Upper bounds on coarsening rates

Robert Kohn, Felix Otto

Research output: Contribution to journalArticle

Abstract

We consider two standard models of surface-energy-driven coarsening: a constant-mobility Cahn-Hilliard equation, whose large-time behavior corresponds to Mullins-Sekerka dynamics; and a degenerate-mobility Cahn-Hilliard equation, whose large-time behavior corresponds to motion by surface diffusion. Arguments based on scaling suggest that the typical length scale should behave as l(t) ∼ t1/3 in the first case and l(t) ∼ t1/4 in the second. We prove a weak, one-sided version of this assertion - showing, roughly speaking, that no solution can coarsen faster than the expected rate. Our result constrains the behavior in a time-averaged sense rather than pointwise in time, and it constrains not the physical length scale but rather the perimeter per unit volume. The argument is simple and robust, combining the basic dissipation relations with an interpolation inequality and an ODE argument.

Original languageEnglish (US)
Pages (from-to)375-395
Number of pages21
JournalCommunications in Mathematical Physics
Volume229
Issue number3
DOIs
StatePublished - Sep 2002

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Coarsening
Cahn-Hilliard Equation
Large Time Behavior
Upper bound
Length Scale
Interpolation Inequality
Surface Diffusion
Surface Energy
Perimeter
Assertion
Standard Model
Dissipation
surface diffusion
Scaling
surface energy
interpolation
Unit
dissipation
Motion
scaling

ASJC Scopus subject areas

  • Mathematical Physics
  • Physics and Astronomy(all)
  • Statistical and Nonlinear Physics

Cite this

Upper bounds on coarsening rates. / Kohn, Robert; Otto, Felix.

In: Communications in Mathematical Physics, Vol. 229, No. 3, 09.2002, p. 375-395.

Research output: Contribution to journalArticle

Kohn, Robert ; Otto, Felix. / Upper bounds on coarsening rates. In: Communications in Mathematical Physics. 2002 ; Vol. 229, No. 3. pp. 375-395.
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