Abstract and Applied Analysis
Volume 2004 (2004), Issue 3, Pages 215-237

Continuum limits of particles interacting via diffusion

Nicholas D. Alikakos,1,2 Giorgio Fusco,3 and Georgia Karali4

1Department of Mathematics, University of Athens, Athens 15784, Greece
2Department of Mathematics, University of North Texas, 76203, TX, USA
3Dipartimento di Mathematica, Universita di L'Aquila, L'Aquila 67010, Italy
4Department of Mathematics, University of Toronto, Toronto M5S 3G3, Canada

Received 5 August 2003

Copyright © 2004 Nicholas D. Alikakos et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider a two-phase system mainly in three dimensions and we examine the coarsening of the spatial distribution, driven by the reduction of interface energy and limited by diffusion as described by the quasistatic Stefan free boundary problem. Under the appropriate scaling we pass rigorously to the limit by taking into account the motion of the centers and the deformation of the spherical shape. We distinguish between two different cases and we derive the classical mean-field model and another continuum limit corresponding to critical density which can be related to a continuity equation obtained recently by Niethammer andOtto. So, the theory of Lifshitz, Slyozov, and Wagner is improved by taking into account the geometry of the spatial distribution.