ISSN:
1432-0916
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
,
Physics
Notes:
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* toc (ϱs). 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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01211070
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