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 and 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), 41-62. doi: doi:10.1007/BF01759381.

[2]

L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.

[3]

A. F. Bennett and P. E. Kloeden, The dissipative quasigeostrophic equations, Mathematika, 28 (1982), 265-285. doi: doi:10.1112/S0025579300010329.

[4]

P. Brune, "Inertiale Mannigfaltigkeiten von stochastischen PDE's mit dynamischen Randbedingungen," Diplomarbeit, Universität Paderborn, 2006.

[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., (). 

[6]

C. Castaing and M. Valadier, "Convex Analysis and Measurable Multifunctions," Springer-Verlag, Berlin, 1977. Lecture Notes in Mathematics, Vol. 580.

[7]

S.-N. Chow and K. Lu, Invariant manifolds for flows in Banach spaces, J. Differential Equations, 74 (1988), 285-317. doi: doi:10.1016/0022-0396(88)90007-1.

[8]

S.-N. Chow, K. Lu, and G. R. Sell, Smoothness of inertial manifolds, J. Math. Anal. Appl., 169 (1992), 283-312. doi: doi:10.1016/0022-247X(92)90115-T.

[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), 315-338. doi: doi:10.3934/dcds.2007.18.315.

[10]

I. D. Chueshov, "Introduction to the Theory of Infinite-Dimensional Dissipative Systems," AKTA, Kharkiv, 2002.

[11]

I. D. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations, J. Dynam. Differential Equations, 13 (2001), 355-380. doi: doi:10.1023/A:1016684108862.

[12]

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

[13]

J. Duan, K. Lu, and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135. doi: doi:10.1214/aop/1068646380.

[14]

J. Duan, K. Lu, and B. Schmalfuss, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynam. Differential Equations, 16 (2004), 949-972. doi: doi:10.1007/s10884-004-7830-z.

[15]

J. Escher, A note on quasilinear parabolic systems with dynamical boundary conditions, In "Progress in Partial Differential Equations: the Metz Surveys," 2 (1992), volume 296 of Pitman Res. Notes Math. Ser., pages 138-148. Longman Sci. Tech., Harlow, 1993.

[16]

J. Escher, Quasilinear parabolic systems with dynamical boundary conditions, Comm. Partial Differential Equations, 18 (1993), 1309-1364. doi: doi:10.1080/03605309308820976.

[17]

G. Francois, Spectral asymptotics stemming from parabolic equations under dynamical boundary conditions, Asymptot. Anal., 46 (2006), 43-52.

[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., (). 

[19]

K. Lu and B. Schmalfuß, Invariant manifolds for stochastic wave equations, J. Differential Equations, 236 (2007), 460-492. doi: doi:10.1016/j.jde.2006.09.024.

[20]

G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," volume 143 of Applied Mathematical Sciences, Springer-Verlag, New York, 2002.

[21]

T. Wanner, Linearization of random dynamical systems, In Dynamics reported, volume 4 of Dynam. Report. Expositions Dynam. Systems (N.S.), pages 203-269. Springer, Berlin, 1995.

[22]

J. Wloka, "Partielle Differentialgleichungen," B. G. Teubner, Stuttgart, 1982.

show all references

References:
[1]

H. Amann and J. Escher, Strongly continuous dual semigroups, Ann. Mat. Pura Appl., 171 (1996), 41-62. doi: doi:10.1007/BF01759381.

[2]

L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.

[3]

A. F. Bennett and P. E. Kloeden, The dissipative quasigeostrophic equations, Mathematika, 28 (1982), 265-285. doi: doi:10.1112/S0025579300010329.

[4]

P. Brune, "Inertiale Mannigfaltigkeiten von stochastischen PDE's mit dynamischen Randbedingungen," Diplomarbeit, Universität Paderborn, 2006.

[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., (). 

[6]

C. Castaing and M. Valadier, "Convex Analysis and Measurable Multifunctions," Springer-Verlag, Berlin, 1977. Lecture Notes in Mathematics, Vol. 580.

[7]

S.-N. Chow and K. Lu, Invariant manifolds for flows in Banach spaces, J. Differential Equations, 74 (1988), 285-317. doi: doi:10.1016/0022-0396(88)90007-1.

[8]

S.-N. Chow, K. Lu, and G. R. Sell, Smoothness of inertial manifolds, J. Math. Anal. Appl., 169 (1992), 283-312. doi: doi:10.1016/0022-247X(92)90115-T.

[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), 315-338. doi: doi:10.3934/dcds.2007.18.315.

[10]

I. D. Chueshov, "Introduction to the Theory of Infinite-Dimensional Dissipative Systems," AKTA, Kharkiv, 2002.

[11]

I. D. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations, J. Dynam. Differential Equations, 13 (2001), 355-380. doi: doi:10.1023/A:1016684108862.

[12]

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

[13]

J. Duan, K. Lu, and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135. doi: doi:10.1214/aop/1068646380.

[14]

J. Duan, K. Lu, and B. Schmalfuss, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynam. Differential Equations, 16 (2004), 949-972. doi: doi:10.1007/s10884-004-7830-z.

[15]

J. Escher, A note on quasilinear parabolic systems with dynamical boundary conditions, In "Progress in Partial Differential Equations: the Metz Surveys," 2 (1992), volume 296 of Pitman Res. Notes Math. Ser., pages 138-148. Longman Sci. Tech., Harlow, 1993.

[16]

J. Escher, Quasilinear parabolic systems with dynamical boundary conditions, Comm. Partial Differential Equations, 18 (1993), 1309-1364. doi: doi:10.1080/03605309308820976.

[17]

G. Francois, Spectral asymptotics stemming from parabolic equations under dynamical boundary conditions, Asymptot. Anal., 46 (2006), 43-52.

[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., (). 

[19]

K. Lu and B. Schmalfuß, Invariant manifolds for stochastic wave equations, J. Differential Equations, 236 (2007), 460-492. doi: doi:10.1016/j.jde.2006.09.024.

[20]

G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," volume 143 of Applied Mathematical Sciences, Springer-Verlag, New York, 2002.

[21]

T. Wanner, Linearization of random dynamical systems, In Dynamics reported, volume 4 of Dynam. Report. Expositions Dynam. Systems (N.S.), pages 203-269. Springer, Berlin, 1995.

[22]

J. Wloka, "Partielle Differentialgleichungen," B. G. Teubner, Stuttgart, 1982.

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