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Uniform moment propagation for the Becker--Döring equations

Part of: Equations of mathematical physics and other areas of application Stability theory Time-dependent statistical mechanics (dynamic and nonequilibrium)

Published online by Cambridge University Press:  27 December 2018

José A. Cãnizo ,
Amit Einav  and
Bertrand Lods
José A. Cãnizo
Affiliation:
Departamento de Matemática Aplicada, Universidad de Granada, Av. Fuentenueva S/N 18071, Granada, Spain
Amit Einav
Affiliation:
Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstrasse 8-10, A-1040 Wien, Österreich ([email protected])
Bertrand Lods
Affiliation:
Departement of Economics and Statistics & Collegio Carlo Alberto,Università degli Studi di Torino, Corso Unione Sovietica, 218/bis 10134 Torino, Italy
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Abstract

We show uniform-in-time propagation of algebraic and stretched exponential moments for the Becker--Döring equations. Our proof is based upon a suitable use of the maximum principle together with known rates of convergence to equilibrium.

Keywords

Becker-Döringpropagation of momentsmaximum principle

MSC classification

Primary: 34D05: Asymptotic properties
Secondary: 35Q82: PDEs in connection with statistical mechanics 82C05: Classical dynamic and nonequilibrium statistical mechanics (general) 82C40: Kinetic theory of gases
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 4 , August 2019 , pp. 995 - 1015
Copyright
Copyright © Royal Society of Edinburgh 2018 

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References

1Alonso, R., Cañizo, J. A., Gamba, I. and Mouhot, C.. A new approach to the creation and propagation of exponential moments in the Boltzmann equation. Comm. Partial Differ. Equ. 38 (2013), 155–169.Google Scholar
2Amann, H.. Ordinary differential equations: an introduction to nonlinear analysis (Volume 13 of De Gruyter studies in Mathematics (Berlin: de Gruyter, 1990).Google Scholar
3Ball, J. M. and Carr, J.. Asymptotic behaviour of solutions to the Becker-Döring equations for arbitrary initial data. Proc. Roy. Soc. Edinburgh Sect. A 108 (1988), 109–116.Google Scholar
4Ball, J. M., Carr, J. and Penrose, O.. The Becker-Döring cluster equations: Basic properties and asymptotic behaviour of solutions. Comm. Math. Phys. 104 (1986), 657–692.Google Scholar
5Becker, R. and Döring, W.. Kinetische Behandlung der Keimbildung in übersättigten Dämpfen. Ann. Phys. 416 (1935), 719–752.Google Scholar
6Bobylev, A. V.. Moment inequalities for the Boltzmann equation and applications to spatially homogeneous problems. J. Statist. Phys. 88 (1996), 1183–1214.Google Scholar
7Cañizo, J. A.. Asymptotic behavior of the generalized Becker-Döring equations for general initial data. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2005), 3731–3745.Google Scholar
8Cañizo, J. A.. Convergence to equilibrium for the discrete Coagulation-Fragmentation equations with detailed balance. J. Statist. Phys. 129 (2007), 1–26.Google Scholar
9Cañizo, J. A. and Lods, B.. Exponential convergence to equilibrium for subcritical solutions of the Becker-Döring equations. J. Differ. Equ. 255 (2013), 905–950.Google Scholar
10Cañizo, J. A., Einav, A. and Lods, B.. Trend to equilibrium for the Becker-Döring equations: An analogue of Cercignani's conjecture. Anal. PDE 10-7 (2017), 1663–1708.Google Scholar
11Jabin, P.-E. and Niethammer, B.. On the rate of convergence to equilibrium in the Becker-Döring equations. J. Differ. Equ. 191 (2003), 518–543.Google Scholar
12Laurençot, P. and Mischler, S.. From the Becker-Döring to the Lifshitz-Slyozov-Wagner equations. J. Statist. Phys. 106 (2002), 957–991.Google Scholar
13Murray, R. W. and Pego, R. L.. Algebraic decay to equilibrium for the Becker–Döring equations. SIAM J. Math. Anal. 48 (2016a), 2819–2842.Google Scholar
14Murray, R. W. and Pego, R. L.. Cutoff estimates for the Becker-Döring equations. Preprint (2016b).Google Scholar
15Niethammer, B.. On the evolution of large clusters in the Becker-Döring model. J. Nonlinear Sci. 13 (2008), 115–122.Google Scholar
16Penrose, O.. Metastable states for the Becker-Döring cluster equations. Comm. Math. Phys. 124 (1989), 515–541.Google Scholar
17Penrose, O.. The Becker-Döring equations at large times and their connection with the LSW theory of coarsening. J. Statist. Phys. 89 (1997), 305–320.Google Scholar
18Schlichting, A.. Macroscopic limit of the Becker-Döring equation via gradient flows. Preprint (2016).Google Scholar
19Slemrod, M.. The Becker-Döring equations, In Modeling in applied sciences (ed. Bellomo, N. and Pulvirenti, M.,pp. 149–171 (Boston: Birkhäuser, Springer, 2000).Google Scholar
20Velázquez, J. J. L.. The Becker-Döring equations and the Lifshitz-Slyozov theory of coarsening. J. Statist. Phys. 92 (1998), 195–236.Google Scholar