December  2021, 26(12): 6311-6337. doi: 10.3934/dcdsb.2021020

Limiting behavior of unstable manifolds for spdes in varying phase spaces

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 Published  December 2021 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.

Citation: Lin Shi, Dingshi Li, Kening Lu. Limiting behavior of unstable manifolds for spdes in varying phase spaces. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6311-6337. doi: 10.3934/dcdsb.2021020
References:
[1]

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

[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.

[3]

P. W. BatesK. 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.

[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.

[5]

P. W. BatesK. 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.

[6]

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

[7]

T. CaraballoI. 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.

[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.

[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.

[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.

[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.

[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.

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

[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.

[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.

[17]

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

[18]

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

[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.

[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.

[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.

[22]

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

[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.

[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.

[25]

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

[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.

[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.

[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.

[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.

[30]

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

[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.

[32]

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

[33]

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

show all references

References:
[1]

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

[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.

[3]

P. W. BatesK. 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.

[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.

[5]

P. W. BatesK. 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.

[6]

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

[7]

T. CaraballoI. 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.

[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.

[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.

[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.

[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.

[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.

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

[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.

[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.

[17]

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

[18]

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

[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.

[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.

[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.

[22]

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

[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.

[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.

[25]

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

[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.

[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.

[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.

[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.

[30]

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

[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.

[32]

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

[33]

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

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