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Pullback Attractors and Shear Flows
2016In this chapter we consider the problem of existence and finite dimensionality of the pullback attractor for a class of two-dimensional turbulent boundary driven flows which naturally appear in lubrication theory. We generalize here the results from Chap. 9 to the non-autonomous problem.
Grzegorz Łukaszewicz, Piotr Kalita
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Pullback Attractors and Statistical Solutions
2016This chapter is devoted to constructions of invariant measures and statistical solutions for non-autonomous Navier–Stokes equations in bounded and certain unbounded domains in \(\mathbb{R}^{2}\).After introducing some basic notions and results concerning attractors in the context of the Navier–Stokes equations, we construct the family of probability ...
Grzegorz Łukaszewicz, Piotr Kalita
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On pullback attractors in for nonautonomous reaction–diffusion equations
Nonlinear Analysis: Theory, Methods & Applications, 2010Summary: Using a recent method based on the concept of the Kuratowski measure of noncompactness of a bounded set together with some new estimates of solutions, we prove the existence of a unique minimal pullback attractor for the evolutionary process associated with a nonautonomous nonlinear reaction-diffusion system in \(H^1_0\) in which the right ...
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Stochastic Synchronization of Random Pullback Attractors
2020In this and in the next chapter we deal with the synchronization of random dynamical systems (RDS). The concept of RDS (see Arnold [4] and the literature cited therein) covers the most important families of dynamical systems with randomness, including random and stochastic ordinary and partial differential equations and random difference equations ...
Igor Chueshov, Björn Schmalfuß
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Pullback Attractors for NonAutonomous Dynamical Systems
2013We study a nonautonomous reaction-diffusion equation with zero Dirichlet boundary condition, in an unbounded domain containing a nonautonomous forcing term taking values in the space H −1, and with a continuous nonlinearity which does not ensure uniqueness of solution. Using results of the theory of set-valued nonautonomous (pullback) dynamical systems,
María Anguiano +3 more
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Pullback Exponential Attractors for Nonautonomous Reaction–Diffusion Equations
International Journal of Bifurcation and Chaos, 2015This paper presents a necessary and sufficient condition to prove the existence of the pullback exponential attractor. The asymptotic a priori estimate method is used to produce an abstract result on the existence of the pullback exponential attractor in a strong space without regularity. The established results are illustrated by applying them to the
Yan, Xingjie, Qi, Wei
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Pullback attractors in nonautonomous difference equations
Journal of Difference Equations and Applications, 2000Nonautonomous difference equations are formulated as cocycles which generalize semigroups corresponding to autonomous difference equations. Pullback attractors are the appropriate generalization of autonomous attractors to cocycles. The existence of a pullback attractor follows when the difference equation cocycle has a pullback absorbing set.
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A uniformly exponential random forward attractor which is not a pullback attractor
Archiv der Mathematik, 2002The author constructs an example of a random forward attractor for a random dynamical system (RDS) that is not a pullback attractor which is one of 3 possible notions of an attractor one can naturally define for RDS (and which all coincide for deterministic dynamical systems).
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