November  2015, 35(11): 5203-5219. doi: 10.3934/dcds.2015.35.5203

Invariant foliations for stochastic partial differential equations with dynamic boundary conditions

1. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China

Received  September 2013 Revised  March 2014 Published  May 2015

Invariant foliations are geometric structures useful for describing and understanding qualitative behaviors of nonlinear dynamical systems. They decompose the state space into regions of different dynamical regimes, and thus help depict dynamics. We investigate invariant foliations for a class of stochastic partial differential equations with random dynamical boundary conditions, and then provide an approximation for these foliations when the noise intensity is sufficiently small.
Citation: Zhongkai Guo. Invariant foliations for stochastic partial differential equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5203-5219. doi: 10.3934/dcds.2015.35.5203
References:
[1]

E. Alòs and S. Bonaccorsi, Spdes with dirichlet white-noise boundary conditions,, Ann. Inst. H. Poincaré Probab. Statist., 38 (2002), 125.  doi: 10.1016/S0246-0203(01)01097-4.  Google Scholar

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H. Amann and J. Escher, Strongly continuous dual semigroups,, Ann. Mat. Pura Appl., 171 (1996), 41.  doi: 10.1007/BF01759381.  Google Scholar

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

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P. Brune and B. Schmalfuss, Inertial manifolds for stochastic PDE with dynamical boundary conditions,, Commun. Pure Appl. Anal., 10 (2011), 831.  doi: 10.3934/cpaa.2011.10.831.  Google Scholar

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T. Caraballo, J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for random and stochastic partial differential equations,, Adv. Nonlinear Stud., 10 (2010), 23.   Google Scholar

[6]

G. Chen, J. Duan and J. Zhang, Slow foliation of a slow-fast stochastic evolutionary system,, J. Funct. Anal., 267 (2014), 2663.  doi: 10.1016/j.jfa.2014.07.031.  Google Scholar

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X. Chen, J. Hale and B. Tan, Invariant foliations for $C^{1}$ semigroups in Banach spaces,, J. Diff. Eqs., 139 (1997), 283.  doi: 10.1006/jdeq.1997.3255.  Google Scholar

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

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I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems,, AKTA, (1999).   Google Scholar

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I. Chueshov and B. Schmalfuss, Qualitative behavior of a class of stochastic parabolic PDEs with dynamcal boundary conditions,, Discrete Contin. Dyn. Syst., 18 (2007), 315.  doi: 10.3934/dcds.2007.18.315.  Google Scholar

[11]

I. Chueshov and B. Schmalfuss, Parabolic stochastic partial differential equations with dynamcial boundary conditions,, Differential Integral Equations, 17 (2004), 751.   Google Scholar

[12]

P. Colli and J. F. Rodrigues, Diffusion through thin layers with high specific heat,, Asymptotic Anal., 3 (1990), 249.   Google Scholar

[13]

A. Du and J. Duan, Invariant manifold reduction for stochastic dynamical systems,, Dynamical Systems and Applications, 16 (2007), 681.   Google Scholar

[14]

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

[15]

K. J. Engel, Spectral theory and generator property for one-sided coupled operator matrices,, Semigroup Forum, 58 (1999), 267.  doi: 10.1007/s002339900020.  Google Scholar

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K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolutions Equations,, Spinger-Verlag, (2000).   Google Scholar

[17]

J. Escher, Quasilinear parabolic systems with dynamical boundary conditions,, Comm. Partial Differential Equations, 18 (1993), 1309.  doi: 10.1080/03605309308820976.  Google Scholar

[18]

J. Escher, A note on quasilinear parabolic systems with dynamical boundary conditions,, Longman Sci. Tech., 296 (1993), 138.   Google Scholar

[19]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical systems, and Bifurcation of Vector Fields,, Springer-Verlag, (1983).  doi: 10.1007/978-1-4612-1140-2.  Google Scholar

[20]

T. Hintermann, Evolution equations with dynamic boundary conditions,, Proc. Roy. Soc. Edinburgh Sect. A, 113 (1989), 43.  doi: 10.1017/S0308210500023945.  Google Scholar

[21]

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

[22]

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

[23]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[24]

J. Ren, J. Duan and C. Jones, Approximation of random slow manifolds and settling of inertial particles under uncertainty,, , ().   Google Scholar

[25]

J. F. Rodrigues, V. A. Solonnikov and F. Yi, On a parabolic system with time derivative in the boundary conditions and related free boundary problems,, Math. Ann., 315 (1999), 61.  doi: 10.1007/s002080050318.  Google Scholar

[26]

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

[27]

