746 research outputs found
Non-self-similar behavior in the LSW theory of Ostwald ripening
The classical Lifshitz-Slyozov-Wagner theory of domain coarsening predicts
asymptotically self-similar behavior for the size distribution of a dilute
system of particles that evolve by diffusional mass transfer with a common mean
field. Here we consider the long-time behavior of measure-valued solutions for
systems in which particle size is uniformly bounded, i.e., for initial measures
of compact support.
We prove that the long-time behavior of the size distribution depends
sensitively on the initial distribution of the largest particles in the system.
Convergence to the classically predicted smooth similarity solution is
impossible if the initial distribution function is comparable to any finite
power of distance to the end of the support. We give a necessary criterion for
convergence to other self-similar solutions, and conditional stability theorems
for some such solutions. For a dense set of initial data, convergence to any
self-similar solution is impossible.Comment: 31 pages, LaTeX2e; Revised version, to appear in J. Stat. Phy
Optimal bounds for self-similar solutions to coagulation equations with product kernel
We consider mass-conserving self-similar solutions of Smoluchowski's
coagulation equation with multiplicative kernel of homogeneity . We establish rigorously that such solutions exhibit a singular behavior
of the form as . This property had been
conjectured, but only weaker results had been available up to now
Dynamics and self-similarity in min-driven clustering
We study a mean-field model for a clustering process that may be described
informally as follows. At each step a random integer is chosen with
probability , and the smallest cluster merges with randomly chosen
clusters. We prove that the model determines a continuous dynamical system on
the space of probability measures supported in , and we establish
necessary and sufficient conditions for approach to self-similar form. We also
characterize eternal solutions for this model via a Levy-Khintchine formula.
The analysis is based on an explicit solution formula discovered by Gallay and
Mielke, extended using a careful choice of time scale
Self-similar solutions with fat tails for a coagulation equation with diagonal kernel
We consider self-similar solutions of Smoluchowski's coagulation equation
with a diagonal kernel of homogeneity . We show that there exists a
family of second-kind self-similar solutions with power-law behavior
as with . To our knowledge
this is the first example of a non-solvable kernel for which the existence of
such a family has been established
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