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Trend to equilibrium for the Becker–Döring equations: an analogue of Cercignani's conjecture

Accepted version
Peer-reviewed

Type

Article

Change log

Authors

Cañizo, JA 
Einav, A 
Lods, B 

Abstract

We investigate the rate of convergence to equilibrium for subcritical solutions to the Becker–Döring equations with physically relevant coagulation and fragmentation coefficients and mild assumptions on the given initial data. Using a discrete version of the log-Sobolev inequality with weights, we show that in the case where the coagulation coefficient grows linearly and the detailed balance coefficients are of typical form, one can obtain a linear functional inequality for the dissipation of the relative free energy. This results in showing Cercignani’s conjecture for the Becker–Döring equations and consequently in an exponential rate of convergence to equilibrium. We also show that for all other typical cases, one can obtain an “almost” Cercignani’s conjecture, which results in an algebraic rate of convergence to equilibrium.

Description

Keywords

Becker–Döring, nucleation, exponential convergence, entropy method

Journal Title

Analysis and PDE

Conference Name

Journal ISSN

2157-5045
1948-206X

Volume Title

10

Publisher

Mathematical Sciences Publishers
Sponsorship
Engineering and Physical Sciences Research Council (EP/L002302/1)
JAC was supported by the Marie-Curie CIG grant KineticCF and the Spanish Ministerio de Econom´ıa y Competitividad / European Regional Development Fund (ERDF/FEDER), project MTM2014-52056-P. AE was supported by EPSRC grant EP/L002302/1. BL acknowledges support of the de Castro Statistics Initiative, Collegio C. Alberto, Moncalieri, Italy.