Abstract
This paper is concerned with the Becker–Döring (BD) system of equations and their relationship to the Lifschitz–Slyozov–Wagner (LSW) equations. A diffusive version of the LSW equations is derived from the BD equations. Existence and uniqueness theorems for this diffusive LSW system are proved. The major part of the paper is taken up with proving that solutions of the diffusive LSW system converge in the zero diffusion limit to solutions of the classical LSW system. In particular, it is shown that the rate of coarsening for the diffusive system converges to the rate of coarsening for the classical system.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ball, J.M., Carr, J., Penrose, O.: The Becker Döring cluster equations: basic properties and asymptotic behavior of solutions. Commun. Math. Phys. 104, 657–692 (1986)
Becker, R., Döring, W.: Kinetische Behandlung der Keimbildung in übersättigten Dämpfen. Ann. Phys. (Leipz.) 24, 719–752 (1935)
Collet, J.-F., Goudon, T.: On solutions of the Lifschitz–Slyozov model. Nonlinearity 13, 1239–1262 (2000)
Dai, S., Pego, R.L.: Universal bounds on coarsening rates for mean-field models of phase transitions. SIAM J. Math. Anal. 37, 347–371 (2005)
Farjoun, Y., Neu, J.C.: Exhaustion of nucleation in a closed system. Phys. Rev. E 78, 051402 (2008). 7 pp.
Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus, 2nd edn. Graduate Texts in Mathematics, vol. 113. Springer, New York (1991)
Kohn, R.V., Otto, F.: Upper bounds on coarsening rates. Commun. Math. Phys. 229, 375–395 (2002)
Laurençot, P.: The Lifschitz–Slyozov–Wagner equation with conserved total volume. SIAM J. Math. Anal. 34, 257–272 (2002)
Laurençot, P., Mischler, S.: From the Becker–Döring to the Lifschitz–Slyozov–Wagner equations. J. Stat. Phys. 106, 957–991 (2002)
Lifschitz, I.M., Slyozov, V.V.: Kinetics of precipitation from supersaturated solid solutions. J. Phys. Chem. Solids 19, 35–50 (1961)
Meerson, B.: Fluctuations provide strong selection in Ostwald ripening. Phys. Rev. E 60, 3072–3075 (1999)
Niethammer, B.: On the evolution of large clusters in the Becker Döring model. J. Nonlinear Sci. 13, 115–155 (2003)
Niethammer, B., Pego, R.L.: Non-self-similar behavior in the LSW theory of Ostwald ripening. J. Stat. Phys. 95, 867–902 (1999)
Niethammer, B., Pego, R.L.: On the initial value problem in the Lifschitz–Slyozov–Wagner theory of Ostwald ripening. SIAM J. Math. Anal. 31, 467–485 (2000)
Niethammer, B., Pego, R.L.: Well-posedness for measure transport in a family of nonlocal domain coarsening models. Indiana Univ. Math. J. 54, 499–530 (2005)
Pego, R.L.: Lectures on dynamics in models of coarsening and coagulation. In: Dynamics in Models of Coarsening, Coagulation, Condensation and Quantization. Lecture Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 9, pp. 1–61. World Scientific, Singapore (2007)
Penrose, O.: The Becker Döring equations at large times and their connection with the LSW theory of coarsening. J. Stat. Phys. 89, 305–320 (1997)
Velázquez, J.J.L.: The Becker–Döring equations and the Lifschitz–Slyozov–Wagner theory of coarsening. J. Stat. Phys. 92, 195–236 (1998)
Wagner, C.: Theorie der alterung von niederschlägen durch umlösen. Z. Elektrochem. 65, 581–591 (1961)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by R. Kohn.
Rights and permissions
About this article
Cite this article
Conlon, J.G. On a Diffusive Version of the Lifschitz–Slyozov–Wagner Equation. J Nonlinear Sci 20, 463–521 (2010). https://doi.org/10.1007/s00332-010-9065-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00332-010-9065-y