Pullback attractors for non-autonomous 2D-Navier–Stokes equations in some unbounded domains

Tomás Caraballo; Grzegorz Łukaszewicz; José Real

doi:10.1016/j.crma.2005.12.015

Dynamical Systems

Pullback attractors for non-autonomous 2D-Navier–Stokes equations in some unbounded domains

Presented by: Haïm Brezis

Tomás Caraballo ¹; Grzegorz Łukaszewicz ²; José Real ¹

¹ Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080 Sevilla, Spain
² University of Warsaw, Institute of Applied Mathematics and Mechanics, Banacha 2, 02-097 Warsaw, Poland

Comptes Rendus. Mathématique, Volume 342 (2006) no. 4, pp. 263-268.

Abstract
Résumé

In this Note we first introduce the concept of pullback asymptotic compactness. Next, we establish a result ensuring the existence of a pullback attractor for a non-autonomous dynamical system under the general assumptions of pullback asymptotic compactness and the existence of a pullback absorbing family of sets. Finally, we prove the existence of a pullback attractor for a non-autonomous 2D Navier–Stokes model in an unbounded domain, a case in which the theory of uniform attractors does not work since the non-autonomous term is quite general.

Dans cette Note, on présente d'abord la notion de compacité asymptotique pullback. On établit ensuite un résultat d'existence d'un attracteur pullback pour un système dynamique non autonome, sous les hypothèses de compacité asymptotique pullback et d'existence d'une famille d'ensembles absorbants au sens pullback. On prouve finalement l'existence d'un attracteur pullback pour un système de Navier–Stokes bidimensionel non autonome dans un domaine non borné, une situation dans laquelle, étant donnée la généralité du terme non autonome, la théorie des attracteurs uniformes ne peut pas être appliquée.

Received: 2005-02-28
Accepted: 2005-12-07
Published online: 2006-01-05

DOI: 10.1016/j.crma.2005.12.015

Author's affiliations:

Tomás Caraballo ¹; Grzegorz Łukaszewicz ²; José Real ¹

¹ Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080 Sevilla, Spain
² University of Warsaw, Institute of Applied Mathematics and Mechanics, Banacha 2, 02-097 Warsaw, Poland

@article{CRMATH_2006__342_4_263_0,
     author = {Tom\'as Caraballo and Grzegorz {\L}ukaszewicz and Jos\'e Real},
     title = {Pullback attractors for non-autonomous {2D-Navier{\textendash}Stokes} equations in some unbounded domains},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {263--268},
     publisher = {Elsevier},
     volume = {342},
     number = {4},
     year = {2006},
     doi = {10.1016/j.crma.2005.12.015},
     language = {en},
}

TY  - JOUR
AU  - Tomás Caraballo
AU  - Grzegorz Łukaszewicz
AU  - José Real
TI  - Pullback attractors for non-autonomous 2D-Navier–Stokes equations in some unbounded domains
JO  - Comptes Rendus. Mathématique
PY  - 2006
SP  - 263
EP  - 268
VL  - 342
IS  - 4
PB  - Elsevier
DO  - 10.1016/j.crma.2005.12.015
LA  - en
ID  - CRMATH_2006__342_4_263_0
ER  -

%0 Journal Article
%A Tomás Caraballo
%A Grzegorz Łukaszewicz
%A José Real
%T Pullback attractors for non-autonomous 2D-Navier–Stokes equations in some unbounded domains
%J Comptes Rendus. Mathématique
%D 2006
%P 263-268
%V 342
%N 4
%I Elsevier
%R 10.1016/j.crma.2005.12.015
%G en
%F CRMATH_2006__342_4_263_0

Tomás Caraballo; Grzegorz Łukaszewicz; José Real. Pullback attractors for non-autonomous 2D-Navier–Stokes equations in some unbounded domains. Comptes Rendus. Mathématique, Volume 342 (2006) no. 4, pp. 263-268. doi : 10.1016/j.crma.2005.12.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.12.015/

References
Cited by

[1] J.M. Ball Global attractors for damped semilinear wave equations, Discrete Contin. Dynam. Systems, Volume 10 (2004) no. 1–2, pp. 31-52

[2] T. Caraballo; J.A. Langa On the upper semicontinuity of cocycle attractors for nonautonomous and random dynamical systems, Dynam. Contin. Discrete Impuls. Systems A, Volume 10 (2003), pp. 491-514

[3] T. Caraballo, G. Łukaszewicz, J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, manuscript, Technical Report no. 141 of the Institute of Applied Mathematics and Mechanics, University of Warsaw

[4] V.V. Chepyzhov; M.I. Vishik Attractors for Equations of Mathematical Physics, Colloq. Publ., vol. 49, American Mathematical Society, Providence, RI, 2002

[5] H. Crauel; A. Debussche; F. Flandoli Random attractors, J. Dynam. Differential Equations, Volume 9 (1995) no. 2, pp. 307-341

[6] P.E. Kloeden; B. Schmalfuss Asymptotic behaviour of nonautonomous difference inclusions, Systems Control Lett., Volume 33 (1998) no. 4, pp. 275-280

[7] J.A. Langa; B. Schmalfuss Finite dimensionality of attractors for non-autonomous dynamical systems given by partial differential equations, Stochastics and Dynamics, Volume 4 (2004) no. 3, pp. 385-404

[8] J.L. Lions Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Gauthier-Villars, Paris, 1969

[9] G. Łukaszewicz; W. Sadowski Uniform attractor for 2D magneto-micropolar fluid flow in some unbounded domains, Z. Angew. Math. Phys., Volume 55 (2004), pp. 1-11

[10] I. Moise; R. Rosa; X. Wang Attractors for noncompact nonautonomous systems via energy equations, Discrete Contin. Dynam. Systems, Volume 10 (2004) no. 1 & 2, pp. 473-496

[11] R. Rosa The global attractor for the 2D Navier–Stokes flow on some unbounded domains, Nonlinear Anal., Volume 32 (1998) no. 1, pp. 71-85

[12] B. Schmalfuss Attractors for non-autonomous dynamical systems (B. Fiedler; K. Gröger; J. Sprekels, eds.), Proc. Equadiff 99, Berlin, World Scientific, 2000, pp. 684-689

[13] G.R. Sell Non-autonomous differential equations and topological dynamics, I, II, Trans. Amer. Math. Soc., Volume 127 (1967), pp. 241-262 (263–283)

[14] R. Temam Navier–Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, 1979

Cited by Sources:

Comments - Policy