Invariant manifolds and the stability of traveling waves in scalar viscous conservation laws

Margaret Beck, C. Eugene Wayne

Research output: Contribution to journalArticle

Abstract

The stability of traveling wave solutions of scalar viscous conservation laws is investigated by decomposing perturbations into three components: two far-field components and one near-field component. The linear operators associated to the far-field components are the constant coefficient operators determined by the asymptotic spatial limits of the original operator. Scaling variables can be applied to study the evolution of these components, allowing for the construction of invariant manifolds and the determination of their temporal decay rate. The large time evolution of the near-field component is shown to be governed by that of the far-field components, thus giving it the same temporal decay rate. We also give a discussion of the relationship between this geometric approach and previous results, which demonstrate that the decay rate of perturbations can be increased by requiring that initial data lie in appropriate algebraically weighted spaces. (C) 2007 Elsevier Inc. All rights reserved.

Original languageEnglish
Pages (from-to)87-116
Number of pages30
JournalJournal of Differential Equations
Volume244
Issue number1
DOIs
Publication statusPublished - 1 Jan 2008

Keywords

  • stability of traveling waves
  • similarity variables
  • invariant manifolds
  • scalar viscous conservation laws
  • weighted function spaces
  • EQUATIONS

Cite this

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Invariant manifolds and the stability of traveling waves in scalar viscous conservation laws. / Beck, Margaret; Wayne, C. Eugene.

In: Journal of Differential Equations, Vol. 244, No. 1, 01.01.2008, p. 87-116.

Research output: Contribution to journalArticle

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AU - Beck, Margaret

AU - Wayne, C. Eugene

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AB - The stability of traveling wave solutions of scalar viscous conservation laws is investigated by decomposing perturbations into three components: two far-field components and one near-field component. The linear operators associated to the far-field components are the constant coefficient operators determined by the asymptotic spatial limits of the original operator. Scaling variables can be applied to study the evolution of these components, allowing for the construction of invariant manifolds and the determination of their temporal decay rate. The large time evolution of the near-field component is shown to be governed by that of the far-field components, thus giving it the same temporal decay rate. We also give a discussion of the relationship between this geometric approach and previous results, which demonstrate that the decay rate of perturbations can be increased by requiring that initial data lie in appropriate algebraically weighted spaces. (C) 2007 Elsevier Inc. All rights reserved.

KW - stability of traveling waves

KW - similarity variables

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KW - weighted function spaces

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