Invariant measures for non-autonomous dissipative dynamical systems

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

Invariant measures for non-autonomous dissipative dynamical systems

Grzegorz Łukaszewicz ^1, and James C. Robinson ^2,

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

Abstract
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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.

Keywords: time-averaged measure, invariant measure., Pullback attractor, Banach limits, Liouville equation.

Mathematics Subject Classification: 35B41, 35D99, 76F2.

Citation: Grzegorz Łukaszewicz, James C. Robinson. Invariant measures for non-autonomous dissipative dynamical systems. Discrete and Continuous Dynamical Systems, 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), 484-498. doi: 10.1016/j.na.2005.03.111.
[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), 263-268. doi: 10.1016/j.crma.2005.12.015.
[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), 761-781. doi: 10.3934/dcdsb.2008.10.761.
[4]	A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Springer, New York, 2012. doi: 10.1007/978-1-4614-4581-4.
[5]	M. Chekroun and N. E. Glatt-Holtz, Invariant measures for dissipative dynamical systems: Abstract results and applications, Comm. Math. Phys., 316 (2012), 723-761. doi: 10.1007/s00220-012-1515-y.
[6]	C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, 2001. doi: 10.1017/CBO9780511546754.
[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), 785-802. doi: 10.3934/cpaa.2009.8.785.
[8]	G. Łukaszewicz, Pullback attractors and statistical solutions for 2-D Navier-Stokes equations, Discrete Continu. Dyn. Syst. Ser. B, 9 (2008), 643-659. doi: 10.3934/dcdsb.2008.9.643.
[9]	G. Łukaszewicz, J. Real and J. C. Robinson, Invariant measures for dissipative systems and generalised banach limits, J. Dynam. Differential Equations, 23 (2011), 225-250. doi: 10.1007/s10884-011-9213-6.
[10]	V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, 1960.
[11]	J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, Cambridge, 2001. doi: 10.1007/978-94-010-0732-0.
[12]	R. Rosa, The global attractor for the 2 D Navier-Stokes flow on some unbounded domains, Nonlinear Anal., 32 (1998), 71-85. doi: 10.1016/S0362-546X(97)00453-7.
[13]	R. Temam, Navier-Stokes equations, Theory and Numerical Analysis, 2nd. ed., North Holland, Amsterdam, 1979.
[14]	A. M. van der Vaart and J. A. Wellner, Weak Convergence and Empirical Processes: With Applications to Statistics, Springer, New York, 2000.
[15]	X. Wang, Upper semi-continuity of stationary statistical properties of dissipative systems, Discrete Contin. Dyn. Syst., 23 (2009), 521-540. doi: 10.3934/dcds.2009.23.521.

show all references

References:

[1]	T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111.
[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), 263-268. doi: 10.1016/j.crma.2005.12.015.
[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), 761-781. doi: 10.3934/dcdsb.2008.10.761.
[4]	A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Springer, New York, 2012. doi: 10.1007/978-1-4614-4581-4.
[5]	M. Chekroun and N. E. Glatt-Holtz, Invariant measures for dissipative dynamical systems: Abstract results and applications, Comm. Math. Phys., 316 (2012), 723-761. doi: 10.1007/s00220-012-1515-y.
[6]	C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, 2001. doi: 10.1017/CBO9780511546754.
[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), 785-802. doi: 10.3934/cpaa.2009.8.785.
[8]	G. Łukaszewicz, Pullback attractors and statistical solutions for 2-D Navier-Stokes equations, Discrete Continu. Dyn. Syst. Ser. B, 9 (2008), 643-659. doi: 10.3934/dcdsb.2008.9.643.
[9]	G. Łukaszewicz, J. Real and J. C. Robinson, Invariant measures for dissipative systems and generalised banach limits, J. Dynam. Differential Equations, 23 (2011), 225-250. doi: 10.1007/s10884-011-9213-6.
[10]	V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, 1960.
[11]	J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, Cambridge, 2001. doi: 10.1007/978-94-010-0732-0.
[12]	R. Rosa, The global attractor for the 2 D Navier-Stokes flow on some unbounded domains, Nonlinear Anal., 32 (1998), 71-85. doi: 10.1016/S0362-546X(97)00453-7.
[13]	R. Temam, Navier-Stokes equations, Theory and Numerical Analysis, 2nd. ed., North Holland, Amsterdam, 1979.
[14]	A. M. van der Vaart and J. A. Wellner, Weak Convergence and Empirical Processes: With Applications to Statistics, Springer, New York, 2000.
[15]	X. Wang, Upper semi-continuity of stationary statistical properties of dissipative systems, Discrete Contin. Dyn. Syst., 23 (2009), 521-540. doi: 10.3934/dcds.2009.23.521.

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