Continuum limits of particles interacting via diffusion. (English) Zbl 1073.35058

Alikakos, Nicholas D.; Fusco, Giorgio; Karali, Georgia

Abstr. Appl. Anal. 2004, No. 3, 215-237 (2004).

Ostwald ripening takes place during the late stages of phase transitions when the larger grains of the new phase grow at the expense of smaller ones, thus resulting in a coarsening of the size distribution. In previous papers by the authors [see N. D. Alikakos, G. Fusco and G. Karali, Commun. Math. Phys. 238, 481–488 (2003; Zbl 1083.82017)] and the references therein), a coupled system of evolution equations were derived for the radii and centers of \(N\) almost spherical grains, the set of grains evolving within a bounded domain \(\Omega\) according to the Mullins-Sekerka dynamics.
Denoting by \(\varepsilon\) a measure of the distortion of the shape of the grains from the spherical shape and performing an appropriate rescaling where the distance between grains scales as \(\varepsilon^\eta\), \(\eta\in (0,1)\), the limit \(\varepsilon\to 0\) is studied for \(\eta\in (0,1/3]\): in the subcritical case \(\eta\in (0,1/3)\), the classical Lifshitz-Slyozov-Wagner (LSW) mean-field model is obtained. In the critical case \(\eta=1/3\), another mean-field model is derived which appears to be a correction to the classical LSW model accounting for the geometry of the spatial distribution. The connection between this corrected model and another correction to the classical LSW model recently derived in [B. Niethammer and F. Otto, Calc. Var. Partial Differ. Equ. 13, 33–68 (2001; Zbl 0988.35021)], is also established.

Reviewer: Philippe Laurençot (Toulouse)

MSC:

35F25	Initial value problems for nonlinear first-order PDEs
34D15	Singular perturbations of ordinary differential equations
45K05	Integro-partial differential equations
82C24	Interface problems; diffusion-limited aggregation in time-dependent statistical mechanics
35B25	Singular perturbations in context of PDEs

Keywords:

Ostwald ripening; Lifshitz-Slyozov-Wagner theory; Mullins-Sekerka model

Citations:

Zbl 0988.35021; Zbl 1083.82017