FRONT MIGRATION IN THE NONLINEAR CAHN-HILLIARD EQUATION
FRONT MIGRATION IN THE NONLINEAR CAHN-HILLIARD EQUATION
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DOI：
10.1098/rspa.1989.0027
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发表时间：
1989-04-08
影响因子：
--
通讯作者：
PEGO, RL
中科院分区：
文献类型：
--
作者：
PEGO, RL
本文用匹配渐近展开法描述了N > 1维空间中非线性Cahn-Hilliard相分离方程的解。当内部过渡层的厚度与层间距离和曲率半径相比很小时，这种展开形式上是有效的。在主要的（最慢的）时间尺度上，界面速度由界面的平均曲率确定，通过非局部关系，该非局部关系与众所周知的凝固的准静态模型中的关系相同，该凝固模型表现出由Mullins和Sekerka（J.appl.Phys.34，323-329（1963））发现的形状不稳定性。在更快的时间尺度上，Cahn-Hilliard方程正则化了经典的两相Stefan问题。两相Stefan问题的相似解应该描述边界层。在附录中严格证明了这种允许亚稳态的相似解的存在唯一性。
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.