholar

[4]

P. W. Bates, K. Lu and C. Zeng, Existence and persistence of invariant manifolds for semiflows in Banach space, Mem. Amer. Math. Soc., 135 (1998), 645. doi: 10.1090/memo/0645.  Google Scholar

[5]

P. W. Bates, K. Lu and C. Zeng, Approximately invariant manifolds and global dynamics of spike states, Invent. Math., 174 (2008), 355-433.  doi: 10.1007/s00222-008-0141-y.  Google Scholar

[6]

A. Bensoussan and F. Flandoli, Stochastic inertial manifold, Stochastics Rep., 53 (1995), 13-39.  doi: 10.1080/17442509508833981.  Google Scholar

[7]

T. Caraballo, I. Chueshov and J. A. Langa, Existence of invariant manifolds for coupled parabolic and hyperbolic stochastic partial differential equations, Nonlinearity, 18 (2005), 747-767.  doi: 10.1088/0951-7715/18/2/015.  Google Scholar

[8]

T. Caraballo, I. D. Chueshov and P. E. Kloeden, Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain, SIAM J. Math. Anal. 38 (2007) 1489–1507. doi: 10.1137/050647281.  Google Scholar

[9]

S.-N. Chow, X.-B. Lin and K. Lu, Smooth invariant foliations in infinite-dimensional spaces, J. Diff. Eqs., 94 (1991) 266–291. doi: 10.1016/0022-0396(91)90093-O.  Google Scholar

[10]

T. V. Girya and I. D. Chueshov, Inertial manifolds and stationary measures for stochastically perturbed dissipative dynamical systems, Sb. Math., 186 (1995) 29–45. doi: 10.1070/SM1995v186n01ABEH000002.  Google Scholar

[11]

I. Chueshov and S. Kuksin, Random kick-forced 3D Navier-Stokes equations in a thin domain, Arch. Ration. Mech. Anal., 188 (2008) 117–153. doi: 10.1007/s00205-007-0068-2.  Google Scholar

[12]

I. Chueshov and S. Kuksin, Stochastic 3D Navier-Stokes equations in a thin domain and its $\alpha$-approximation, Phys. D, 237 (2008) 1352–1367. doi: 10.1016/j.physd.2008.03.012.  Google Scholar

[13] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimension, University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.  Google Scholar
[14]

G. Da Prato and A. Debussche, Construction of stochastic inertial manifolds using backward integration, Stochastics Rep., 59 (1996) 305–324. doi: 10.1080/17442509608834094.  Google Scholar

[15]

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

[16]

J. Duan, K. Lu and B. Schmalfuß, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynam. Diff. Eqns., 16 (2004) 949–972. doi: 10.1007/s10884-004-7830-z.  Google Scholar

[17]

J. K. Hale and G. Raugel, Reaction-diffusion equation on the thin domain, J. Math. pures et. appl., 71 (1992) 33–95.  Google Scholar

[18]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, volume 840 of Lecture Notes in Mathematics, Springer-Verlag, New York, 1981.  Google Scholar

[19]

D. A. Jones, A. M. Stuart and E. S. Titi, Persistence of invariant sets for dissipative evolution equations, J. Math. Anal. Appl., 219 (1998) 479–502. doi: 10.1006/jmaa.1997.5847.  Google Scholar

[20]

D. Li, B. Wang and X. Wang, Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains, J. Diff. Eqs., 262 (2017) 1575–1602. doi: 10.1016/j.jde.2016.10.024.  Google Scholar

[21]

D. Li, K. Lu, B. Wang, and X. Wang, Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains, Discrete Contin. Dyn. Syst., 38 (2018) 187–208. doi: 10.3934/dcds.2018009.  Google Scholar

[22]

K. Lu and B. Schmalfuß, Invariant foliations for stochastic partial differential equations, Stoch. Dyn., 8 (2008) 505–518. doi: 10.1142/S0219493708002421.  Google Scholar

[23]

K. Lu and B. Schmalfuß, Invariant manifolds for stochastic wave equations, J. Diff. Eqs., 236 (2007) 460–492. doi: 10.1016/j.jde.2006.09.024.  Google Scholar

[24]

S.-E. A. Mohammed and M. K. R. Scheutzow, The stable manifold theorem for stochastic differential equations., The Annals of Probability, 27 (1999) 615–652, . doi: 10.1214/aop/1022677380.  Google Scholar

[25]

S. Mohammed, T. Zhang and H. Zhao, The stable manifold theorem for semilinear SPDEs, Memoirs of AMS, , 196 (2008) 1–105. Google Scholar

[26]

P. S. Ngiamsunthorn, Invariant manifolds for parabolic equations under perturbation of the domain, Nonlinear Analysis TMA, 80 (2013) 23–48. doi: 10.1016/j.na.2012.12.001.  Google Scholar

[27]

M. Prizzi and K. P. Rybakowski, The effect of domain squeezing upon the dynamics of reaction-diffusion equations, J. Diff. Eqs. 173 (2001) 271–320. doi: 10.1006/jdeq.2000.3917.  Google Scholar

[28]

M. Prizzi and K. P. Rybakowski, Inertial manifolds on squeezed domains, J. Dynam. Diff. Eqs., 15 (2003) 1–48. doi: 10.1023/A:1026151910637.  Google Scholar

[29]

M. Prizzi and K. P. Rybakowski, On inertial manifolds for reaction-diffusion equations on genuinely high-dimensional thin domains, Studia Math., 154 (2003) 253–275. doi: 10.4064/sm154-3-6.  Google Scholar

[30]

E. Santamaría, Distance of Attractors of Evolutionary Equations, Universidad Complutense de Madrid, Ph.D thesis, 2014. Google Scholar

[31]

B. Schmalfuß, A random fixed point theorem and the random graph transformation, J. Math. Anal. Appl., 225 (1998) 91–113. doi: 10.1006/jmaa.1998.6008.  Google Scholar

[32]

N. Varchon, Domain perturbation and invariant manifolds, J. Evol. Equ., 12 (2012) 547–569. doi: 10.1007/s00028-012-0144-4.  Google Scholar

[33]

T. Wanner, Linearization of random dynamical systems, Dynamics Rep., 4 (1995) 203–269.  Google Scholar

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