Necessary and sufficient conditions for the existence of global attractors for semigroups and applications

Abstract

First we establish some necessary and sufficient conditions for the existence of the global attractor of an infinite dimensional dynamical system, using the measure of noncompactness. Then we give a new method/recipe for proving the existence of the global attractor. The main advantage of this new method/recipe is that one needs only to verify a necessary compactness condition with the same type of energy estimates as those for establishing the absorbing set. In other words, one doesn't need to obtain estimates in function spaces of higher regularity. In particular, this property is useful when higher regularity is not available, as demonstrated in the example on the Navier-Stokes equations on nonsmooth domains.

Attractors of Local Semiflows on Topological Spaces

“…The attractor theories in metric spaces (especially nonlocally compact metric spaces) were fully developed in the past decades for both autonomous and nonautonomous systems [1,3,4,8,10,13,16,[19][20][21]. Here we are interested in the case where the phase space is a topological one that may not be metrizable.…”

Notes on the Global Attractors for Semigroup

“…In this section, firstly we recall some basic definitions in [1,7], then we show that the two necessary and sufficient conditions for the existence of global attractors for semigroups are equivalent directly. Definition 3.1 Let M be a complete metric space.…”

“…However, it is difficult or even impossible to verify the uniform compactness of the semigroup for many problems. In [1], the authors use the measure of noncompactness of a set to introduce a new concept of compactness called  -limit compact, then they show that there exists a global attractor for a semigroup if and only if: 0 C 1) there is an absorbing set, and 2) the semigroup is  -limit compact.…”

“…Our main motivation of this paper is to prove that asymptotically compact   -limit compact, and then we prove that the conditions in [1,7] are equivalent directly in…”

Global attractor of the extended Fisher-Kolmogorov equation in H k spaces

“…So, many authors are interested in the existence of global attractors such as Hale, Temam, among others [13][14][15][16][17][18][19][20][21][22][23].…”