# The Library

### Invariant measures for dissipative systems and generalised Banach limits

Tools

Łukaszewicz, Grzegorz, Real, José and Robinson, James C. (James Cooper), 1969-.
(2011)
*Invariant measures for dissipative systems and generalised Banach limits.*
Journal of Dynamics and Differential Equations, Volume 23
(Number 2).
pp. 225-250.
ISSN 1040-7294

**Full text not available from this repository.**

Official URL: http://dx.doi.org/10.1007/s10884-011-9213-6

## Abstract

Inspired by a theory due to Foias and coworkers (see, for example, Foias et al. Navier-Stokes equations and turbulence, Cambridge University Press, Cambridge, 2001) and recent work of Wang (Disc Cont Dyn Sys 23:521-540, 2009), we show that the generalised Banach limit can be used to construct invariant measures for continuous dynamical systems on metric spaces that have compact attracting sets, taking limits evaluated along individual trajectories. We also show that if the space is a reflexive separable Banach space, or if the dynamical system has a compact absorbing set, then rather than taking limits evaluated along individual trajectories, we can take an ensemble of initial conditions: the generalised Banach limit can be used to construct an invariant measure based on an arbitrary initial probability measure, and any invariant measure can be obtained in this way. We thus propose an alternative to the classical Krylov-Bogoliubov construction, which we show is also applicable in this situation.

Item Type: | Journal Article |
---|---|

Subjects: | Q Science > QA Mathematics |

Divisions: | Faculty of Science > Mathematics |

Library of Congress Subject Headings (LCSH): | Invariant measures, Banach spaces, Dynamics, Attractors (Mathematics) |

Journal or Publication Title: | Journal of Dynamics and Differential Equations |

Publisher: | Springer |

ISSN: | 1040-7294 |

Date: | 2011 |

Volume: | Volume 23 |

Number: | Number 2 |

Page Range: | pp. 225-250 |

Identification Number: | 10.1007/s10884-011-9213-6 |

Status: | Peer Reviewed |

Publication Status: | Published |

Funder: | Poland , Spain. Ministerio de Ciencia e Innovación (MICINN), Andalusia (Spain). Consejería de Innovación, Ciencia y Empresa , Engineering and Physical Sciences Research Council (EPSRC) |

References: | 1. Aliprantis, Ch.D., Border, K.C.: Infinite Dimensional Analysis. A Hitchhiker’s Guide, 3rd edn. Springer, Berlin (2006) 2. Anguiano, M., Caraballo, T., Real, J.: Existence of pullback attractor for a reaction-diffusion equation in some unbounded domains with non-autonomous forcing term in H−1. Int. J. Bifur. Chaos 20, 2645– 2656 (2010) 3. Babin, A.V., Vishik, M.I.: Attractors of Evolution Equations. North–Holland, Amsterdam (1992) 4. Cholewa, J., Dłotko, T.: Global Attractors in Abstract Parabolic Problems. Cambridge University Press, Cambridge (2000) 5. Crauel, H.: Random point attractors versus random set attractors. J. Lond. Math. Soc. 63, 413–427 (2001) 6. Da Prato,G., Zabczyk, J.:Ergodicity for Infinite Dimensional Systems. Cambridge University Press, Cambridge (1996) 7. Foias, C.: Statistical study of Navier–Stokes equations. I. Rend. Sem. Mat. Univ. Padova 48, 219– 348 (1972) 8. Foias,C.: Statistical study of Navier–Stokes equations. II. Rend. Sem. Mat. Univ. Padova 49, 9–123 (1973) 9. Foias, C., Manley, O., Rosa, R., Temam, R.: Navier-Stokes Equations and Turbulence. Encyclopedia of Mathematics and its Applications, vol. 83. Cambridge University Press, Cambridge (2001) 10. Hale, J.K.: Asymptotic Behavior of Dissipative Systems. AMS, Providence (1988) 11. Haraux, A.: Systèmes dynamiques dissipatifs et applications. Masson, Paris (1990) 12. Kryloff, N., Bogoliouboff, N.: La théorie générale de la mesure dans son application à l’étude des systèmes dynamiques de la mécanique non linéaire. Ann. Math. 38, 65–113 (1937) 13. Langa, J.A., Robinson, J.C.: Determining asymptotic behavior from the dynamics on attracting sets. J. Dyn. Differ. Equ. 11, 319–331 (1999) 14. Nemytskii, V.V., Stepanov, V.V.: Qualitative Theory of Differential Equations. Princeton University Press, Princeton (1960) 15. Robinson, J.C.: Infinite-Dimensional Dynamical Systems. Cambridge University Press, Cambridge (2001) 16. Rosa, R.: The global attractor for the 2D Navier–Stokes flow on some unbounded domains. Nonlinear Anal. 32, 71–85 (1998) 17. Temam, R.: Infinite-Dimensional Dynamical Systems inMechanics and Physics. Springer, Berlin (1988) 18. van der Vaart, A.W., Wellner, J.A.: Weak Convergence and Empirical Processes: With Applications to Statistics. Springer, New York (2000) 19. Walters, P.: An introduction to Ergodic Theory. Graduate Texts in Mathematics. Springer, New York (1982) 20. Wang, X.: Upper semi-continuity of stationary statistical properties of dissipative systems. Disc. Cont. Dyn. Sys. 23, 521–540 (2009) |

URI: | http://wrap.warwick.ac.uk/id/eprint/41549 |

Data sourced from Thomson Reuters' Web of Knowledge

### Actions (login required)

View Item |