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Trend to equilibrium
for the Becker–Döring equations: an analogue of Cercignani's
conjecture
José A. Cañizo, Amit Einav and Bertrand Lods |
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Vol. 10 (2017), No. 7, 1663–1708
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Abstract |
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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. |
Keywords
Becker–Döring, nucleation, exponential convergence, entropy
method
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Mathematical Subject Classification 2010
Primary: 34D05, 35Q82, 82C05, 82C40
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Milestones
Received: 24 December 2016
Revised: 28 March 2017
Accepted: 29 May 2017
Published: 1 August 2017
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Authors |
| José A. Cañizo | |
| Departamento de Matemática
Aplicada Universidad de Granada Av. Fuentenueva S/N 18071 Granada Spain |
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| Amit Einav | |
| Department of Pure Mathematics and
Mathematical Statistics University of Cambridge Cambridge CB3 0WB United Kingdom |
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| Bertrand Lods | |
| Department of Economics and
Statistics & Collegio Carlo Alberto Università degli Studi di Torino Corso Unione Sovietica, 218/bis 10134 Torino Italy |
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