A Kinetic Model for Grain Growth

TL;DR

This paper analyzes a kinetic model for grain growth based on the von Neumann-Mullins law, establishing well-posedness through approximation and mathematical estimates.

Contribution

It provides the first rigorous proof of existence of solutions for Fradkov's kinetic grain growth model with a nonlinear, nonlocal coupling.

Findings

Existence of solutions established via finite-dimensional approximation.

Key estimates prevent mass escape to infinity in finite time.

The model's well-posedness is proven under certain conditions.

Abstract

We provide a well-posedness analysis of a kinetic model for grain growth introduced by Fradkov which is based on the von Neumann-Mullins law. The model consists of an infinite number of transport equations with a tri-diagonal coupling modelling topological changes in the grain configuration. Self-consistency of this kinetic model is achieved by introducing a coupling weight which leads to a nonlinear and nonlocal system of equations. We prove existence of solutions by approximation with finite dimensional systems. Key ingredients in passing to the limit are suitable super-solutions, a bound from below on the total mass, and a tightness estimate which ensures that no mass is transported to infinity in finite time.