Grain boundary motion arising from the gradient flow of the Aviles-Giga functional

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Abstract

This paper considers the singular limit of the equation Θt=-εΔ2Θ+ε-1∇·([|∇Θ|2-1] ∇Θ). Grain boundaries (limiting discontinuities in ∇Θ) form networks that coarsen over time. A matched asymptotic analysis is used to derive a free boundary problem consisting of curve motion coupled along hyperbolic characteristics and junction conditions. An intermediate boundary layer near extrema junctions is discovered, along with the relevant nonlocal junction conditions. The limiting dynamics can be viewed in the context of a gradient flow of the sharp interface energy on an attracting manifold. Dynamic scaling of the long-time coarsening process can be explained by dimensional analysis of the reduced problem.

Original languageEnglish (US)
Pages (from-to)80-98
Number of pages19
JournalPhysica D: Nonlinear Phenomena
Volume215
Issue number1
DOIs
StatePublished - Mar 1 2006

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Keywords

  • Gradient flows
  • Grain boundaries
  • Matched asymptotic expansion

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics

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