October  2014, 34(10): 4211-4222. doi: 10.3934/dcds.2014.34.4211

Invariant measures for non-autonomous dissipative dynamical systems

1. 

University of Warsaw, Institute of Applied Mathematics and Mechanics, Banacha 2, 02-097 Warsaw

2. 

Mathematics Institute, University of Warwick, Coventry, CV4 7AL., United Kingdom

Received  December 2008 Revised  November 2009 Published  April 2014

Given a non-autonomous process $U(\cdot,\cdot)$ on a complete separable metric space $X$ that has a pullback attractor $A(\cdot)$, we construct a family of invariant Borel probability measures $\{\mu_t\}_{t\in \mathbb{R}}$: the measures satisfy ${\rm supp }\,{\mu_t}\subset A(t)$ for all $t\in \mathbb{R}$ and the invariance property $\mu_t(E)=\mu_\tau(U(t,\tau)^{-1}E)$ for every Borel set $E\in X$. Our construction uses the generalised Banach limit. We then show that a Liouville-type equation holds for the evolution of $\mu_t$ under the process $U(\cdot,\cdot)$ generated by the ordinary differential equation $u_t=F(t,u)$ on a Banach space, and apply our theory to the non-autonomous 2D Navier--Stokes equations on unbounded domains satisfying a Poincaré inequality.
Citation: Grzegorz Łukaszewicz, James C. Robinson. Invariant measures for non-autonomous dissipative dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4211-4222. doi: 10.3934/dcds.2014.34.4211
References:
[1]

T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems,, Nonlinear Anal., 64 (2006), 484.  doi: 10.1016/j.na.2005.03.111.  Google Scholar

[2]

T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains,, C. R. Math. Acad. Sci. Paris, 342 (2006), 263.  doi: 10.1016/j.crma.2005.12.015.  Google Scholar

[3]

T. Caraballo, P. E. Kloeden and J. Real, Invariant measures and statistical solutions of the globally modified Navier-Stokes equations,, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 761.  doi: 10.3934/dcdsb.2008.10.761.  Google Scholar

[4]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems,, Springer, (2012).  doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[5]

M. Chekroun and N. E. Glatt-Holtz, Invariant measures for dissipative dynamical systems: Abstract results and applications,, Comm. Math. Phys., 316 (2012), 723.  doi: 10.1007/s00220-012-1515-y.  Google Scholar

[6]

C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence,, Cambridge University Press, (2001).  doi: 10.1017/CBO9780511546754.  Google Scholar

[7]

P. E. Kloeden, P. Marín-Rubio and J. Real, Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations,, Commun. Pure Appl. Anal., 8 (2009), 785.  doi: 10.3934/cpaa.2009.8.785.  Google Scholar

[8]

G. Łukaszewicz, Pullback attractors and statistical solutions for 2-D Navier-Stokes equations,, Discrete Continu. Dyn. Syst. Ser. B, 9 (2008), 643.  doi: 10.3934/dcdsb.2008.9.643.  Google Scholar

[9]

G. Łukaszewicz, J. Real and J. C. Robinson, Invariant measures for dissipative systems and generalised banach limits,, J. Dynam. Differential Equations, 23 (2011), 225.  doi: 10.1007/s10884-011-9213-6.  Google Scholar

[10]

V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations,, Princeton University Press, (1960).   Google Scholar

[11]

J. C. Robinson, Infinite-Dimensional Dynamical Systems,, Cambridge University Press, (2001).  doi: 10.1007/978-94-010-0732-0.  Google Scholar

[12]

R. Rosa, The global attractor for the 2 D Navier-Stokes flow on some unbounded domains,, Nonlinear Anal., 32 (1998), 71.  doi: 10.1016/S0362-546X(97)00453-7.  Google Scholar

[13]

R. Temam, Navier-Stokes equations, Theory and Numerical Analysis,, 2nd. ed., (1979).   Google Scholar

[14]

A. M. van der Vaart and J. A. Wellner, Weak Convergence and Empirical Processes: With Applications to Statistics,, Springer, (2000).   Google Scholar

[15]

X. Wang, Upper semi-continuity of stationary statistical properties of dissipative systems,, Discrete Contin. Dyn. Syst., 23 (2009), 521.  doi: 10.3934/dcds.2009.23.521.  Google Scholar

show all references

References:
[1]

T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems,, Nonlinear Anal., 64 (2006), 484.  doi: 10.1016/j.na.2005.03.111.  Google Scholar

[2]

T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains,, C. R. Math. Acad. Sci. Paris, 342 (2006), 263.  doi: 10.1016/j.crma.2005.12.015.  Google Scholar

[3]

T. Caraballo, P. E. Kloeden and J. Real, Invariant measures and statistical solutions of the globally modified Navier-Stokes equations,, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 761.  doi: 10.3934/dcdsb.2008.10.761.  Google Scholar

[4]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems,, Springer, (2012).  doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[5]

M. Chekroun and N. E. Glatt-Holtz, Invariant measures for dissipative dynamical systems: Abstract results and applications,, Comm. Math. Phys., 316 (2012), 723.  doi: 10.1007/s00220-012-1515-y.  Google Scholar

[6]

C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence,, Cambridge University Press, (2001).  doi: 10.1017/CBO9780511546754.  Google Scholar

[7]

P. E. Kloeden, P. Marín-Rubio and J. Real, Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations,, Commun. Pure Appl. Anal., 8 (2009), 785.  doi: 10.3934/cpaa.2009.8.785.  Google Scholar

