Self-similar solutions for the LSW model with encounters

TL;DR

This paper rigorously proves the existence and uniqueness of an exponentially decaying self-similar solution for the LSW model with encounters, providing mathematical validation for the model's long-time behavior.

Contribution

It establishes the existence and isolation of a self-similar solution for the LSW model with encounters using fixed-point methods and asymptotic analysis.

Findings

Existence of an exponentially decaying self-similar solution.

The solution is isolated in the function space.

Rigorous mathematical proof using fixed-point and asymptotic techniques.

Abstract

The LSW model with encounters has been suggested by Lifshitz and Slyozov as a regularization of their classical mean-field model for domain coarsening to obtain universal self-similar long-time behavior. We rigorously establish that an exponentially decaying self-similar solution to this model exist, and show that this solutions is isolated in a certain function space. Our proof relies on setting up a suitable fixed-point problem in an appropriate function space and careful asymptotic estimates of the solution to a corresponding homogeneous problem.