Upper Bound on the Coarsening Rate for an Epitaxial Growth Model

Robert Kohn, Xiaodong Yan

Research output: Contribution to journalArticle

Abstract

We study a specific example of energy-driven coarsening in two space dimensions. The energy is ∫|∇∇u|2 + (1 - |∇u|2)2; the evolution is the fourth-order PDE representing steepest descent. This equation has been proposed as a model of epitaxial growth for systems with slope selection. Numerical simulations and heuristic arguments indicate that the standard deviation of u grows like t 1/3, and the energy per unit area decays like t-1/3. We prove a weak, one-sided version of the latter statement: The time-averaged energy per unit area decays no faster than t-1/3. Our argument follows a strategy introduced by Kohn and Otto in the context of phase separation, combining (i) a dissipation relation, (ii) an isoperimetric inequality, and (iii) an ODE lemma. The interpolation inequality is new and rather subtle; our proof is by contradiction, relying on recent compactness results for the Aviles-Giga energy.

Original languageEnglish (US)
Pages (from-to)1549-1564
Number of pages16
JournalCommunications on Pure and Applied Mathematics
Volume56
Issue number11
DOIs
StatePublished - Nov 2003

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Epitaxial Growth
Coarsening
Growth Model
Epitaxial growth
Phase separation
Interpolation
Upper bound
Computer simulation
Energy
Decay
Interpolation Inequality
Unit
Isoperimetric Inequality
Steepest Descent
Phase Separation
Standard deviation
Fourth Order
Compactness
Dissipation
Lemma

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Upper Bound on the Coarsening Rate for an Epitaxial Growth Model. / Kohn, Robert; Yan, Xiaodong.

In: Communications on Pure and Applied Mathematics, Vol. 56, No. 11, 11.2003, p. 1549-1564.

Research output: Contribution to journalArticle

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