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Necessary and sufficient conditions for the existence of global attractors for semigroups and applications. (English) Zbl 1028.37047
Summary: 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 does not 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.

MSC:
37L30Attractors and their dimensions, Lyapunov exponents
35Q30Stokes and Navier-Stokes equations
76D05Navier-Stokes equations (fluid dynamics)