Long-time asymptotic of the Lifshitz-Slyozov equation with nucleation

TL;DR

This paper analyzes the long-term behavior of the Lifshitz-Slyozov equation with nucleation, showing that size distributions tend to degenerate states at zero size over time, with convergence rates supported by numerical evidence.

Contribution

It demonstrates the asymptotic approach to degenerate states for a class of rate functions using entropy-based monotonicity, and provides numerical insights into convergence rates.

Findings

Size distributions approach zero size states over time

Convergence rate is algebraic in time

Results are relevant for interpreting experimental data

Abstract

We consider the Lifshitz-Slyozov model with inflow boundary conditions of nucleation type. We show that for a collection of representative rate functions the size distributions approach degenerate states concentrated at zero size for sufficiently large times. The proof relies on monotonicity properties of some quantities associated to an entropy functional. Moreover, we give numerical evidence on the fact that the convergence rate to the goal state is algebraic in time. Besides their mathematical interest, these results can be relevant for the interpretation of experimental data.