Abstract
We obtain an entropy functional for the Lifshitz–Slyozov system. This can be used to investigate the time asymptotics of the system. In particular, we describe situations in which the monomers concentration either tend to 0 or saturate as time becomes large. The latter situation can be excluded under assumptions on the support of the initial data.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
I. M. Lifshitz and V. V. Slyozov, The kinetics of precipitation from supersaturated solid solutions, J. Phys. Chem. Solids 19:35–50 (1961).
V. V. Sagalovich and V. V. Slyozov, Diffusive decomposition of solid solutions, Sov. Phys. Usp. 30:23–44 (1987).
I. M. Lifshitz and L. Pitaevski, Cinétique Physique, Coll. Physique Théorique, Landau-Lifshitz (Mir, 1990).
B. Niethammer and R. Pego, On the initial-value problem in the Lifshitz-Slyozov-Wagner theory of Ostwald ripening, SIAM J. Math. Anal. 31 :467–485 (2000).
J. F. Collet and T. Goudon, On solutions of the Lifshitz-Slyozov model, Nonlinearity 13:1239–1262 (2000).
P. Laurençot, Weak solutions to the Lifshitz-Slyozov-Wagner equation, Indiana Univ. Math. J. 50:1319–1346 (2001).
J. F. Collet and T. Goudon, Lifshitz-Slyozov equations: The model with encounters, Transp. Theory Stat. Phys. 28:545–573 (1999).
P. Laurençot, The Lifshitz-Slyozov equation with encounters, Math. Models Methods Appl. Sci. 11:731–748 (2001).
J. F. Collet and S. Hariz, A modified version of the Lifshitz-Slyozov model, Appl. Math. Letters 12:81–85 (1999).
S. Hariz, Une version modifiée du modèle de Lifshitz-Slyozov: existence, unicité de la solution, simulation numérique, Thesis, Université de Nice-Sophia Antipolis, Juin 1999.
O. Penrose, The Becker-Döring equations at large times and their connection with the LSW theory of coarsening, J. Stat. Phys. 89:305–320 (1997).
J. F. Collet, T. Goudon, F. Poupaud, and A. Vasseur, The Becker-Döring system and its Lifshitz-Slyozov limit, to appear in SIAM J. Appl. Math.
J. F. Collet, T. Goudon, S. Hariz, F. Poupaud, and A. Vasseur, Some recent results on the kinetic theory of phase transitions, to appear in IMA Proceedings.
B. Niethammer and R. Pego, Non-self-similar behavior in the LSW theory of Ostwald ripening, J. Stat. Phys. 95:867–902 (1999).
B. Niethammer and R. Pego, The LSW model for domain coarsening: asymptotic behavior for conserved total mass, J. Stat. Phys. 104:1113–1144 (2001).
L. Brown, A new examination of classical coarsening theory, Acta Metall. 37:71–77 (1989).
B. Meerson and P. Sasorov, Domain stability, competition, growth, and selection in globally constrained bistable systems, Phys. Rev. E 53:3491–3494 (1996).
J. Carr and O. Penrose, Asymptotic behavior of solutions to a simplified Lifshitz-Slyozov equation, Physica D 124:166–176 (1998).
J. A. Carrillo and T. Goudon, A numerical study on large-time asymptotics of the Lifshitz-Slyozov system, Rapport INRIA n. 4287 (2001).
C. Dellacherie and P. A. Meyer, Probabilités et potentiel, Chapitres I-IV. Publications de l'Institut de Mathématique de l'Université de Strasbourg (Hermann, 1975).
R. Di Perna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98:511–547 (1989).
R. Edwards, Functional analysis, Theory and Applications (Dover, 1994).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Collet, JF., Goudon, T. & Vasseur, A. Some Remarks on Large-Time Asymptotic of the Lifshitz–Slyozov Equations. Journal of Statistical Physics 108, 341–359 (2002). https://doi.org/10.1023/A:1015404021853
Issue Date:
DOI: https://doi.org/10.1023/A:1015404021853