Abstract
In the classical theory of domain coarsening the particles of the coarsening phase evolve by diffusional mass transfer with a mean field. We study the long-time behavior of measure-valued solutions with compact support to this model coupled with the constraint of conserved total mass, including mean-field mass. Unlike the case of conserved volume fraction, this system has no precisely self-similar solutions, and sufficiently low supersaturation can lead to the finite-time extinction of all particles. We find a new explicit family of asymptotically self-similar solutions, and in case that the largest particle size is unbounded we establish results similar to the volume-conserved case. These include necessary criteria for asymptotic self-similarity, and sensitive dependence of long-time behavior on the distribution of largest particles in the system.
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REFERENCES
N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Encycl. Math. Appl. v. 27 (Cambridge Univ. Press, Cambridge, 1987).
J. Carr and O. Penrose, Asymptotic behaviour in a simplified Lifshitz-Slyozov equation, Physica D 124:166-176 (1998).
J.-F. Collet and T. Goudon, On solutions of the Lifshitz-Slyozov model, Nonlinearity 13:1239-1262 (2000).
H. Federer, Geometric Measure Theory (Spinger-Verlag, New York, 1969).
A. Friedman and B. Ou, A model of crystal precipitation, J. Math. Anal. Appl. 137:550-575 (1989).
A. Friedman, B. Ou, and D. Ross, Crystal Precipitation with Discrete Initial Data, J. Math. Anal. Appl. 137:576-590 (1989).
B. Giron, B. Meerson, and P. V. Sasorov, Weak selection and stability of localized distributions in Ostwald ripening, Phys. Rev. E 58:4213-4217 (1998).
S. C. Hardy and P. W. Voorhees, Ostwald Ripening in a system with a high volume fraction of coarsening phase, Metallurgical Trans. A 19:2713-2721 (1988).
Ph. Laurencot, On solutions to the Lifshitz-Slyozov model, Ind. Univ. Math. J., to appear.
I. M. Lifshitz and V. V. Slyozov, The kinetics of precipitation from supersaturated solid solutions, J. Phys. Chem. Solids 19:35-50 (1961).
B. Meerson and P. V. Sasorov, Domain stability, competition, growth and selection in globally constrained bistable systems, Phys. Rev. E 53:3491-4 (1996).
B. Niethammer, Derivation of the LSW theory for Ostwald ripening by homogenization methods, Arch. Rat. Mech. Anal. 147:119-178 (1999).
B. Niethammer and R. L. Pego, On the initial value problem in the Lifshitz-Slyozov-Wagner theory of Ostwald ripening, SIAM J. Math. Anal. 31:467-485 (2000).
B. Niethammer and R. L. Pego, Non-self-similar behavior in the LSW theory of Ostwald Ripening, J. Stat. Phys. 95:867-902 (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).
E. Seneta, Regularly Varying Functions, Lec. Notes in Math. Vol. 508 (Springer-Verlag, New York, 1976).
J. J. L. Velázquez, The Becker-Döring equations and the Lifshitz-Slyozov theory of coarsening, J. Stat. Phys. 92:195-236 (1998).
P. W. Voorhees, The theory of Ostwald ripening, J. Stat. Phys. 38:231-252 (1985).
P. W. Voorhees, Ostwald ripening of two-phase mixtures, Ann. Rev. Mater. Sci. 22: 197-215 (1992).
C. Wagner, Theorie der Alterung von Niederschlägen durch Umlösen, Z. Elektrochem. 65:581-594 (1961).
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Niethammer, B., Pego, R.L. The LSW Model for Domain Coarsening: Asymptotic Behavior for Conserved Total Mass. Journal of Statistical Physics 104, 1113–1144 (2001). https://doi.org/10.1023/A:1010405812125
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DOI: https://doi.org/10.1023/A:1010405812125