Upper bounds on coarsening rates

Robert V. Kohn, Felix Otto

Research output: Contribution to journalArticlepeer-review


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
Issue number3
StatePublished - Sep 2002

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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