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Self–similarity and instability in classical mean–field models for domain coarsening
Author(s) -
Niethammer B.,
Pego R.L.
Publication year - 2002
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/1617-7061(200203)1:1<407::aid-pamm407>3.0.co;2-b
Subject(s) - similarity (geometry) , convergence (economics) , bounded function , mathematics , domain (mathematical analysis) , statistical physics , similarity solution , self similarity , instability , distribution (mathematics) , mathematical analysis , physics , computer science , geometry , thermodynamics , quantum mechanics , artificial intelligence , image (mathematics) , boundary layer , economics , economic growth
In the classical theory by Lifshitz, Slyozov and Wagner (LSW) coarsening of a dilute system of particles is modelled by a nonlocal transport equation for the particle size distribution. LSW predict that the asymptotic behavior for large times is self–similar and that a particular self–similar profile is approached. In this talk we discuss rigorous results on the long–time behavior of solutions for several variants of this model. For systems in which particle size is uniformly bounded these results establish a sensitive dependence on the data and thus in general do not confirm the predictions by LSW. More precisely we prove that convergence to the classically predicted similarity solution is impossible if the initial distribution is comparable to any finite power of distance to the end of the support. In addition we give a necessary criterion for convergence to other self–similar solutions, which implies non–self–similar asymptotics for a dense set of data.