Mathematical analysis of a coarsening model with local interactions

TL;DR

This paper analyzes a one-dimensional particle coarsening model with local interactions, proving existence of solutions, exploring non-uniqueness, and providing estimates and insights into the system's statistical behavior.

Contribution

It offers a rigorous mathematical framework for a coarsening model with local interactions, including existence proofs and non-uniqueness analysis, complemented by heuristic and numerical insights.

Findings

Existence of solutions under density assumptions

Non-uniqueness due to particle vanishing and local interactions

Upper coarsening estimate and statistical property discussion

Abstract

We consider particles on a one-dimensional lattice whose evolution is governed by nearest-neighbor interactions where particles that have reached size zero are removed from the system. Concentrating on configurations with infinitely many particles, we prove existence of solutions under a reasonable density assumption on the initial data and show that the vanishing of particles and the localized interactions can lead to non-uniqueness. Moreover, we provide a rigorous upper coarsening estimate and discuss the generic statistical properties as well as some non-generic behavior of the evolution by means of heuristic arguments and numerical observations.