The Becker-Döring cluster equations: Basic properties and asymptotic behaviour of solutions

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Abstract

Existence and uniqueness results are established for solutions to the Becker-Döring cluster equations. The density ?{variant} is shown to be a conserved quantity. Under hypotheses applying to a model of a quenched binary alloy the asymptotic behaviour of solutions with rapidly decaying initial data is determined. Denoting the set of equilibrium solutions by c(?{variant}), 0 ? ?{variant} ? ?{variant}s, the principal result is that if the initial density ?{variant}0 ? ?{variant}s then the solution converges strongly to c(?{variant}o), while if ?{variant}0 > ?{variant}s the solution converges weak* to c(?{variant}s). In the latter case the excess density ?{variant}0-?{variant}s corresponds to the formation of larger and larger clusters, i.e. condensation. The main tools for studying the asymptotic behaviour are the use of a Lyapunov function with desirable continuity properties, obtained from a known Lyapunov function by the addition of a special multiple of the density, and a maximum principle for solutions. © 1986 Springer-Verlag.

Original languageEnglish
Pages (from-to)657-692
Number of pages36
JournalCommunications in Mathematical Physics
Volume104
Issue number4
DOIs
Publication statusPublished - Dec 1986

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