Spectral comparison between the second and the fourth order equations of conservative type with non-local terms

Isamu Ohnishi, Yasumasa Nishiura

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

We present a spectral comparison theorem between the second and the fourth order equations of conservative type with non-local terms. Nonlocal effects arise naturally due to the long-range spatial connectivity in polymer problems or to the difference of relaxation times for phase separation problems with stress effect. If such nonlocal effects are built into the usual Cahn-Hilliard dynamics, we have the fourth order equations with nonlocal terms. We introduce the second order conservative equations with the same nonlocal terms as the fourth order ones. The aim is to show that both the second and the fourth order equations have the same set of steady states and their stability properties also coincide with each other. This reduction from the fourth order to the second order is quite useful in applications. In fact a simple and new proof for the instability of n-layered solution of the Cahn-Hilliard equation is given with the aid of this reduction.

Original languageEnglish
Pages (from-to)253-262
Number of pages10
JournalJapan Journal of Industrial and Applied Mathematics
Volume15
Issue number2
DOIs
Publication statusPublished - 1998 Jan 1
Externally publishedYes

Keywords

  • Critical eigenvalues
  • Spectral comparison
  • The Cahn-Hilliard equations
  • The conservative reaction diffusion equations
  • Variational method

ASJC Scopus subject areas

  • Engineering(all)
  • Applied Mathematics

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