TL;DR
This survey reviews the development and current state of attractor theory in dissipative PDEs, covering classical results, modern extensions, and various types of attractors, with illustrative examples.
Contribution
It provides a comprehensive overview of classical and recent results in attractor theory for dissipative PDEs, including new insights into non-autonomous and random attractors.
Findings
Summary of classical attractor results
Discussion of modern attractor types and properties
Illustrative examples and counter-examples
Abstract
This survey is dedicated to the 100th anniversary of Mark Iosifovich Vishik and is based on a number of mini-courses taught by the author at University of Surrey (UK) and Lanzhou University (China). It discusses the classical and modern results of the theory of attractors for dissipative PDEs including attractors for autonomous and non-autonomous equations, dynamical systems in general topological spaces, various types of trajectory, pullback and random attractors, exponential attractors, determining functionals and inertial manifolds as well as the dimension theory for the above mentioned classes of attractors. The theoretical results are illustrated by a number of clarifying examples and counter-examples.
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Taxonomy
TopicsStability and Controllability of Differential Equations · Experimental and Theoretical Physics Studies · Mathematical Dynamics and Fractals