Caraballo, Tomás; Łukaszewicz, Grzegorz; Real, José Pullback attractors for non-autonomous 2D-Navier–Stokes equations in some unbounded domains. (English) Zbl 1085.37054 C. R., Math., Acad. Sci. Paris 342, No. 4, 263-268 (2006). Summary: We first introduce the concept of pullback asymptotic compactness. Next, we establish a result ensuring the existence of a pullback attractor for a nonautonomous 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 nonautonomous 2D Navier-Stokes model in an unbounded domain, a case in which the theory of uniform attractors does not work since the nonautonomous term is quite general. Cited in 78 Documents MSC: 37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems 35Q30 Navier-Stokes equations 35B41 Attractors 37N10 Dynamical systems in fluid mechanics, oceanography and meteorology 37B55 Topological dynamics of nonautonomous systems 74D05 Linear constitutive equations for materials with memory Keywords:pullback asymptotic compactness; pullback attractor; nonautonomous dynamical system; pullback absorbing family of sets; uniform attractors PDFBibTeX XMLCite \textit{T. Caraballo} et al., C. R., Math., Acad. Sci. Paris 342, No. 4, 263--268 (2006; Zbl 1085.37054) Full Text: DOI References: [1] Ball, J. M., Global attractors for damped semilinear wave equations, Discrete Contin. Dynam. Systems, 10, 1-2, 31-52 (2004) · Zbl 1056.37084 [2] Caraballo, T.; Langa, J. A., On the upper semicontinuity of cocycle attractors for nonautonomous and random dynamical systems, Dynam. Contin. Discrete Impuls. Systems A, 10, 491-514 (2003) · Zbl 1035.37013 [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; 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 · Zbl 1128.37019 [4] Chepyzhov, V. V.; Vishik, M. I., Attractors for Equations of Mathematical Physics, Colloq. Publ., vol. 49 (2002), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0986.35001 [5] Crauel, H.; Debussche, A.; Flandoli, F., Random attractors, J. Dynam. Differential Equations, 9, 2, 307-341 (1995) · Zbl 0884.58064 [6] Kloeden, P. E.; Schmalfuss, B., Asymptotic behaviour of nonautonomous difference inclusions, Systems Control Lett., 33, 4, 275-280 (1998) · Zbl 0902.93043 [7] Langa, J. A.; Schmalfuss, B., Finite dimensionality of attractors for non-autonomous dynamical systems given by partial differential equations, Stochastics and Dynamics, 4, 3, 385-404 (2004) · Zbl 1057.37069 [8] Lions, J. L., Quelques méthodes de résolution des problèmes aux limites non linéaires (1969), Dunod, Gauthier-Villars: Dunod, Gauthier-Villars Paris · Zbl 0189.40603 [9] Łukaszewicz, G.; Sadowski, W., Uniform attractor for 2D magneto-micropolar fluid flow in some unbounded domains, Z. Angew. Math. Phys., 55, 1-11 (2004) [10] Moise, I.; Rosa, R.; Wang, X., Attractors for noncompact nonautonomous systems via energy equations, Discrete Contin. Dynam. Systems, 10, 1 & 2, 473-496 (2004) · Zbl 1060.35023 [11] Rosa, R., The global attractor for the 2D Navier-Stokes flow on some unbounded domains, Nonlinear Anal., 32, 1, 71-85 (1998) · Zbl 0901.35070 [12] Schmalfuss, B., Attractors for non-autonomous dynamical systems, (Fiedler, B.; Gröger, K.; Sprekels, J., Proc. Equadiff 99, Berlin (2000), World Scientific), 684-689 · Zbl 0971.37038 [13] Sell, G. R., Non-autonomous differential equations and topological dynamics, I, II, Trans. Amer. Math. Soc., 127, 241-262 (1967), 263-283 · Zbl 0189.39602 [14] Temam, R., Navier-Stokes Equations, Theory and Numerical Analysis (1979), North-Holland: North-Holland Amsterdam · Zbl 0426.35003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.