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The obstacle problem for quasilinear stochastic PDEs with non-homogeneous operator
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 |
References:
[1] |
E. Alòs and S. Bonaccorsi, Spdes with dirichlet white-noise boundary conditions, Ann. Inst. H. Poincaré Probab. Statist., 38 (2002), 125-154.
doi: 10.1016/S0246-0203(01)01097-4. |
[2] |
H. Amann and J. Escher, Strongly continuous dual semigroups, Ann. Mat. Pura Appl., 171 (1996), 41-62.
doi: 10.1007/BF01759381. |
[3] |
L. Arnold, Random Dynamical Systems, Springer-Verlag, New York, 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), 831-846.
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), 23-52. |
[6] |
G. Chen, J. Duan and J. Zhang, Slow foliation of a slow-fast stochastic evolutionary system, J. Funct. Anal., 267 (2014), 2663-2697.
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), 283-318.
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), 266-291.
doi: 10.1016/0022-0396(91)90093-O. |
[9] |
I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, AKTA, Kharkiv, 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), 315-338.
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), 751-780. |
[12] |
P. Colli and J. F. Rodrigues, Diffusion through thin layers with high specific heat, Asymptotic Anal., 3 (1990), 249-263. |
[13] |
A. Du and J. Duan, Invariant manifold reduction for stochastic dynamical systems, Dynamical Systems and Applications, 16 (2007), 681-696. |
[14] |
J. Duan, K. Lu and B. Schmalfuss, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynamics and Diff. Eqns., 16 (2004), 949-972.
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), 267-295.
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), 1309-1346.
doi: 10.1080/03605309308820976. |
[18] |
J. Escher, A note on quasilinear parabolic systems with dynamical boundary conditions, Longman Sci. Tech., 296 (1993), 138-148. |
[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), 43-60.
doi: 10.1017/S0308210500023945. |
[21] |
K. Lu and B. Schmafuss, Invariant foliation for stochastic partial differential equations, Stoch. Dyn., 8 (2008), 505-518.
doi: 10.1142/S0219493708002421. |
[22] |
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. |
[23] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 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), 61-95.
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), 042702, 12pp.
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), 1150011, 12pp.
doi: 10.1142/S0219493712003614. |
[28] |
T. Wanner, Linearization of random dynamical systmes, Dynamics Reported, 4 (1995), 203-269. |
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), 125-154.
doi: 10.1016/S0246-0203(01)01097-4. |
[2] |
H. Amann and J. Escher, Strongly continuous dual semigroups, Ann. Mat. Pura Appl., 171 (1996), 41-62.
doi: 10.1007/BF01759381. |
[3] |
L. Arnold, Random Dynamical Systems, Springer-Verlag, New York, 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), 831-846.
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), 23-52. |
[6] |
G. Chen, J. Duan and J. Zhang, Slow foliation of a slow-fast stochastic evolutionary system, J. Funct. Anal., 267 (2014), 2663-2697.
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), 283-318.
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), 266-291.
doi: 10.1016/0022-0396(91)90093-O. |
[9] |
I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, AKTA, Kharkiv, 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), 315-338.
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), 751-780. |
[12] |
P. Colli and J. F. Rodrigues, Diffusion through thin layers with high specific heat, Asymptotic Anal., 3 (1990), 249-263. |
[13] |
A. Du and J. Duan, Invariant manifold reduction for stochastic dynamical systems, Dynamical Systems and Applications, 16 (2007), 681-696. |
[14] |
J. Duan, K. Lu and B. Schmalfuss, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynamics and Diff. Eqns., 16 (2004), 949-972.
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), 267-295.
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), 1309-1346.
doi: 10.1080/03605309308820976. |
[18] |
J. Escher, A note on quasilinear parabolic systems with dynamical boundary conditions, Longman Sci. Tech., 296 (1993), 138-148. |
[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), 43-60.
doi: 10.1017/S0308210500023945. |
[21] |
K. Lu and B. Schmafuss, Invariant foliation for stochastic partial differential equations, Stoch. Dyn., 8 (2008), 505-518.
doi: 10.1142/S0219493708002421. |
[22] |
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. |
[23] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 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), 61-95.
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), 042702, 12pp.
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), 1150011, 12pp.
doi: 10.1142/S0219493712003614. |
[28] |
T. Wanner, Linearization of random dynamical systmes, Dynamics Reported, 4 (1995), 203-269. |
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