May  2011, 10(3): 831-846. doi: 10.3934/cpaa.2011.10.831

Inertial manifolds for stochastic pde with dynamical boundary conditions

1. 

Institut für Mathematik, University of Paderborn, 33098 Paderborn, Germany, Germany

Received  April 2009 Revised  October 2009 Published  December 2010

In this article we investigate the dynamics of stochastic partial differential equations with dynamical boundary conditions. We prove that such a problem with Lipschitz continuous non--linearity generates a random dynamical system. The main result is to show that this random dynamical system has an inertial manifold. Under additional assumptions on the non--linearity this manifold is differentiable.
Citation: Peter Brune, Björn Schmalfuss. Inertial manifolds for stochastic pde with dynamical boundary conditions. Communications on Pure & Applied Analysis, 2011, 10 (3) : 831-846. doi: 10.3934/cpaa.2011.10.831
References:
[1]

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

[2]

L. Arnold, "Random Dynamical Systems,", Springer Monographs in Mathematics, (1998). Google Scholar

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A. F. Bennett and P. E. Kloeden, The dissipative quasigeostrophic equations,, Mathematika, 28 (1982), 265. doi: doi:10.1112/S0025579300010329. Google Scholar

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P. Brune, "Inertiale Mannigfaltigkeiten von stochastischen PDE's mit dynamischen Randbedingungen,", Diplomarbeit, (2006). Google Scholar

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T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuß and J. Valero, Multivalued non-autonomous and random dynamical systems, pullback and random attractors, functional stochastic equations, conjugacy method., Submitted., (). Google Scholar

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C. Castaing and M. Valadier, "Convex Analysis and Measurable Multifunctions,", Springer-Verlag, (1977). Google Scholar

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S.-N. Chow and K. Lu, Invariant manifolds for flows in Banach spaces,, J. Differential Equations, 74 (1988), 285. doi: doi:10.1016/0022-0396(88)90007-1. Google Scholar

[8]

S.-N. Chow, K. Lu, and G. R. Sell, Smoothness of inertial manifolds,, J. Math. Anal. Appl., 169 (1992), 283. doi: doi:10.1016/0022-247X(92)90115-T. Google Scholar

[9]

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

[10]

I. D. Chueshov, "Introduction to the Theory of Infinite-Dimensional Dissipative Systems,", AKTA, (2002). Google Scholar

[11]

I. D. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations,, J. Dynam. Differential Equations, 13 (2001), 355. doi: doi:10.1023/A:1016684108862. Google Scholar

[12]

G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions," volume 44 of Encyclopedia of Mathematics and its Applications,, Cambridge University Press, (1992). Google Scholar

[13]

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

[14]

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

[15]

J. Escher, A note on quasilinear parabolic systems with dynamical boundary conditions,, In, (1992), 138. Google Scholar

[16]

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

[17]

G. Francois, Spectral asymptotics stemming from parabolic equations under dynamical boundary conditions,, Asymptot. Anal., 46 (2006), 43. Google Scholar

[18]

M. J. Garrido Atienza, K. Lu, and B. Schmalfuß, Unstable manifolds for a stochastic partial differential equation driven by a fractional Brownian motion,, Manuscript., (). Google Scholar

[19]

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

[20]

G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," volume 143 of Applied Mathematical Sciences,, Springer-Verlag, (2002). Google Scholar

[21]

T. Wanner, Linearization of random dynamical systems,, In Dynamics reported, (1995), 203. Google Scholar

[22]

J. Wloka, "Partielle Differentialgleichungen,", B. G. Teubner, (1982). Google Scholar

show all references

References:
[1]

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

[2]

L. Arnold, "Random Dynamical Systems,", Springer Monographs in Mathematics, (1998). Google Scholar

[3]

A. F. Bennett and P. E. Kloeden, The dissipative quasigeostrophic equations,, Mathematika, 28 (1982), 265. doi: doi:10.1112/S0025579300010329. Google Scholar

[4]

P. Brune, "Inertiale Mannigfaltigkeiten von stochastischen PDE's mit dynamischen Randbedingungen,", Diplomarbeit, (2006). Google Scholar

[5]

T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuß and J. Valero, Multivalued non-autonomous and random dynamical systems, pullback and random attractors, functional stochastic equations, conjugacy method., Submitted., (). Google Scholar

[6]

C. Castaing and M. Valadier, "Convex Analysis and Measurable Multifunctions,", Springer-Verlag, (1977). Google Scholar

[7]

S.-N. Chow and K. Lu, Invariant manifolds for flows in Banach spaces,, J. Differential Equations, 74 (1988), 285. doi: doi:10.1016/0022-0396(88)90007-1. Google Scholar

[8]

S.-N. Chow, K. Lu, and G. R. Sell, Smoothness of inertial manifolds,, J. Math. Anal. Appl., 169 (1992), 283. doi: doi:10.1016/0022-247X(92)90115-T. Google Scholar

[9]

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

[10]

I. D. Chueshov, "Introduction to the Theory of Infinite-Dimensional Dissipative Systems,", AKTA, (2002). Google Scholar

[11]

I. D. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations,, J. Dynam. Differential Equations, 13 (2001), 355. doi: doi:10.1023/A:1016684108862. Google Scholar

[12]

G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions," volume 44 of Encyclopedia of Mathematics and its Applications,, Cambridge University Press, (1992). Google Scholar

[13]

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

[14]

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

[15]

J. Escher, A note on quasilinear parabolic systems with dynamical boundary conditions,, In, (1992), 138. Google Scholar

[16]

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

[17]

G. Francois, Spectral asymptotics stemming from parabolic equations under dynamical boundary conditions,, Asymptot. Anal., 46 (2006), 43. Google Scholar

[18]

M. J. Garrido Atienza, K. Lu, and B. Schmalfuß, Unstable manifolds for a stochastic partial differential equation driven by a fractional Brownian motion,, Manuscript., (). Google Scholar

[19]

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

[20]

G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," volume 143 of Applied Mathematical Sciences,, Springer-Verlag, (2002). Google Scholar

[21]

T. Wanner, Linearization of random dynamical systems,, In Dynamics reported, (1995), 203. Google Scholar

[22]

J. Wloka, "Partielle Differentialgleichungen,", B. G. Teubner, (1982). Google Scholar

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