[8]

G. Łukaszewicz, Pullback attractors and statistical solutions for 2-D Navier-Stokes equations,, Discrete Continu. Dyn. Syst. Ser. B, 9 (2008), 643.  doi: 10.3934/dcdsb.2008.9.643.  Google Scholar

[9]

G. Łukaszewicz, J. Real and J. C. Robinson, Invariant measures for dissipative systems and generalised banach limits,, J. Dynam. Differential Equations, 23 (2011), 225.  doi: 10.1007/s10884-011-9213-6.  Google Scholar

[10]

V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations,, Princeton University Press, (1960).   Google Scholar

[11]

J. C. Robinson, Infinite-Dimensional Dynamical Systems,, Cambridge University Press, (2001).  doi: 10.1007/978-94-010-0732-0.  Google Scholar

[12]

R. Rosa, The global attractor for the 2 D Navier-Stokes flow on some unbounded domains,, Nonlinear Anal., 32 (1998), 71.  doi: 10.1016/S0362-546X(97)00453-7.  Google Scholar

[13]

R. Temam, Navier-Stokes equations, Theory and Numerical Analysis,, 2nd. ed., (1979).   Google Scholar

[14]

A. M. van der Vaart and J. A. Wellner, Weak Convergence and Empirical Processes: With Applications to Statistics,, Springer, (2000).   Google Scholar

[15]

X. Wang, Upper semi-continuity of stationary statistical properties of dissipative systems,, Discrete Contin. Dyn. Syst., 23 (2009), 521.  doi: 10.3934/dcds.2009.23.521.  Google Scholar

[1]

Petr Kůrka. On the measure attractor of a cellular automaton. Conference Publications, 2005, 2005 (Special) : 524-535. doi: 10.3934/proc.2005.2005.524

[2]

Ammari Zied, Liard Quentin. On uniqueness of measure-valued solutions to Liouville's equation of Hamiltonian PDEs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 723-748. doi: 10.3934/dcds.2018032

[3]

Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3015-3027. doi: 10.3934/dcdsb.2016085

[4]

Yan Wang, Guanggan Chen. Invariant measure of stochastic fractional Burgers equation with degenerate noise on a bounded interval. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3121-3135. doi: 10.3934/cpaa.2019140

[5]

Jonathan C. Mattingly, Etienne Pardoux. Invariant measure selection by noise. An example. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4223-4257. doi: 10.3934/dcds.2014.34.4223

[6]

Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114-118.

[7]

Zeqi Zhu, Caidi Zhao. Pullback attractor and invariant measures for the three-dimensional regularized MHD equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1461-1477. doi: 10.3934/dcds.2018060

[8]

Paola Mannucci, Claudio Marchi, Nicoletta Tchou. Asymptotic behaviour for operators of Grushin type: Invariant measure and singular perturbations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (1) : 119-128. doi: 10.3934/dcdss.2019008

[9]

Boris Kalinin, Anatole Katok. Measure rigidity beyond uniform hyperbolicity: invariant measures for cartan actions on tori. Journal of Modern Dynamics, 2007, 1 (1) : 123-146. doi: 10.3934/jmd.2007.1.123

[10]

Simon Lloyd, Edson Vargas. Critical covering maps without absolutely continuous invariant probability measure. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2393-2412. doi: 10.3934/dcds.2019101

[11]

Piotr Gwiazda, Sander C. Hille, Kamila Łyczek, Agnieszka Świerczewska-Gwiazda. Differentiability in perturbation parameter of measure solutions to perturbed transport equation. Kinetic & Related Models, 2019, 12 (5) : 1093-1108. doi: 10.3934/krm.2019041

[12]

David Gómez-Castro, Juan Luis Vázquez. The fractional Schrödinger equation with singular potential and measure data. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7113-7139. doi: 10.3934/dcds.2019298

[13]

Jawad Al-Khal, Henk Bruin, Michael Jakobson. New examples of S-unimodal maps with a sigma-finite absolutely continuous invariant measure. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 35-61. doi: 10.3934/dcds.2008.22.35

[14]

Boris Kalinin, Anatole Katok, Federico Rodriguez Hertz. Errata to "Measure rigidity beyond uniform hyperbolicity: Invariant measures for Cartan actions on tori" and "Uniqueness of large invariant measures for $\Zk$ actions with Cartan homotopy data". Journal of Modern Dynamics, 2010, 4 (1) : 207-209. doi: 10.3934/jmd.2010.4.207

[15]

Welington Cordeiro, Manfred Denker, Xuan Zhang. On specification and measure expansiveness. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1941-1957. doi: 10.3934/dcds.2017082

[16]

Welington Cordeiro, Manfred Denker, Xuan Zhang. Corrigendum to: On specification and measure expansiveness. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3705-3706. doi: 10.3934/dcds.2018160

[17]

Tomasz Downarowicz, Yonatan Gutman, Dawid Huczek. Rank as a function of measure. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2741-2750. doi: 10.3934/dcds.2014.34.2741

[18]

Francesco Di Plinio, Gregory S. Duane, Roger Temam. Time-dependent attractor for the Oscillon equation. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 141-167. doi: 10.3934/dcds.2011.29.141

[19]

Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194

[20]

Zari Dzalilov, Iradj Ouveysi, Alexander Rubinov. An extended lifetime measure for telecommunication network. Journal of Industrial & Management Optimization, 2008, 4 (2) : 329-337. doi: 10.3934/jimo.2008.4.329

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (19)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]