Front migration in the nonlinear Cahn-Hilliard equation 论文

详细信息

发表期刊/会议

Proceedings of the Royal Society of London A Mathematical and Physical Sciences

发表日期

1989-04-08

发表年份

1989

摘要

Abstract The method of matched asymptotic expansions is used to describe solutions of the nonlinear Cahn-Hilliard equation for phase separation in N > 1 space dimensions. The expansion is formally valid when the thickness of internal transition layers is small compared with the distance separating layers and with their radii of curvature. On the dominant (slowest) timescale the interface velocity is determined by the mean curvature of the interface, by a non-local relation which is identical to that in a well-known quasi-static model of solidification, which exhibits a shape instability discovered by Mullins & Sekerka (J. appl. Phys. 34, 323-329 (1963)). On a faster timescale, the Cahn-Hilliard equation regularizes a classic two-phase Stefan problem. Similarity solutions of the two-phase Stefan problem should describe boundary layers. Existence and uniqueness of such similarity solutions which admit metastable states is proved rigorously in an appendix.