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Communications in Mathematical Physics

, Volume 104, Issue 4, pp 657–692 | Cite as

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

  • J. M. Ball
  • J. Carr
  • O. Penrose
Article

Abstract

Existence and uniqueness results are established for solutions to the Becker-Döring cluster equations. The density ϱ 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 byc(ϱ), 0 ≦ ϱ ≦ ϱ s , the principal result is that if the initial density ϱ0 ≦ ϱ s then the solution converges strongly toc(ϱo), while if ϱ0 > ϱ s the solution converges weak* tocs). In the latter case the excess density ϱ0–ϱ 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.

Keywords

Neural Network Initial Data Asymptotic Behaviour Nonlinear Dynamics Basic Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • J. M. Ball
    • 1
  • J. Carr
    • 1
  • O. Penrose
    • 2
  1. 1.Department of MathematicsHeriot-Watt UniversityEdinburghScotland
  2. 2.Faculty of MathematicsThe Open UniversityMilton Keynes

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