The role of self-similarity in singularities of PDEs

JG Eggers, MA Fontelos

Research output: Contribution to journalArticle (Academic Journal)

87 Citations (Scopus)

Abstract

We survey rigorous, formal and numerical results on the formation of point-like singularities (or blow-up) for a wide range of evolution equations. We use a similarity transformation of the original equation with respect to the blow-up point, such that self-similar behaviour is mapped to the fixed point of a dynamical system. We point out that analysing the dynamics close to the fixed point is a useful way of characterizing the singularity, in that the dynamics frequently reduces to very few dimensions. As far as we are aware, examples from the literature either correspond to stable fixed points, low-dimensional centre-manifold dynamics, limit cycles or travelling waves. For each 'class' of singularity, we give detailed examples
Translated title of the contributionThe role of self-similarity in singularities of PDEs
Original languageEnglish
Pages (from-to)R1 - R44
Number of pages44
JournalNonlinearity
Volume22, issue 1
DOIs
Publication statusPublished - Jan 2009

Bibliographical note

Publisher: IOP

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