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


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.


Neural Network Initial Data Asymptotic Behaviour Nonlinear Dynamics Basic Property 
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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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