# American Institute of Mathematical Sciences

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. [2] H. Amann and J. Escher, Strongly continuous dual semigroups,, Ann. Mat. Pura Appl., 171 (1996), 41. doi: 10.1007/BF01759381. [3] L. Arnold, Random Dynamical Systems,, Springer-Verlag, (1998). doi: 10.1007/978-3-662-12878-7. [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. [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. [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. [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. [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. [9] I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems,, AKTA, (1999). [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. [11] I. Chueshov and B. Schmalfuss, Parabolic stochastic partial differential equations with dynamcial boundary conditions,, Differential Integral Equations, 17 (2004), 751. [12] P. Colli and J. F. Rodrigues, Diffusion through thin layers with high specific heat,, Asymptotic Anal., 3 (1990), 249. [13] A. Du and J. Duan, Invariant manifold reduction for stochastic dynamical systems,, Dynamical Systems and Applications, 16 (2007), 681. [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. [15] K. J. Engel, Spectral theory and generator property for one-sided coupled operator matrices,, Semigroup Forum, 58 (1999), 267. doi: 10.1007/s002339900020. [16] K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolutions Equations,, Spinger-Verlag, (2000). [17] J. Escher, Quasilinear parabolic systems with dynamical boundary conditions,, Comm. Partial Differential Equations, 18 (1993), 1309. doi: 10.1080/03605309308820976. [18] J. Escher, A note on quasilinear parabolic systems with dynamical boundary conditions,, Longman Sci. Tech., 296 (1993), 138. [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. [20] T. Hintermann, Evolution equations with dynamic boundary conditions,, Proc. Roy. Soc. Edinburgh Sect. A, 113 (1989), 43. doi: 10.1017/S0308210500023945. [21] K. Lu and B. Schmafuss, Invariant foliation for stochastic partial differential equations,, Stoch. Dyn., 8 (2008), 505. doi: 10.1142/S0219493708002421. [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. [23] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983). doi: 10.1007/978-1-4612-5561-1. [24] J. Ren, J. Duan and C. Jones, Approximation of random slow manifolds and settling of inertial particles under uncertainty,, , (). [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. [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. [27] X. Sun, X. Kan and J. Duan, Approximation of invariant foliations for stochastic dynamical systems,, Stoch. Dyn., 12 (2012). doi: 10.1142/S0219493712003614. [28] T. Wanner, Linearization of random dynamical systmes,, Dynamics Reported, 4 (1995), 203.

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##### 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. [2] H. Amann and J. Escher, Strongly continuous dual semigroups,, Ann. Mat. Pura Appl., 171 (1996), 41. doi: 10.1007/BF01759381. [3] L. Arnold, Random Dynamical Systems,, Springer-Verlag, (1998). doi: 10.1007/978-3-662-12878-7. [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. [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. [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. [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. [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. [9] I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems,, AKTA, (1999). [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. [11] I. Chueshov and B. Schmalfuss, Parabolic stochastic partial differential equations with dynamcial boundary conditions,, Differential Integral Equations, 17 (2004), 751. [12] P. Colli and J. F. Rodrigues, Diffusion through thin layers with high specific heat,, Asymptotic Anal., 3 (1990), 249. [13] A. Du and J. Duan, Invariant manifold reduction for stochastic dynamical systems,, Dynamical Systems and Applications, 16 (2007), 681. [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. [15] K. J. Engel, Spectral theory and generator property for one-sided coupled operator matrices,, Semigroup Forum, 58 (1999), 267. doi: 10.1007/s002339900020. [16] K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolutions Equations,, Spinger-Verlag, (2000). [17] J. Escher, Quasilinear parabolic systems with dynamical boundary conditions,, Comm. Partial Differential Equations, 18 (1993), 1309. doi: 10.1080/03605309308820976. [18] J. Escher, A note on quasilinear parabolic systems with dynamical boundary conditions,, Longman Sci. Tech., 296 (1993), 138. [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. [20] T. Hintermann, Evolution equations with dynamic boundary conditions,, Proc. Roy. Soc. Edinburgh Sect. A, 113 (1989), 43. doi: 10.1017/S0308210500023945. [21] K. Lu and B. Schmafuss, Invariant foliation for stochastic partial differential equations,, Stoch. Dyn., 8 (2008), 505. doi: 10.1142/S0219493708002421. [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. [23] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983). doi: 10.1007/978-1-4612-5561-1. [24] J. Ren, J. Duan and C. Jones, Approximation of random slow manifolds and settling of inertial particles under uncertainty,, , (). [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. [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. [27] X. Sun, X. Kan and J. Duan, Approximation of invariant foliations for stochastic dynamical systems,, Stoch. Dyn., 12 (2012). doi: 10.1142/S0219493712003614. [28] T. Wanner, Linearization of random dynamical systmes,, Dynamics Reported, 4 (1995), 203.
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