December  2017, 37(12): 6383-6403. doi: 10.3934/dcds.2017277

Random pullback exponential attractors: General existence results for random dynamical systems in Banach spaces

1. 

Departamento Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, C/ Tarfia s/n, 41012 Sevilla, Spain

2. 

Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universität Graz, Heinrichstr. 36, 8010 Graz, Austria

* Corresponding author: Stefanie Sonner

Received  March 2017 Revised  July 2017 Published  August 2017

Fund Project: The first author was partially supported by FEDER (EU) and Ministerio de Economía y Competitividad (Spain) grant MTM2015-63723-P and by the Junta de Andalucía under the Proyecto de Excelencia P12-FQM-1492.

We derive general existence theorems for random pullback exponential attractors and deduce explicit bounds for their fractal dimension. The results are formulated for asymptotically compact random dynamical systems in Banach spaces.

Citation: Tomás Caraballo, Stefanie Sonner. Random pullback exponential attractors: General existence results for random dynamical systems in Banach spaces. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6383-6403. doi: 10.3934/dcds.2017277
References:
[1]

T. Caraballo and S. Sonner, Random exponential attractors for stochastic damped wave equations, in preparation.

[2]

A. N. Carvalho and S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: Theoretical results, Comm. Pure Appl. Anal., 12 (2013), 3047-3071.  doi: 10.3934/cpaa.2013.12.3047.

[3]

A. N. Carvalho and S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: properties and applications, Comm. Pure Appl. Anal., 13 (2014), 1141-1165.  doi: 10.3934/cpaa.2014.13.1141.

[4]

I. Chueshov, Monotone Random Systems Theory and Applications, Lecture Notes in Math., 1779, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.

[5]

H. Crauel and P. E. Kloeden, Nonautonomous and random attractors, Jahresber. Dtsch. Math.-Ver., 117 (2015), 173-206.  doi: 10.1365/s13291-015-0115-0.

[6]

R. Czaja and M. A. Efendiev, Pullback exponential attractors for nonautonomous equations part Ⅰ: Semilinear parabolic equations, J. Math. Anal. Appl., 381 (2011), 748-765.  doi: 10.1016/j.jmaa.2011.03.053.

[7]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, Research in Applied Mathematics, Masson, Paris, John Wiley & Sons, Ltd., Chichester, 1994.

[8]

D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers and Differential Operators, Cambridge University Press, New York, 1996. doi: 10.1017/CBO9780511662201.

[9]

M. A. EfendievS. Zelik and A. Miranville, Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems, Proc. R. Soc. Edinburgh Sect. A, 135 (2005), 703-730.  doi: 10.1017/S030821050000408X.

[10]

A. N. Kolmogorov and V. M. Tihomirov, ε-entropy and ε-capacity of sets in functional spaces, Amer. Math. Soc. Transl. Ser. 2, 17 (1961), 277-364. 

[11]

J. A. LangaA. Miranville and J. Real, Pullback exponential attractors, Discrete Contin. Dyn. Syst., 26 (2010), 1329-1357.  doi: 10.3934/dcds.2010.26.1329.

[12]

A. Shirikyan and S. Zelik, Exponential attractors for random dynamical systems and applications, Stoch. Partial Differ. Equ. Anal. Comput., 1 (2013), 241-281.  doi: 10.1007/s40072-013-0007-1.

[13]

S. Zhou, Random exponential attractor for cocycle and application to non-autonomous stochastic lattice systems with multiplicative noise, J. Differential Equations, 263 (2017), 2247-2279.  doi: 10.1016/j.jde.2017.03.044.

show all references

References:
[1]

T. Caraballo and S. Sonner, Random exponential attractors for stochastic damped wave equations, in preparation.

[2]

A. N. Carvalho and S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: Theoretical results, Comm. Pure Appl. Anal., 12 (2013), 3047-3071.  doi: 10.3934/cpaa.2013.12.3047.

[3]

A. N. Carvalho and S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: properties and applications, Comm. Pure Appl. Anal., 13 (2014), 1141-1165.  doi: 10.3934/cpaa.2014.13.1141.

[4]

I. Chueshov, Monotone Random Systems Theory and Applications, Lecture Notes in Math., 1779, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.

[5]

H. Crauel and P. E. Kloeden, Nonautonomous and random attractors, Jahresber. Dtsch. Math.-Ver., 117 (2015), 173-206.  doi: 10.1365/s13291-015-0115-0.

[6]

R. Czaja and M. A. Efendiev, Pullback exponential attractors for nonautonomous equations part Ⅰ: Semilinear parabolic equations, J. Math. Anal. Appl., 381 (2011), 748-765.  doi: 10.1016/j.jmaa.2011.03.053.

[7]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, Research in Applied Mathematics, Masson, Paris, John Wiley & Sons, Ltd., Chichester, 1994.

[8]

D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers and Differential Operators, Cambridge University Press, New York, 1996. doi: 10.1017/CBO9780511662201.

[9]

M. A. EfendievS. Zelik and A. Miranville, Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems, Proc. R. Soc. Edinburgh Sect. A, 135 (2005), 703-730.  doi: 10.1017/S030821050000408X.

[10]

A. N. Kolmogorov and V. M. Tihomirov, ε-entropy and ε-capacity of sets in functional spaces, Amer. Math. Soc. Transl. Ser. 2, 17 (1961), 277-364. 

[11]

J. A. LangaA. Miranville and J. Real, Pullback exponential attractors, Discrete Contin. Dyn. Syst., 26 (2010), 1329-1357.  doi: 10.3934/dcds.2010.26.1329.

[12]

A. Shirikyan and S. Zelik, Exponential attractors for random dynamical systems and applications, Stoch. Partial Differ. Equ. Anal. Comput., 1 (2013), 241-281.  doi: 10.1007/s40072-013-0007-1.

[13]

S. Zhou, Random exponential attractor for cocycle and application to non-autonomous stochastic lattice systems with multiplicative noise, J. Differential Equations, 263 (2017), 2247-2279.  doi: 10.1016/j.jde.2017.03.044.

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