Self-Similar Solutions to a Kinetic Model for Grain Growth

We prove the existence of self-similar solutions to the Fradkov model for two-dimensional grain growth, which consists of an infinite number of nonlocally coupled transport equations for the number densities of grains with given area and number of neighbors (topological class). For the proof we intr...

Full description

Saved in:
Bibliographic Details
Published in:Journal of nonlinear science 2012-06, Vol.22 (3), p.399-427
Main Authors: Herrmann, Michael, Laurençot, Philippe, Niethammer, Barbara
Format: Article
Language:English
Subjects:
Analysis
Classical Mechanics
Economic Theory/Quantitative Economics/Mathematical Methods
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Theoretical
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We prove the existence of self-similar solutions to the Fradkov model for two-dimensional grain growth, which consists of an infinite number of nonlocally coupled transport equations for the number densities of grains with given area and number of neighbors (topological class). For the proof we introduce a finite maximal topological class and study an appropriate upwind discretization of the time-dependent problem in self-similar variables. We first show that the resulting finite-dimensional dynamical system admits nontrivial steady states. We then let the discretization parameter tend to zero and prove that the steady states converge to a compactly supported self-similar solution for a Fradkov model with finitely many equations. In a third step we let the maximal topology class tend to infinity and obtain self-similar solutions to the original system that decay exponentially. Finally, we use the upwind discretization to compute self-similar solutions numerically.
ISSN:0938-8974
1432-1467
DOI:10.1007/s00332-011-9122-1