Abstract and Applied Analysis
Volume 2004 (2004), Issue 3, Pages 215-237
doi:10.1155/S1085337504310080
Continuum limits of particles interacting via diffusion
Nicholas D. Alikakos1
, Giorgio Fusco2
and Georgia Karali3
1Department of Mathematics, University of Athens, Athens 15784, Greece
2Dipartimento di Mathematica, Universita di L'Aquila, L'Aquila 67010, Italy
3Department of Mathematics, University of Toronto, Toronto M5S 3G3, Canada
Abstract
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.