Pullback attractors for asymptotically compact non-autonomous dynamical systems

By Tomás Caraballo, G. Łukaszewicz, José Real. 2006

434 citations recorded by the source corpus.

Abstract First, we introduce the concept of pullback asymptotically compact non-autonomous dynamical system as an extension of the similar concept in the autonomous framework. Our definition is different from that of asymptotic compactness already used in the theory of random and non-autonomous dynamical systems (as developed by Crauel, Flandoli, Kloeden, Schmalfuss, amongst others) which means the existence of a (random or time-dependent) family of compact attracting sets. Next, we prove 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. This attractor is minimal and, in most practical applications, it is unique. Finally, we illustrate the theory with a 2D Navier–Stokes model in an unbounded domain.

Subfields: Computational Theory and Mathematics, Computer Networks and Communications, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/63af71c607b969cdb4a5f6d75e440b1fb83a08d3

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Citations in the corpus

Non--autonomous and random attractors for delay random semilinear equations without uniqueness (2008, background): “[13], Cheban [17], Chueshov [19], Crauel and Flandoli [20], Flandoli and Schmalfuß [21], Kloeden [24], Kloeden and Schmalfuß [25], Robinson [28], Schmalfuß[30], amongst many others).”
Pullback Exponential Attractors for Non-autonomous Lattice Systems (2012, background): “Global attractors, uniform attractors and pullback attractors (or kernel sections) all play important roles in the study of asymptotic behavior of autonomous and non-autonomous infinite dynamical systems [1–10].”
Pullback attractors for a class of non-autonomous nonclassical diffusion equations (2010, background): “The theory of pullback attractors has been developed for both non-autonomous and random dynamical systems and it has been shown to be very useful in the understanding of the dynamics of non-autonomous dynamical systems (see [10] and references therein).”
Sufficient and Necessary Criteria for Existence of Pullback Attractors for Non-compact Random Dynamical Systems (2012, background): “The existence of pullback attractors for such Φ has been investigated by many authors in [9, 10, 35] and the references therein.”
Pullback attractors for a semilinear heat equation in a non-cylindrical domain (2008, methodology): “This will be done in terms of the process formulation of a non-autonomous dynamical system rather than the cocycle formalism (see [6,12]) since the former is more appropriate in the present context.”
Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions (2010, background): “[4], [5] and [6], Chueshov [8], Flandoli and Schmalfuß [9], Kloeden [13], Kloeden and Schmalfuß [14], Robinson [16], Schmalfuß [17], amongst many others.”
Attractors for differential equations with unbounded delays (2007, background): “It is based in the paper [ 8 ], although there it is stated in another framework of dynamical systems: tempered sets.”
On the existence of pullback attractors for non-autonomous reaction–diffusion equations (2008, positive): “However, as we know, even given the more general external force, we can also obtain the pullback attractors for many problems, see 3 , 4 for Navier–Stokes equations, 2 for wave equations, and 1 for reaction–diffusion equations.”
Existence of pullback attractors for pullback asymptotically compact processes (2010, background): “In that case, the introduced definitions of dissipation and asymptotic compactness should be changed accordingly to cope with these other classes (see [2] for more details).”
Pullback Attractors for 2D Navier-Stokes Equations with Delays and Their Regularity (2013, background): “3 Abstract results on minimal pullback attractors In this section we recall briefly some results from [9] about the existence of minimal pullback attractors (see also [2, 3, 18]).”