Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-12T00:08:28.534Z Has data issue: false hasContentIssue false

Inertial manifolds and finite-dimensional reduction for dissipative PDEs*

Published online by Cambridge University Press:  01 December 2014

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

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
Copyright
Copyright © Royal Society of Edinburgh 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.