3 research outputs found
Rates of convergence to scaling profiles in a submonolayer deposition model and the preservation of memory of the initial condition
We establish rates of convergence of solutions to scaling (or similarity)
profiles in a coagulation type system modelling submonolayer deposition. We
prove that, although all memory of the initial condition is lost in the
similarity limit, information about the large cluster tail of the initial
condition is preserved in the rate of approach to the similarity profile. The
proof relies in a change of variables that allows for the decoupling of the
original infinite system of ordinary differential equations into a closed
two-dimensional nonlinear system for the monomer--bulk dynamics and a lower
triangular infinite dimensional linear one for the cluster dynamics. The
detailed knowledge of the long time monomer concentration, which was obtained
earlier by Costin et al. in (O. Costin, M. Grinfeld, K.P. O'Neill and H. Park,
Long-time behaviour of point islands under fixed rate deposition, Commun. Inf.
Syst. 13, (2), (2013), pp.183-200) using asymptotic methods and is rederived
here by center manifold arguments, is then used for the asymptotic evaluation
of an integral representation formula for the concentration of -clusters.
The use of higher order expressions, both for the Stirling expansion and for
the monomer evolution at large times allow us to obtain, not only the
similarity limit, but also the rate at which it is approached.Comment: Revised according to referee's suggestions; to be published in SIAM
J. Math. Ana
On the convergence to critical scaling profiles in submonolayer deposition models
In this work we study the rate of convergence to similarity profiles in a
mean field model for the deposition of a submonolayer of atoms in a crystal
facet, when there is a critical minimal size for the stability of the
formed clusters. The work complements recently published related results by the
same authors in which the rate of convergence was studied outside of a critical
direction in the cluster size vs. time plane. In this paper
we consider a different similarity variable, ,
corresponding to an inner expansion of that critical direction, and prove the
convergence of solutions to a similarity profile when with fixed, as well as the rate at which the limit is
approached.Comment: Dedicated to the memory of Jack Car