Remarks on the Fractal Dimension of Bi-space Global and Exponential Attractors
Year: 2008
Type: article
Abstract: Bi-spaces global and exponential attractors for the time continuous dynamical systems are considered and the bounds on their fractal dimension are discussed in the context of the smoothing properties of the system between ap- propriately chosen function spaces. A unifled analytic semigroup approach to abstract parabolic equations is described and applications to the sample prob- lems are given. Sunto. In questo lavoro sono considerate le nozioni di attrattori globali ed espo- nenziali \bi-spaces per sistemi dinamici continui, e discusse limitazioni relative alla loro dimensione frattale in spazi di funzioni opportuni. Inoltre, viene pre- sentato un approccio uniflcato per lo studio di problemi parabolici astratti, nella teoria dei semigruppi analitici, ed alcuni esempi sono mostrati. 1. Introductory notes Attractors for the dynamical systems governed by partial difierential equations in inflnite dimensional Banach spaces have been considered by several authors within past few decades; see (3, 8, 11, 16, 18, 28, 1, 10, 12, 14, 21) and references therein. A general approach leading to the description of their topological dimension have essentially been developed, allowing to obtain much of the relevant information about the asymptotic behavior of the systems corresponding to a number of physical equations in both Hilbert and Banach function space setting. In this article we reconstruct the results of (1, 3, 11, 12) concerning the bi-spaces global and exponential attractors and provide simultaneously certain bounds on their fractal dimension, which are a straightforward consequence of the smoothing properties of the dynamical system acting between appropriately chosen function spaces. These mentioned properties are known to be typical for problems that fall into a class of abstract parabolic equations in Banach spaces. Since the latter class contains many relevant physical equations, it is reasonable to describe the speciflc 'dissipative and smoothing mechanism' that leads to the existence of the global and exponential attractors and provides the bounds on their fractal dimension within the unifled analytic semigroup approach of (17).
Authors: Jan W. Cholewa, Radosław Czaja, Gianluca Mola
Language: Italian
Subfield: Control and Systems Engineering
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W54584309