The evolution of a crystal surface: Analysis of a one-dimensional step train connecting two facets in the ADL regime

Hala Al Hajj Shehadeh, Robert V. Kohn, Jonathan Weare

Research output: Contribution to journalArticle

Abstract

We study the evolution of a monotone step train separating two facets of a crystal surface. The model is one-dimensional and we consider only the attachmentdetachment-limited regime. Starting with the well-known ODEs for the velocities of the steps, we consider the system of ODEs giving the evolution of the "discrete slopes." It is the l2-steepest-descent of a certain functional. Using this structure, we prove that the solution exists for all time and is asymptotically self-similar. We also discuss the continuum limit of the discrete self-similar solution, characterizing it variationally, identifying its regularity, and discussing its qualitative behavior. Our approach suggests a PDE for the slope as a function of height and time in the continuum setting. However, existence, uniqueness, and asymptotic self-similarity remain open for the continuum version of the problem.

Original languageEnglish (US)
Pages (from-to)1771-1784
Number of pages14
JournalPhysica D: Nonlinear Phenomena
Volume240
Issue number21
DOIs
StatePublished - Oct 15 2011

Keywords

  • Epitaxial relaxation
  • Facet
  • Self similar solution
  • Steepest descent

ASJC Scopus subject areas

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

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