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

Cañizo, José; Einav, Amit; Lods, Bertrand

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.

Citation

equations : an analogue of Cercignani’s conjecture. Analysis & PDE, 10(7), 1663–1708. https://doi.org/10.2140/apde.2017.10.1663

• Journal Article Type: Article
• Acceptance Date: May 29, 2017
• Online Publication Date: Aug 1, 2017
• Publication Date: Aug 1, 2017
• Deposit Date: Nov 16, 2020
• Journal: Analysis and PDE
• Print ISSN: 2157-5045
• Electronic ISSN: 1948-206X
• Publisher: Mathematical Sciences Publishers (MSP)
• Volume: 10
• Issue: 7
• Pages: 1663–1708
• DOI: https://doi.org/10.2140/apde.2017.10.1663
• Public URL: https://durham-repository.worktribe.com/output/1250998