Self-similar Solutions to a Kinetic Model for Grain Growth

TL;DR

This paper proves the existence of self-similar solutions for a kinetic model of grain growth, using discretization and limiting processes, and supports findings with numerical computations.

Contribution

It introduces a method to establish self-similar solutions for an infinite coupled system by discretization and limiting arguments, advancing understanding of grain growth models.

Findings

Existence of self-similar solutions proven mathematically.

Numerical methods successfully compute these solutions.

Solutions exhibit exponential decay at infinity.

Abstract

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 neighbours (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 differential system has nontrivial steady states. Afterwards we 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.