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Inertial manifolds and finite-dimensional reduction for dissipative PDEs*

Published online by Cambridge University Press:  01 December 2014

Sergey Zelik
Sergey Zelik*
Affiliation:
Department of Mathematics, University of Surrey, Guildford GU2 7XH, UK, (s.zelik@surrey.ac.uk) Lobachevsky State University of Nizhny Novgorod, ul. Ulyanova 10, Nizhny Novgorod 603005, Russia
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Abstract

This paper is devoted to the problem of finite-dimensional reduction for parabolic partial differential equations. We give a detailed exposition of the classical theory of inertial manifolds as well as various attempts to generalize it based on the so-called Mañé projection theorems. The recent counter-examples showing that the underlying dynamics may in a sense be infinite dimensional if the spectral gap condition is violated, as well as a discussion of the most important open problems, are also included.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 144 , Issue 6 , December 2014 , pp. 1245 - 1327
Copyright
Copyright © Royal Society of Edinburgh 2014 

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Footnotes

*

This paper is a late addition to the papers surveying active areas in partial differential equations, published in issues 141.2 and 142.6, which were based on a series of mini-courses held in Edinburgh from 2010 to 2013.

References

* This paper is a late addition to the papers surveying active areas in partial differential equations, published in issues 141.2 and 142.6, which were based on a series of mini-courses held in Edinburgh from 2010 to 2013.