Cutoff estimates for the Becker-D\"oring equations

TL;DR

This paper analyzes the spectral properties and cutoff behavior of solutions to the linearized Becker-Döring equations with subcritical mass, providing detailed estimates on the lifetime of perturbations near equilibrium.

Contribution

It characterizes the spectrum of the linearized operator and establishes precise cutoff estimates for perturbations, extending previous work on the trend toward equilibrium.

Findings

The spectrum includes the entire imaginary axis in polynomially weighted spaces.

Established upper and lower bounds on perturbation lifetimes.

Provided detailed cutoff estimates for solutions near equilibrium.

Abstract

This paper continues the authors' previous study (SIAM J. Math. Anal., 2016) of the trend toward equilibrium of the Becker-D\"oring equations with subcritical mass, by characterizing certain fine properties of solutions to the linearized equation. In particular, we partially characterize the spectrum of the linearized operator, showing that it contains the entire imaginary axis in polynomially weighted spaces. Moreover, we prove detailed cutoff estimates that establish upper and lower bounds on the lifetime of a class of perturbations to equilibrium.