X. Sun, X. Kan and J. Duan, Approximation of invariant foliations for stochastic dynamical systems,, Stoch. Dyn., 12 (2012).  doi: 10.1142/S0219493712003614.  Google Scholar

[28]

T. Wanner, Linearization of random dynamical systmes,, Dynamics Reported, 4 (1995), 203.   Google Scholar

show all references

References:
[1]

E. Alòs and S. Bonaccorsi, Spdes with dirichlet white-noise boundary conditions,, Ann. Inst. H. Poincaré Probab. Statist., 38 (2002), 125.  doi: 10.1016/S0246-0203(01)01097-4.  Google Scholar

[2]

H. Amann and J. Escher, Strongly continuous dual semigroups,, Ann. Mat. Pura Appl., 171 (1996), 41.  doi: 10.1007/BF01759381.  Google Scholar

[3]

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

[4]

P. Brune and B. Schmalfuss, Inertial manifolds for stochastic PDE with dynamical boundary conditions,, Commun. Pure Appl. Anal., 10 (2011), 831.  doi: 10.3934/cpaa.2011.10.831.  Google Scholar

[5]

T. Caraballo, J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for random and stochastic partial differential equations,, Adv. Nonlinear Stud., 10 (2010), 23.   Google Scholar

[6]

G. Chen, J. Duan and J. Zhang, Slow foliation of a slow-fast stochastic evolutionary system,, J. Funct. Anal., 267 (2014), 2663.  doi: 10.1016/j.jfa.2014.07.031.  Google Scholar

[7]

X. Chen, J. Hale and B. Tan, Invariant foliations for $C^{1}$ semigroups in Banach spaces,, J. Diff. Eqs., 139 (1997), 283.  doi: 10.1006/jdeq.1997.3255.  Google Scholar

[8]

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

[9]

I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems,, AKTA, (1999).   Google Scholar

[10]

I. Chueshov and B. Schmalfuss, Qualitative behavior of a class of stochastic parabolic PDEs with dynamcal boundary conditions,, Discrete Contin. Dyn. Syst., 18 (2007), 315.  doi: 10.3934/dcds.2007.18.315.  Google Scholar

[11]

I. Chueshov and B. Schmalfuss, Parabolic stochastic partial differential equations with dynamcial boundary conditions,, Differential Integral Equations, 17 (2004), 751.   Google Scholar

[12]

P. Colli and J. F. Rodrigues, Diffusion through thin layers with high specific heat,, Asymptotic Anal., 3 (1990), 249.   Google Scholar

[13]

A. Du and J. Duan, Invariant manifold reduction for stochastic dynamical systems,, Dynamical Systems and Applications, 16 (2007), 681.   Google Scholar

[14]

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

[15]

K. J. Engel, Spectral theory and generator property for one-sided coupled operator matrices,, Semigroup Forum, 58 (1999), 267.  doi: 10.1007/s002339900020.  Google Scholar

[16]

K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolutions Equations,, Spinger-Verlag, (2000).   Google Scholar

[17]

J. Escher, Quasilinear parabolic systems with dynamical boundary conditions,, Comm. Partial Differential Equations, 18 (1993), 1309.  doi: 10.1080/03605309308820976.  Google Scholar

[18]

J. Escher, A note on quasilinear parabolic systems with dynamical boundary conditions,, Longman Sci. Tech., 296 (1993), 138.   Google Scholar

[19]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical systems, and Bifurcation of Vector Fields,, Springer-Verlag, (1983).  doi: 10.1007/978-1-4612-1140-2.  Google Scholar

[20]

T. Hintermann, Evolution equations with dynamic boundary conditions,, Proc. Roy. Soc. Edinburgh Sect. A, 113 (1989), 43.  doi: 10.1017/S0308210500023945.  Google Scholar

[21]

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

[22]

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

[23]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[24]

J. Ren, J. Duan and C. Jones, Approximation of random slow manifolds and settling of inertial particles under uncertainty,, , ().   Google Scholar

[25]

J. F. Rodrigues, V. A. Solonnikov and F. Yi, On a parabolic system with time derivative in the boundary conditions and related free boundary problems,, Math. Ann., 315 (1999), 61.  doi: 10.1007/s002080050318.  Google Scholar

[26]

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

[27]

X. Sun, X. Kan and J. Duan, Approximation of invariant foliations for stochastic dynamical systems,, Stoch. Dyn., 12 (2012).  doi: 10.1142/S0219493712003614.  Google Scholar

[28]

T. Wanner, Linearization of random dynamical systmes,, Dynamics Reported, 4 (1995), 203.   Google Scholar

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