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Pullback attractors for non-autonomous 2D-Navier–Stokes equations in some unbounded domains. (English) Zbl 1085.37054

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.

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
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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
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