Exponential convergence to equilibrium for subcritical solutions of the Becker-D\"oring equations

TL;DR

This paper establishes that subcritical solutions of the Becker-Döring equations converge exponentially fast to equilibrium, with the decay rate determined by spectral gap analysis, improving previous convergence results.

Contribution

It provides a new spectral analysis of the linearized Becker-Döring equations in Hilbert and weighted  spaces, leading to quantitative exponential convergence results.

Findings

Exponential convergence to equilibrium for subcritical solutions.

Spectral gap bounds for the linearized operator.

Quantitative decay rates based on spectral analysis.

Abstract

We prove that any subcritical solution to the Becker-D\"{o}ring equations converges exponentially fast to the unique steady state with same mass. Our convergence result is quantitative and we show that the rate of exponential decay is governed by the spectral gap for the linearized equation, for which several bounds are provided. This improves the known convergence result by Jabin & Niethammer (see ref. [14]). Our approach is based on a careful spectral analysis of the linearized Becker-D\"oring equation (which is new to our knowledge) in both a Hilbert setting and in certain weighted $\ell^1$ spaces. This spectral analysis is then combined with uniform exponential moment bounds of solutions in order to obtain a convergence result for the nonlinear equation.