The geometry of the phase diffusion equation

Research output: Contribution to journalArticle

38 Citations (Scopus)

Abstract

The Cross-Newell phase diffusion equation, τ(|k|)ΘT = -∇ · (B(|k|) · k), k = ∇ Θ, and its regularization describes natural patterns and defects far from onset in large aspect ratio systems with rotational symmetry. In this paper we construct explicit solutions of the unregularized equation and suggest candidates for its weak solutions. We confirm these ideas by examining a fourth-order regularized equation in the limit of infinite aspect ratio. The stationary solutions of this equation include the minimizers of a free energy, and we show these minimizers are remarkably well-approximated by a second-order "self-dual" equation. Moreover, the self-dual solutions give upper bounds for the free energy which imply the existence of weak limits for the asymptotic minimizers. In certain cases, some recent results of Jin and Kohn [28] combined with these upper bounds enable us to demonstrate that the energy of the asymptotic minimizers converges to that of the self-dual solutions in a viscosity limit.

Original languageEnglish (US)
Pages (from-to)223-274
Number of pages52
JournalJournal of Nonlinear Science
Volume10
Issue number2
StatePublished - Mar 2000

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Minimizer
Diffusion equation
Dual Solutions
Free energy
Geometry
Aspect ratio
geometry
Aspect Ratio
Free Energy
aspect ratio
Upper bound
Weak Limit
Fourth-order Equations
Rotational symmetry
free energy
Stationary Solutions
Explicit Solution
Viscosity
Weak Solution
Regularization

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mathematics(all)
  • Applied Mathematics
  • Mathematical Physics
  • Statistical and Nonlinear Physics

Cite this

The geometry of the phase diffusion equation. / Ercolani, Nicholas M; Indik, Robert A; Newell, Alan C; Passot, T.

In: Journal of Nonlinear Science, Vol. 10, No. 2, 03.2000, p. 223-274.

Research output: Contribution to journalArticle

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